Fermat-Zahl – Wikipedia

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width:13.254ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{\;\!2^{n}}+1}"></dd></dl> mit einer <a href="/wiki/Ganze_Zahl" title="Ganze Zahl">ganzen Zahl</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}">. Die ersten Fermat-Zahlen lauten 3, 5 und 17. Im August 1640 vermutete Fermat fälschlicherweise, dass alle Zahlen dieser Form (die später nach ihm benannt wurden) <a href="/wiki/Primzahl" title="Primzahl">Primzahlen</a> seien.<a href="#cite_note-Fermat1640-1">[1]</a> Dies wurde jedoch 1732 von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> widerlegt, der zeigte, dass schon die sechste Fermatzahl F5 durch 641 teilbar ist.<a href="#cite_note-2">[2]</a> Man kennt außer den ersten fünf (3, 5, 17, 257, 65537) derzeit keine weitere Fermat-Zahl, die gleichzeitig Primzahl ist, und vermutet, dass es außer diesen fünf Zahlen auch keine weitere gibt (siehe Abschnitt <a class="mw-selflink-fragment" href="#Warum_es_wahrscheinlich_keine_weiteren_Fermat-Primzahlen_gibt">weiter unten</a>). <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Fermat-Zahlen">1 Fermat-Zahlen</a></li> <li class="toclevel-1 tocsection-2"><a href="#Fermatsche_Primzahlen">2 Fermatsche Primzahlen</a></li> <li class="toclevel-1 tocsection-3"><a href="#Faktorisierungsergebnisse_von_Fermat-Zahlen">3 Faktorisierungsergebnisse von Fermat-Zahlen</a></li> <li class="toclevel-1 tocsection-4"><a href="#Eigenschaften">4 Eigenschaften</a></li> <li class="toclevel-1 tocsection-5"><a href="#Ungelöste_Probleme">5 Ungelöste Probleme</a></li> <li class="toclevel-1 tocsection-6"><a href="#Warum_es_wahrscheinlich_keine_weiteren_Fermat-Primzahlen_gibt">6 Warum es wahrscheinlich keine weiteren Fermat-Primzahlen gibt</a></li> <li class="toclevel-1 tocsection-7"><a href="#Geometrische_Anwendung_der_Fermatschen_Primzahlen">7 Geometrische Anwendung der Fermatschen Primzahlen</a></li> <li class="toclevel-1 tocsection-8"><a href="#Verallgemeinerte_Fermatsche_Zahlen">8 Verallgemeinerte Fermatsche Zahlen</a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Verallgemeinerte_Fermatsche_Zahlen_der_Form_Fn(b)">8.1 Verallgemeinerte Fermatsche Zahlen der Form Fn(b)</a></li> <li class="toclevel-2 tocsection-10"><a href="#Liste_der_Primzahlen_der_Form_Fn(b)">8.2 Liste der Primzahlen der Form Fn(b)</a></li> <li class="toclevel-2 tocsection-11"><a href="#Die_10_größten_bekannten_verallgemeinerten_Fermatschen_Primzahlen">8.3 Die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen</a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Siehe_auch">9 Siehe auch</a></li> <li class="toclevel-1 tocsection-13"><a href="#Literatur">10 Literatur</a></li> <li class="toclevel-1 tocsection-14"><a href="#Weblinks">11 Weblinks</a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise">12 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Fermat-Zahlen">Fermat-Zahlen</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=1" title="Abschnitt bearbeiten: Fermat-Zahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Fermat-Zahlen">Quelltext bearbeiten</a>]</div> Die ersten Fermat-Zahlen lauten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=3,\,F_{1}=5,\,F_{2}=17,\,F_{3}=257}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>17</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>257</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=3,\,F_{1}=5,\,F_{2}=17,\,F_{3}=257}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24689b20ffe2c502d804f6119182b481b2f5d675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.989ex; height:2.509ex;" alt="{\displaystyle F_{0}=3,\,F_{1}=5,\,F_{2}=17,\,F_{3}=257}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}=65537}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>65537</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}=65537}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbe6707f5e992f9fd5e1a63c211263808854fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.46ex; height:2.509ex;" alt="{\displaystyle F_{4}=65537}">.<a href="#cite_note-OEISA000215-3">[3]</a> Eine etwas längere Liste bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{14}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{14}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577ecbeb03253e6527194c8f7472a3503208316b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{14}}"> findet man in der folgenden aufklappbaren Box. <style data-mw-deduplicate="TemplateStyles:r248673343">.mw-parser-output div.NavFrame{border-width:1px;border-style:solid;border-left-color:var(--dewiki-rahmenfarbe1);border-right-color:var(--dewiki-rahmenfarbe1);border-top-color:var(--dewiki-rahmenfarbe1);border-bottom-color:var(--dewiki-rahmenfarbe1);clear:both;font-size:95%;margin-top:1.5em;min-height:0;padding:2px;text-align:center}.mw-parser-output div.NavPic{float:left;padding:2px}.mw-parser-output div.NavHead{background-color:var(--dewiki-hintergrundfarbe5);font-weight:bold}.mw-parser-output div.NavFrame:after{clear:both;content:"";display:block}.mw-parser-output div.NavFrame+div.NavFrame,.mw-parser-output div.NavFrame+link+div.NavFrame,.mw-parser-output div.NavFrame+style+div.NavFrame{margin-top:-1px}.mw-parser-output .NavToggle{float:right;font-size:x-small}@media screen{html.skin-theme-clientpref-night .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}</style><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der ersten 15 Fermat-Zahlen</div> <div class="NavContent"> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>n </th> <th style="white-space:nowrap;">Dezimal- stellen von Fn </th> <th style="text-align:left">          Fn </th></tr> <tr> <td>00</td> <td style="text-align:right;">1 </td> <td style="text-align:left;">3 </td></tr> <tr> <td>01</td> <td style="text-align:right;">1 </td> <td style="text-align:left;">5 </td></tr> <tr> <td>02</td> <td style="text-align:right;">2 </td> <td style="text-align:left;">17 </td></tr> <tr> <td>03</td> <td style="text-align:right;">3 </td> <td style="text-align:left;">257 </td></tr> <tr> <td>04</td> <td style="text-align:right;">5 </td> <td style="text-align:left;">65.537 </td></tr> <tr> <td>05</td> <td style="text-align:right;">10 </td> <td style="text-align:left;">4.294.967.297 </td></tr> <tr> <td>06</td> <td style="text-align:right;">20 </td> <td style="text-align:left;">18.446.744.073.709.551.617 </td></tr> <tr> <td>07</td> <td style="text-align:right;">39 </td> <td style="text-align:left;">340.282.366.920.938.463.463.374.607.431.768.211.457 </td></tr> <tr> <td>08</td> <td style="text-align:right;">78 </td> <td style="text-align:left;">115.792.089.237.316.195.423.570.985.008.687.907.853.269.984.665.640.564.039.457.584.007.913.129.639.937 </td></tr> <tr> <td>09</td> <td style="text-align:right;">155 </td> <td style="text-align:left;">13.407.807.929.942.597.099.574.024.998.205.846.127.479.365.820.592.393.377.723.561.443.721.764.030.073.546.976.801.874.298.166.903.427.690.031.858.186.486.050.853.753.882.811.946.569.946.433.649.006.084.097 </td></tr> <tr> <td>10</td> <td style="text-align:right;">309 </td> <td style="text-align:left;">179.769.313.486.231.590.772.930.519.078.902.473.361.797.697.894.230.657.273.430.081.157.732.675.805.500.963.132.708.477.322.407.536.021.120.113.879.871.393.357.658.789.768.814.416.622.492.847.430.639.474.124.377.767.893.424.865.485.276.302.219.601.246.094.119.453.082.952.085.005.768.838.150.682.342.462.881.473.913.110.540.827.237.163.350.510.684.586.298.239.947.245.938.479.716.304.835.356.329.624.224.137.217 </td></tr> <tr> <td>11</td> <td style="text-align:right;">617 </td> <td style="text-align:left;">32.317.006.071.311.007.300.714.876.688.669.951.960.444.102.669.715.484.032.130.345.427.524.655.138.867.890.893.197.201.411.522.913.463.688.717.960.921.898.019.494.119.559.150.490.921.095.088.152.386.448.283.120.630.877.367.300.996.091.750.197.750.389.652.106.796.057.638.384.067.568.276.792.218.642.619.756.161.838.094.338.476.170.470.581.645.852.036.305.042.887.575.891.541.065.808.607.552.399.123.930.385.521.914.333.389.668.342.420.684.974.786.564.569.494.856.176.035.326.322.058.077.805.659.331.026.192.708.460.314.150.258.592.864.177.116.725.943.603.718.461.857.357.598.351.152.301.645.904.403.697.613.233.287.231.227.125.684.710.820.209.725.157.101.726.931.323.469.678.542.580.656.697.935.045.997.268.352.998.638.215.525.166.389.437.335.543.602.135.433.229.604.645.318.478.604.952.148.193.555.853.611.059.596.230.657 </td></tr> <tr> <td>12</td> <td style="text-align:right;">1234 </td> <td style="text-align:left;">1.044.388.881.413.152.506.691.752.710.716.624.382.579.964.249.047.383.780.384.233.483.283.953.907.971.557.456.848.826.811.934.997.558.340.890.106.714.439.262.837.987.573.438.185.793.607.263.236.087.851.365.277.945.956.976.543.709.998.340.361.590.134.383.718.314.428.070.011.855.946.226.376.318.839.397.712.745.672.334.684.344.586.617.496.807.908.705.803.704.071.284.048.740.118.609.114.467.977.783.598.029.006.686.938.976.881.787.785.946.905.630.190.260.940.599.579.453.432.823.469.303.026.696.443.059.025.015.972.399.867.714.215.541.693.835.559.885.291.486.318.237.914.434.496.734.087.811.872.639.496.475.100.189.041.349.008.417.061.675.093.668.333.850.551.032.972.088.269.550.769.983.616.369.411.933.015.213.796.825.837.188.091.833.656.751.221.318.492.846.368.125.550.225.998.300.412.344.784.862.595.674.492.194.617.023.806.505.913.245.610.825.731.835.380.087.608.622.102.834.270.197.698.202.313.169.017.678.006.675.195.485.079.921.636.419.370.285.375.124.784.014.907.159.135.459.982.790.513.399.611.551.794.271.106.831.134.090.584.272.884.279.791.554.849.782.954.323.534.517.065.223.269.061.394.905.987.693.002.122.963.395.687.782.878.948.440.616.007.412.945.674.919.823.050.571.642.377.154.816.321.380.631.045.902.916.136.926.708.342.856.440.730.447.899.971.901.781.465.763.473.223.850.267.253.059.899.795.996.090.799.469.201.774.624.817.718.449.867.455.659.250.178.329.070.473.119.433.165.550.807.568.221.846.571.746.373.296.884.912.819.520.317.457.002.440.926.616.910.874.148.385.078.411.929.804.522.981.857.338.977.648.103.126.085.903.001.302.413.467.189.726.673.216.491.511.131.602.920.781.738.033.436.090.243.804.708.340.403.154.190.337 </td></tr> <tr> <td>13</td> <td style="text-align:right;">2467 </td> <td style="text-align:left;">1.090.748.135.619.415.929.462.984.244.733.782.862.448.264.161.996.232.692.431.832.786.189.721.331.849.119.295.216.264.234.525.201.987.223.957.291.796.157.025.273.109.870.820.177.184.063.610.979.765.077.554.799.078.906.298.842.192.989.538.609.825.228.048.205.159.696.851.613.591.638.196.771.886.542.609.324.560.121.290.553.901.886.301.017.900.252.535.799.917.200.010.079.600.026.535.836.800.905.297.805.880.952.350.501.630.195.475.653.911.005.312.364.560.014.847.426.035.293.551.245.843.928.918.752.768.696.279.344.088.055.617.515.694.349.945.406.677.825.140.814.900.616.105.920.256.438.504.578.013.326.493.565.836.047.242.407.382.442.812.245.131.517.757.519.164.899.226.365.743.722.432.277.368.075.027.627.883.045.206.501.792.761.700.945.699.168.497.257.879.683.851.737.049.996.900.961.120.515.655.050.115.561.271.491.492.515.342.105.748.966.629.547.032.786.321.505.730.828.430.221.664.970.324.396.138.635.251.626.409.516.168.005.427.623.435.996.308.921.691.446.181.187.406.395.310.665.404.885.739.434.832.877.428.167.407.495.370.993.511.868.756.359.970.390.117.021.823.616.749.458.620.969.857.006.263.612.082.706.715.408.157.066.575.137.281.027.022.310.927.564.910.276.759.160.520.878.304.632.411.049.364.568.754.920.967.322.982.459.184.763.427.383.790.272.448.438.018.526.977.764.941.072.715.611.580.434.690.827.459.339.991.961.414.242.741.410.599.117.426.060.556.483.763.756.314.527.611.362.658.628.383.368.621.157.993.638.020.878.537.675.545.336.789.915.694.234.433.955.666.315.070.087.213.535.470.255.670.312.004.130.725.495.834.508.357.439.653.828.936.077.080.978.550.578.912.967.907.352.780.054.935.621.561.090.795.845.172.954.115.972.927.479.877.527.738.560.008.204.118.558.930.004.777.748.727.761.853.813.510.493.840.581.861.598.652.211.605.960.308.356.405.941.821.189.714.037.868.726.219.481.498.727.603.653.616.298.856.174.822.413.033.485.438.785.324.024.751.419.417.183.012.281.078.209.729.303.537.372.804.574.372.095.228.703.622.776.363.945.290.869.806.258.422.355.148.507.571.039.619.387.449.629.866.808.188.769.662.815.778.153.079.393.179.093.143.648.340.761.738.581.819.563.002.994.422.790.754.955.061.288.818.308.430.079.648.693.232.179.158.765.918.035.565.216.157.115.402.992.120.276.155.607.873.107.937.477.466.841.528.362.987.708.699.450.152.031.231.862.594.203.085.693.838.944.657.061.346.236.704.234.026.821.102.958.954.951.197.087.076.546.186.622.796.294.536.451.620.756.509.351.018.906.023.773.821.539.532.776.208.676.978.589.731.966.330.308.893.304.665.169.436.185.078.350.641.568.336.944.530.051.437.491.311.298.834.367.265.238.595.404.904.273.455.928.723.949.525.227.184.617.404.367.854.754.610.474.377.019.768.025.576.605.881.038.077.270.707.717.942.221.977.090.385.438.585.844.095.492.116.099.852.538.903.974.655.703.943.973.086.090.930.596.963.360.767.529.964.938.414.598.185.705.963.754.561.497.355.827.813.623.833.288.906.309.004.288.017.321.424.808.663.962.671.333.528.009.232.758.350.873.059.614.118.723.781.422.101.460.198.615.747.386.855.096.896.089.189.180.441.339.558.524.822.867.541.113.212.638.793.675.567.650.340.362.970.031.930.023.397.828.465.318.547.238.244.232.028.015.189.689.660.418.822.976.000.815.437.610.652.254.270.163.595.650.875.433.851.147.123.214.227.266.605.403.581.781.469.090.806.576.468.950.587.661.997.186.505.665.475.715.792.897 </td></tr> <tr> <td>14</td> <td style="text-align:right;">4933 </td> <td style="text-align:left;">1.189.731.495.357.231.765.085.759.326.628.007.130.763.444.687.096.510.237.472.674.821.233.261.358.180.483.686.904.488.595.472.612.039.915.115.437.484.839.309.258.897.667.381.308.687.426.274.524.698.341.565.006.080.871.634.366.004.897.522.143.251.619.531.446.845.952.345.709.482.135.847.036.647.464.830.984.784.714.280.967.845.614.138.476.044.338.404.886.122.905.286.855.313.236.158.695.999.885.790.106.357.018.120.815.363.320.780.964.323.712.757.164.290.613.406.875.202.417.365.323.950.267.880.089.067.517.372.270.610.835.647.545.755.780.793.431.622.213.451.903.817.859.630.690.311.343.850.657.539.360.649.645.193.283.178.291.767.658.965.405.285.113.556.134.369.793.281.725.888.015.908.414.675.289.832.538.063.419.234.888.599.898.980.623.114.025.121.674.472.051.872.439.321.323.198.402.942.705.341.366.951.274.739.014.593.816.898.288.994.445.173.400.364.617.928.377.138.074.411.345.791.848.573.595.077.170.437.644.191.743.889.644.885.377.684.738.322.240.608.239.079.061.399.475.675.334.739.784.016.491.742.621.485.229.014.847.672.335.977.897.158.397.334.226.349.734.811.441.653.077.758.250.988.926.030.894.789.604.676.153.104.257.260.141.806.823.027.588.003.441.951.455.327.701.598.071.281.589.597.169.413.965.608.439.504.983.171.255.062.282.026.626.200.048.042.149.808.200.002.060.993.433.681.237.623.857.880.627.479.727.072.877.482.838.438.705.048.034.164.633.337.013.385.405.998.040.701.908.662.387.301.605.018.188.262.573.723.766.279.240.798.931.717.708.807.901.740.265.407.930.976.419.648.877.869.604.017.517.691.938.687.988.088.008.944.251.258.826.969.688.364.194.133.945.780.157.844.364.946.052.713.655.454.906.327.187.428.531.895.100.278.695.119.323.496.808.703.630.436.193.927.592.692.344.820.812.834.297.364.478.686.862.064.169.042.458.555.136.532.055.050.508.189.891.866.846.863.799.917.647.547.291.371.573.500.701.015.197.559.097.453.040.033.031.520.683.518.216.494.195.636.696.077.748.110.598.284.901.343.611.469.214.274.121.810.495.077.979.275.556.645.164.983.850.062.051.066.517.084.647.369.464.036.640.569.339.464.837.172.183.352.956.873.912.042.640.003.611.618.789.278.195.710.052.094.562.761.306.703.551.840.330.110.645.101.995.435.167.626.688.669.627.763.820.604.342.480.357.906.415.354.212.732.946.756.073.006.907.088.870.496.125.050.068.156.659.252.761.297.664.065.498.347.492.661.798.824.062.312.210.409.274.584.565.587.264.846.417.650.160.123.175.874.034.726.261.957.289.081.466.197.651.553.830.744.424.709.698.634.753.627.770.356.227.126.145.052.549.125.229.448.040.149.114.795.681.359.875.968.512.808.575.244.271.871.455.454.084.894.986.155.020.794.806.980.939.215.658.055.319.165.641.681.105.966.454.159.951.476.908.583.129.721.503.298.816.585.142.073.061.480.888.021.769.818.338.417.129.396.878.371.459.575.846.052.583.142.928.447.249.703.698.548.125.295.775.920.936.450.022.651.427.249.949.580.708.203.966.082.847.550.921.891.152.133.321.048.011.973.883.636.577.825.533.325.988.852.156.325.439.335.021.315.312.134.081.390.451.021.255.363.707.903.495.916.963.125.924.201.167.877.190.108.935.255.914.539.488.216.897.117.943.269.373.608.639.074.472.792.751.116.715.127.106.396.425.081.353.553.137.213.552.890.539.802.602.978.645.319.795.100.976.432.939.091.924.660.228.878.912.900.654.210.118.287.298.298.707.382.159.717.184.569.540.515.403.029.173.307.292.454.391.789.568.674.219.640.761.451.173.600.617.752.186.991.913.366.837.033.887.201.582.071.625.868.247.133.104.513.315.097.274.713.442.728.340.606.642.890.406.496.636.104.443.217.752.811.227.470.029.162.858.093.727.701.049.646.499.540.220.983.981.932.786.613.204.254.226.464.243.689.610.107.429.923.197.638.681.545.837.561.773.535.568.984.536.053.627.234.424.277.105.760.924.864.023.781.629.665.526.314.910.906.960.488.073.475.217.005.121.136.311.870.439.925.762.508.666.032.566.213.750.416.695.719.919.674.223.210.606.724.721.373.471.234.021.613.540.712.188.239.909.701.971.943.944.347.480.314.217.903.886.317.767.779.921.539.892.177.334.344.368.907.550.318.800.833.546.852.344.370.327.089.284.147.501.640.589.448.482.001.254.237.386.680.074.457.341.910.933.774.891.959.681.016.516.069.106.149.905.572.425.810.895.586.938.833.067.490.204.900.368.624.166.301.968.553.005.687.040.285.095.450.484.840.073.528.643.826.570.403.767.157.286.512.380.255.109.954.518.857.013.476.588.189.300.004.138.849.715.883.139.866.071.547.574.816.476.727.635.116.435.462.804.401.112.711.392.529.180.570.794.193.422.686.818.353.212.799.068.972.247.697.191.474.268.157.912.195.973.794.192.807.298.886.952.361.100.880.264.258.801.320.928.040.011.928.153.970.801.130.741.339.550.003.299.015.924.978.259.936.974.358.726.286.143.980.520.112.454.369.271.114.083.747.919.007.803.406.596.321.353.417.004.068.869.443.405.472.140.675.963.640.997.405.009.225.803.505.672.726.465.095.506.267.339.268.892.424.364.561.897.661.906.898.424.186.770.491.035.344.080.399.248.327.097.911.712.881.140.170.384.182.058.601.614.758.284.200.750.183.500.329.358.499.691.864.066.590.539.660.709.069.537.381.601.887.679.046.657.759.654.588.001.937.117.771.344.698.326.428.792.622.894.338.016.112.445.533.539.447.087.462.049.763.409.147.542.099.248.815.521.395.929.388.007.711.172.017.894.897.793.706.604.273.480.985.161.028.815.458.787.911.160.979.113.422.433.557.549.170.905.442.026.397.275.695.283.207.305.331.845.419.990.749.347.810.524.006.194.197.200.591.652.147.867.193.696.254.337.864.981.603.833.146.354.201.700.628.817.947.177.518.115.217.674.352.016.511.172.347.727.727.075.220.056.177.748.218.928.597.158.346.744.541.337.107.358.427.757.919.660.562.583.883.823.262.178.961.691.787.226.118.865.632.764.934.288.772.405.859.754.877.759.869.235.530.653.929.937.901.193.611.669.007.472.354.746.360.764.601.872.442.031.379.944.139.824.366.828.698.790.212.922.996.174.192.728.625.891.720.057.612.509.349.100.482.545.964.152.046.477.925.114.446.500.732.164.109.099.345.259.799.455.690.095.576.788.686.397.487.061.948.854.749.024.863.607.921.857.834.205.793.797.188.834.779.656.273.479.112.388.585.706.424.836.379.072.355.410.286.787.018.527.401.653.934.219.888.361.061.949.671.961.055.068.686.961.468.019.035.629.749.424.086.587.195.041.004.404.915.266.476.272.761.070.511.568.387.063.401.264.136.517.237.211.409.916.458.796.347.624.949.215.904.533.937.210.937.520.465.798.300.175.408.017.538.862.312.719.042.361.037.129.338.896.586.028.150.046.596.078.872.444.365.564.480.545.689.033.575.955.702.988.396.719.744.528.212.984.142.578.483.954.005.084.264.327.730.840.985.420.021.409.069.485.412.320.805.268.520.094.146.798.876.110.414.583.170.390.473.982.488.899.228.091.818.213.934.288.295.679.717.369.943.152.460.447.027.290.669.964.066.817 </td></tr></tbody></table> </div> </div> Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}\approx F_{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}\approx F_{n}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54efd5879ad8597462d3f254627393d8ea455130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.781ex; height:2.843ex;" alt="{\displaystyle F_{n+1}\approx F_{n}^{2}}"> hat die Fermatzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfbe34f204a6b7b01dd49571e6b287c2bdf7735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.814ex; height:2.509ex;" alt="{\displaystyle F_{n+1}}"> doppelt so viele oder um eine weniger als doppelt so viele Stellen wie ihr Vorgänger <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}">. <div class="mw-heading mw-heading2"><h2 id="Fermatsche_Primzahlen">Fermatsche Primzahlen</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=2" title="Abschnitt bearbeiten: Fermatsche Primzahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Fermatsche Primzahlen">Quelltext bearbeiten</a>]</div> Die Idee hinter Fermatschen Primzahlen ist der Satz, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"> und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ff5e7cd68aa70f4449905e0dd1a9724a0b5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.691ex; height:2.343ex;" alt="{\displaystyle k=2^{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"> prim sein kann: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1\in \mathbb {P} \ \Rightarrow \ k=0\lor \exists n\in \mathbb {N} _{0}\colon k=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mtext> </mtext> <mo stretchy="false">⇒</mo> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>∨</mo> <mi mathvariant="normal">∃</mi> <mi>n</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>k</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1\in \mathbb {P} \ \Rightarrow \ k=0\lor \exists n\in \mathbb {N} _{0}\colon k=2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98684605aea78ec8c6d6951bba4988c08fdef18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.33ex; height:3.009ex;" alt="{\displaystyle 2^{k}+1\in \mathbb {P} \ \Rightarrow \ k=0\lor \exists n\in \mathbb {N} _{0}\colon k=2^{n}}"></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis des Satzes:</div> <div class="NavContent" style="text-align:left"> Beweis durch <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">Widerspruch</a>: Man führt die Annahme, dass das zu Beweisende falsch sei, zu einem Widerspruch. <dl><dd>Annahme: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> ist prim und die Hochzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> hat einen ungeraden Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d6b5c226641ad3def8e65631f0eb463d9c67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}">.</dd> <dd>Dann gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1=2^{{\frac {k}{c}}\cdot c}+1=(2^{\frac {k}{c}})^{c}+1^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>c</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1=2^{{\frac {k}{c}}\cdot c}+1=(2^{\frac {k}{c}})^{c}+1^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5347cfb69967afa562936f0903f031dc62893692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.176ex; height:3.843ex;" alt="{\displaystyle 2^{k}+1=2^{{\frac {k}{c}}\cdot c}+1=(2^{\frac {k}{c}})^{c}+1^{c}}"></dd></dl></dd> <dd>mit einer ganzen Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k/c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c12409d0fbcf72f19b1c869fe426a298513599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.381ex; height:2.843ex;" alt="{\displaystyle k/c}">. Nach Annahme ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ungerade, also ist diese Summe <a href="/wiki/Binomische_Formeln#Der_binomische_Lehrsatz" title="Binomische Formeln">bekanntlich</a> durch die Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {k}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {k}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ee6fc1f31da2b3b17073f8a25c3f93c5aa071d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.929ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {k}{c}}+1}"> der beiden Basen teilbar: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1=(2^{\frac {k}{c}})^{c}+1^{c}=(2^{\frac {k}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (2^{\frac {k}{c}})^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1=(2^{\frac {k}{c}})^{c}+1^{c}=(2^{\frac {k}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (2^{\frac {k}{c}})^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32748564ed75a9cb9a0f5a04021a3b6bd6d9c376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.864ex; height:7.676ex;" alt="{\displaystyle 2^{k}+1=(2^{\frac {k}{c}})^{c}+1^{c}=(2^{\frac {k}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (2^{\frac {k}{c}})^{j}}"></dd></dl></dd> <dd>Weil die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> prim ist, muss ihr Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {k}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {k}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ee6fc1f31da2b3b17073f8a25c3f93c5aa071d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.929ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {k}{c}}+1}"> gleich 1 oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> sein. Aber in Widerspruch dazu ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {k}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {k}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ee6fc1f31da2b3b17073f8a25c3f93c5aa071d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.929ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {k}{c}}+1}"> (wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {k}{c}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {k}{c}}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c511c6f6d171439931b8a333dfbb81e3be835844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.187ex; height:3.343ex;" alt="{\displaystyle 2^{\frac {k}{c}}>0}">) größer als 1 und (wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0\Rightarrow k/c<k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo><</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0\Rightarrow k/c<k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4708fade16dd20a6a5ad5630f05e38552b1d7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.776ex; height:2.843ex;" alt="{\displaystyle k>0\Rightarrow k/c<k}">) kleiner als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}">. Die Annahme, dass sowohl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> prim ist als auch, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> positiv ist und einen ungeraden Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d6b5c226641ad3def8e65631f0eb463d9c67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}"> hat, muss daher fallengelassen werden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"> kann nur prim sein, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> gleich 0 oder gleich einer Zweierpotenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"> ist, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> Die Umkehrung dieses Satzes, dass also nicht nur (wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{0}+1=1+1=2\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{0}+1=1+1=2\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e66d1f363467be246efa393d2d6962e68910491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.005ex; height:2.843ex;" alt="{\displaystyle 2^{0}+1=1+1=2\in \mathbb {P} }"> offensichtlich) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{0}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{0}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcf3f3b7b5c08e2cde4e0567fc97a0ab6f37d10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.22ex; height:2.843ex;" alt="{\displaystyle 2^{0}+1}">, sondern auch jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"> prim sei, ist falsch. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f58df5f4307605e8fa07ae29d6262393b3b0c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{0}}"> bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8718a2df1e70bea3cd21ab9e0cd45dc354818451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{4}}"> sind sogar die einzigen bisher bekannten Fermatschen Primzahlen. Schon Fermat zeigte, dass diese ersten fünf Fermat-Zahlen Primzahlen sind, und vermutete 1640, dass dies auf alle Fermat-Zahlen zutreffe. Diese Vermutung wurde aber schon 1732 von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> einfach widerlegt, indem er mit 641 einen <a href="/wiki/Teilbarkeit#Definition" title="Teilbarkeit">echten Teiler</a> von F5 = 4.294.967.297 fand.<a href="#cite_note-euler-4">[4]</a> Man vermutet inzwischen, dass außer den ersten fünf keine weiteren Fermatschen Primzahlen existieren. Diese Vermutung beruht auf statistischen Abschätzungen: Der <a href="/wiki/Primzahlsatz" title="Primzahlsatz">Primzahlsatz</a> besagt, dass die Anzahl der Primzahlen, die nicht größer als x sind, näherungsweise gleich x / ln x ist. Die <a href="/wiki/Primzahlsatz#Zahlenbeispiele" title="Primzahlsatz">Primzahldichte</a> oder Wahrscheinlichkeit dafür, dass Fn als ungerade Zahl eine Primzahl ist, beträgt daher näherungsweise 2 / ln Fn ≈ 3 / 2n. Die Wahrscheinlichkeit, dass die Fermatzahl Fn oder eine der folgenden Fermatzahlen eine Primzahl ist, ergibt sich durch Summation der <a href="/wiki/Geometrische_Reihe" title="Geometrische Reihe">geometrische Reihe</a> ungefähr zu 6 / 2n. Für verbliebene weder teilweise noch vollständig faktorisierte Fermat-Zahlen ist diese Wahrscheinlichkeit mit etwa 6 ⸱ 10−10 mittlerweile aber sehr klein geworden. <div class="mw-heading mw-heading2"><h2 id="Faktorisierungsergebnisse_von_Fermat-Zahlen">Faktorisierungsergebnisse von Fermat-Zahlen</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=3" title="Abschnitt bearbeiten: Faktorisierungsergebnisse von Fermat-Zahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Faktorisierungsergebnisse von Fermat-Zahlen">Quelltext bearbeiten</a>]</div> Die Zahlen F0 bis F4 sind, wie schon Fermat erkannt hat, Primzahlen: <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>n </th> <th>Fermat-Primzahl Fn </th></tr> <tr> <td>00</td> <td align="right">3 </td></tr> <tr> <td>01</td> <td align="right">5 </td></tr> <tr> <td>02</td> <td align="right">17 </td></tr> <tr> <td>03</td> <td align="right">257 </td></tr> <tr> <td>04</td> <td align="right">65537 </td></tr></tbody></table> Die Zahlen F5 bis F11 sind entgegen der Vermutung Fermats zusammengesetzt. Sie sind bereits vollständig faktorisiert:<a href="#cite_note-Status-5">[5]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der zusammengesetzten und vollständig faktorisierten Fermat-Zahlen</div> <div class="NavContent" style="text-align:left"> <table class="wikitable" style="margin-left:2em; max-width:20em;"> <tbody><tr class="hintergrundfarbe6"> <th>n </th> <th>Entdecker der Faktoren </th> <th>Primfaktorenzerlegung von Fn </th></tr> <tr> <td>05 </td> <td><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1732) </td> <td>4.294.967.297 (10 Stellen)  = 641 (3 Stellen)  × 6.700.417 (7 Stellen) </td></tr> <tr> <td>06 </td> <td><a href="/wiki/Thomas_Clausen_(Astronom)" title="Thomas Clausen (Astronom)">Clausen</a> (1855), <a href="/wiki/Fortun%C3%A9_Landry" title="Fortuné Landry">Landry</a> & <a href="/w/index.php?title=H._Le_Lasseur&action=edit&redlink=1" class="new" title="H. Le Lasseur (Seite nicht vorhanden)">Le Lasseur</a> (1880) </td> <td>18.446.744.073.709.551.617 (20 Stellen)  = 274.177 (6 Stellen)  × 67.280.421.310.721 (14 Stellen) </td></tr> <tr> <td>07 </td> <td><a href="/w/index.php?title=Michael_A._Morrison&action=edit&redlink=1" class="new" title="Michael A. Morrison (Seite nicht vorhanden)">Morrison</a> & <a href="/wiki/John_Brillhart" title="John Brillhart">Brillhart</a> (1970)<a href="#cite_note-morrison-6">[6]</a> </td> <td>340.282.366.920.938.463.463.374.607.431.768.211.457 (39 Stellen)  = 59.649.589.127.497.217 (17 Stellen)  × 5.704.689.200.685.129.054.721 (22 Stellen) </td></tr> <tr> <td>08 </td> <td><a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent</a> & <a href="/wiki/John_M._Pollard" title="John M. Pollard">Pollard</a> (1980) </td> <td>115.792.089.237.316.195.423.570.985.008.687.907.853.269.984.665.640.564.039.457.584.007.913.129.639.937 (78 Stellen)  = 1.238.926.361.552.897 (16 Stellen)  × 93.461.639.715.357.977.769.163.558.199.606.896.584.051.237.541.638.188.580.280.321 (62 Stellen) </td></tr> <tr> <td>09 </td> <td><a href="/w/index.php?title=Alfred_Edward_Western&action=edit&redlink=1" class="new" title="Alfred Edward Western (Seite nicht vorhanden)">Western</a> (1903), <a href="/wiki/Arjen_Lenstra" title="Arjen Lenstra">Lenstra</a> & <a href="/wiki/Mark_S._Manasse" title="Mark S. Manasse">Manasse</a> (1990) </td> <td>13.407.807.929.942.597.099.574.024.998.205.846.127.479.365.820.592.393.377.723.561.443.721.764.030.073.546.976.801.874.298.166.903.427.690.031. 858.186.486.050.853.753.882.811.946.569.946.433.649.006.084.097 (155 Stellen)  = 2.424.833 (7 Stellen)  × 7.455.602.825.647.884.208.337.395.736.200.454.918.783.366.342.657 (49 Stellen)  × 741.640.062.627.530.801.524.787.141.901.937.474.059.940.781.097.519.023.905.821.316.144.415.759.504.705.008.092.818.711.693.940.737 (99 Stellen) </td></tr> <tr> <td>10 </td> <td><a href="/wiki/John_L._Selfridge" title="John L. Selfridge">Selfridge</a> (1953), <a href="/wiki/John_Brillhart" title="John Brillhart">Brillhart</a> (1962), <a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent</a> (1995) </td> <td>179.769.313.486.231.590.772.930 … 304.835.356.329.624.224.137.217 (309 Stellen)  = 45.592.577 (8 Stellen)  × 6.487.031.809 (10 Stellen)  × 4.659.775.785.220.018.543.264.560.743.076.778.192.897 (40 Stellen)  × 130.439.874.405.488.189.727.484 … 806.217.820.753.127.014.424.577 (252 Stellen) </td></tr> <tr> <td>11 </td> <td><a href="/wiki/Allan_Joseph_Champneys_Cunningham" title="Allan Joseph Champneys Cunningham">Cunningham</a> (1899), <a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent</a> & <a href="/wiki/Fran%C3%A7ois_Morain" title="François Morain">Morain</a> (1988) </td> <td>32.317.006.071.311.007.300.714.8 … 193.555.853.611.059.596.230.657 (617 Stellen)  = 319.489 (6 Stellen)  × 974.849 (6 Stellen)  × 167.988.556.341.760.475.137 (21 Stellen)  × 3.560.841.906.445.833.920.513 (22 Stellen)  × 173.462.447.179.147.555.430.258 … 491.382.441.723.306.598.834.177 (564 Stellen) </td></tr></tbody></table> </div> </div> Ab F12 ist keine Fermat-Zahl mehr vollständig faktorisiert. Die ersten acht lauten: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der ersten acht der nur teilweise faktorisierten Fermat-Zahlen</div> <div class="NavContent" style="text-align:left"> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>n </th> <th>Entdecker der Faktoren </th> <th>Primfaktorenzerlegung von Fn </th></tr> <tr> <td>12 </td> <td><a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Lucas</a> & <a href="/w/index.php?title=Iwan_Michejewitsch_Perwuschin&action=edit&redlink=1" class="new" title="Iwan Michejewitsch Perwuschin (Seite nicht vorhanden)">Perwuschin</a> (1877), <a href="/w/index.php?title=Alfred_Edward_Western&action=edit&redlink=1" class="new" title="Alfred Edward Western (Seite nicht vorhanden)">Western</a> (1903), <a href="/w/index.php?title=John_C._Hallyburton_Jr.&action=edit&redlink=1" class="new" title="John C. Hallyburton Jr. (Seite nicht vorhanden)">Hallyburton</a> & <a href="/wiki/John_Brillhart" title="John Brillhart">Brillhart</a> (1974), <a href="/w/index.php?title=Robert_Baillie&action=edit&redlink=1" class="new" title="Robert Baillie (Seite nicht vorhanden)">Baillie</a> (1986), <a href="/w/index.php?title=Michael_Vang&action=edit&redlink=1" class="new" title="Michael Vang (Seite nicht vorhanden)">Vang</a>, <a href="/w/index.php?title=Paul_Zimmermann_(Informatiker)&action=edit&redlink=1" class="new" title="Paul Zimmermann (Informatiker) (Seite nicht vorhanden)">Zimmermann</a> & <a href="/w/index.php?title=Alexander_Kruppa&action=edit&redlink=1" class="new" title="Alexander Kruppa (Seite nicht vorhanden)">Kruppa</a> (2010) </td> <td>1.044.388.881.413.152.506.691.752.710.716 … 340.403.154.190.337 (1234 Stellen)  = 114.689 (6 Stellen)  × 26.017.793 (8 Stellen)  × 63.766.529 (8 Stellen)  × 190.274.191.361 (12 Stellen)  × 1.256.132.134.125.569 (16 Stellen)  ×  568.630.647.535.356.955.169.033.410.940.867.804.839.360.742.060.818.433 (54 Stellen)  × zusammengesetzte Zahl (1133 Stellen) </td></tr> <tr> <td>13 </td> <td><a href="/w/index.php?title=John_C._Hallyburton_Jr.&action=edit&redlink=1" class="new" title="John C. Hallyburton Jr. (Seite nicht vorhanden)">Hallyburton</a> & <a href="/wiki/John_Brillhart" title="John Brillhart">Brillhart</a> (1974), <a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall</a> (1991), <a href="/wiki/Richard_P._Brent" title="Richard P. Brent">Brent</a> (1995) </td> <td>1.090.748.135.619.415.929.462.984.244.733 … 665.475.715.792.897 (2467 Stellen)  =  2.710.954.639.361 (13 Stellen)  ×  2.663.848.877.152.141.313 (19 Stellen)  ×  3.603.109.844.542.291.969 (19 Stellen)  ×  319.546.020.820.551.643.220.672.513 (27 Stellen)  × zusammengesetzte Zahl (2391 Stellen) </td></tr> <tr> <td>14 </td> <td><a href="/w/index.php?title=Tapio_Rajala&action=edit&redlink=1" class="new" title="Tapio Rajala (Seite nicht vorhanden)">Rajala</a> & <a href="/wiki/George_Woltman" title="George Woltman">Woltman</a> (2010) </td> <td>1.189.731.495.357.231.765.085.759.326.628 … 290.669.964.066.817 (4933 Stellen)  =  116.928.085.873.074.369.829.035.993.834.596.371.340.386.703.423.373.313 (54 Stellen)  × zusammengesetzte Zahl (4880 Stellen) </td></tr> <tr> <td>15 </td> <td><a href="/wiki/Maurice_Kraitchik" title="Maurice Kraitchik">Kraitchik</a> (1925), <a href="/w/index.php?title=Gary_Gostin&action=edit&redlink=1" class="new" title="Gary Gostin (Seite nicht vorhanden)">Gostin</a> (1987), <a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall</a> & <a href="/w/index.php?title=C._van_Halewyn&action=edit&redlink=1" class="new" title="C. van Halewyn (Seite nicht vorhanden)">van Halewyn</a> (1997) </td> <td>1.415.461.031.044.954.789.001.553.027.744 … 104.633.712.377.857 (9865 Stellen)  =  1.214.251.009 (10 Stellen)  ×  2.327.042.503.868.417 (16 Stellen)  ×  168.768.817.029.516.972.383.024.127.016.961 (33 Stellen)  × zusammengesetzte Zahl (9808 Stellen) </td></tr> <tr> <td>16 </td> <td><a href="/wiki/John_Selfridge" class="mw-redirect" title="John Selfridge">Selfridge</a> (1953), <a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall</a> & <a href="/w/index.php?title=Karl_Dilcher&action=edit&redlink=1" class="new" title="Karl Dilcher (Seite nicht vorhanden)">Dilcher</a> (1996) </td> <td>2.003.529.930.406.846.464.979.072.351.560 … 895.905.719.156.737 (19729 Stellen)  =  825.753.601 (9 Stellen)  ×  188.981.757.975.021.318.420.037.633 (27 Stellen)  × zusammengesetzte Zahl (19694 Stellen) </td></tr> <tr> <td>17 </td> <td><a href="/w/index.php?title=Gary_Gostin&action=edit&redlink=1" class="new" title="Gary Gostin (Seite nicht vorhanden)">Gostin</a> (1978), <a href="/w/index.php?title=David_Bessell&action=edit&redlink=1" class="new" title="David Bessell (Seite nicht vorhanden)">Bessell</a> & <a href="/wiki/George_Woltman" title="George Woltman">Woltman</a> (2011) </td> <td>4.014.132.182.036.063.039.166.060.606.038 … 318.570.934.173.697 (39457 Stellen)  =  31.065.037.602.817 (14 Stellen)  ×  7.751.061.099.802.522.589.358.967.058.392.886.922.693.580.423.169 (49 Stellen)  × zusammengesetzte Zahl (39395 Stellen) </td></tr> <tr> <td>18 </td> <td><a href="/w/index.php?title=Alfred_Edward_Western&action=edit&redlink=1" class="new" title="Alfred Edward Western (Seite nicht vorhanden)">Western</a> (1903), <a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall</a>, <a href="/w/index.php?title=Richard_J._McIntosh&action=edit&redlink=1" class="new" title="Richard J. McIntosh (Seite nicht vorhanden)">McIntosh</a> & <a href="/w/index.php?title=Claude_Tardif&action=edit&redlink=1" class="new" title="Claude Tardif (Seite nicht vorhanden)">Tardif</a> (1999) </td> <td>16.113.257.174.857.604.736.195.721.184.520 … 349.934.298.300.417 (78914 Stellen)  =  13.631.489 (8 Stellen)  ×  81.274.690.703.860.512.587.777 (23 Stellen)  × zusammengesetzte Zahl (78884 Stellen) </td></tr> <tr> <td>19 </td> <td><a href="/wiki/Hans_Riesel" title="Hans Riesel">Riesel</a> (1962), <a href="/w/index.php?title=Claude_Wrathall&action=edit&redlink=1" class="new" title="Claude Wrathall (Seite nicht vorhanden)">Wrathall</a> (1963), <a href="/w/index.php?title=David_Bessell&action=edit&redlink=1" class="new" title="David Bessell (Seite nicht vorhanden)">Bessell</a> & <a href="/wiki/George_Woltman" title="George Woltman">Woltman</a> (2009) </td> <td>259.637.056.783.100.077.612.659.649.572.688 … 528.226.185.773.057 (157827 Stellen)  =  70.525.124.609 (11 Stellen)  ×  646.730.219.521 (12 Stellen)  ×  37.590.055.514.133.754.286.524.446.080.499.713 (35 Stellen)  × zusammengesetzte Zahl (157770 Stellen) </td></tr></tbody></table> </div> </div> Von F12 bis F32 und von einigen größeren Fermat-Zahlen ist bekannt, dass sie zusammengesetzt sind – hauptsächlich, weil ein oder mehrere Faktoren gefunden wurden. Von zwei Fermat-Zahlen (F20 und F24) kennt man zwar keinen Faktor, hat aber auf andere Art gezeigt, dass sie zusammengesetzt sind.<a href="#cite_note-YoungBuell-7">[7]</a><a href="#cite_note-Crandall-8">[8]</a> Für F14 wurde am 3. Februar 2010 ein Faktor veröffentlicht,<a href="#cite_note-MersenneForumF14-9">[9]</a> für F22 am 25. März 2010.<a href="#cite_note-MersenneForumF22-10">[10]</a> Die kleinste Fermat-Zahl, von der bislang nicht bekannt ist, ob sie prim oder zusammengesetzt ist, ist F33. Diese Zahl hat 2.585.827.973 Stellen. Insgesamt weiß man von den ersten 50 Fermat-Zahlen nur von 10 nicht, ob sie zusammengesetzt sind oder nicht.<a href="#cite_note-11">[11]</a> F18.233.954 ist die größte Fermat-Zahl, von der ein Faktor bekannt ist, nämlich die Primzahl 7 · 218.233.956 + 1. Dieser Faktor wurde am 5. Oktober 2020 von Ryan Propper mit Computer-Programmen von Geoffrey Reynolds, Jean Penné und Jim Fougeron entdeckt und hat 5.488.969 Stellen. Die Fermat-Zahl F18.233.954 selbst hat allerdings mehr als 105.488.966 Stellen.<a href="#cite_note-Rekordfaktor-12">[12]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Versuch einer anschaulichen Erklärung der Größe der Fermat-Zahl F18.233.954</div> <div class="NavContent" style="text-align:left"> Es gibt keine sinnvolle Methode, sich die Menge an Papier, die man benötigt sie aufzuschreiben – oder gar die Zahl selber – vorzustellen: Selbst mit den hypothetisch kleinsten Teilchen aufgeschrieben, ist das Universum spätestens mit F615 vollgeschrieben und für jeden weiteren Schritt bis F18233954 würde sich der Platz zum Aufschreiben jeweils verdoppeln. Nur hat man mit F615 ja quasi damit noch nicht mal richtig angefangen! Ein wissenschaftlicher Taschenrechner würde eine etwa 27 Kilometer lange Zeile oder alternativ eine 27 Meter mal 10 Meter große Tafel allein für das Anschreiben der Anzahl der Stellen, also von 105488966, als Dezimalzahl benötigen. </div> </div> Insgesamt weiß man von 325 Fermat-Zahlen, dass sie zusammengesetzt sind. 370 Primfaktoren sind bisher bekannt (Stand: 6. Juli 2024).<a href="#cite_note-Status-5">[5]</a><a href="#cite_note-MathWorld-13">[13]</a> Der folgenden Tabelle kann man entnehmen, in welchem Intervall wie viele zusammengesetzte Fermat-Zahlen bekannt sind (Stand: 6. Juli 2024): <table class="wikitable" style="text-align:center; border:0"> <caption>Anteil der Fermat-Zahlen, die nachweislich keine Primzahlen sind </caption> <tbody><tr class="hintergrundfarbe6"> <th>n </th> <th style="line-height:120%">bekannt zusammen- gesetzt </th> <th>Anteil </th> <td rowspan="8" style="border:0; background:white"> </td> <th>n </th> <th style="line-height:120%">bekannt zusammen- gesetzt </th> <th>Anteil </th></tr> <tr> <td>05 ≤ n ≤ 32</td> <td>028</td> <td>100,0 % </td> <td>10001 ≤ n ≤ 50000</td> <td>38</td> <td>0,09500 % </td></tr> <tr> <td>033 ≤ n ≤ 100</td> <td>032</td> <td>047,1 % </td> <td>050001 ≤ n ≤ 100000</td> <td>11</td> <td>0,02200 % </td></tr> <tr> <td>101 ≤ n ≤ 500</td> <td>064</td> <td>016,0 % </td> <td>100001 ≤ n ≤ 500000</td> <td>27</td> <td>0,00675 % </td></tr> <tr> <td>0501 ≤ n ≤ 1000</td> <td>022</td> <td>004,4 % </td> <td>0500001 ≤ n ≤ 1000000</td> <td>07</td> <td>0,00140 % </td></tr> <tr> <td>1001 ≤ n ≤ 5000</td> <td>053</td> <td>001,3 % </td> <td>1000001 ≤ n ≤ 5000000</td> <td>13</td> <td>0,00033 % </td></tr> <tr> <td>05001 ≤ n ≤ 10000</td> <td>027</td> <td>000,5 % </td> <td>05000001 ≤ n ≤ 20000000</td> <td>03</td> <td>0,00006 % </td></tr> <tr class="sortbottom hintergrundfarbe6"> <td>TOTAL</td> <td>226</td> <td>002,3 % </td> <td>TOTAL</td> <td>99</td> <td>0,00050 % </td></tr></tbody></table> Die kleinsten 25 Fermat-Primfaktoren sind die folgenden: <dl><dd>3, 5, 17, 257, 641, 65.537, 114.689, 274.177, 319.489, 974.849, 2.424.833, 6.700.417, 13.631.489, 26.017.793, 45.592.577, 63.766.529, 167.772.161, 825.753.601, 1.214.251.009, 6.487.031.809, 70.525.124.609, 190.274.191.361, 646.730.219.521, 2.710.954.639.361, 2.748.779.069.441, … (Folge <a href="//oeis.org/A023394" class="extiw" title="oeis:A023394">A023394</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> Um von einer Fermat-Zahl nachzuweisen, dass sie zusammengesetzt ist, benutzt man in der Regel den <a href="/wiki/P%C3%A9pin-Test" title="Pépin-Test">Pépin-Test</a> und den <a href="/wiki/Zahlk%C3%B6rpersieb" title="Zahlkörpersieb">Suyama-Test</a>, die beide besonders auf diese Zahlen zugeschnitten und sehr schnell sind. Die folgenden 16 Primfaktoren von Fermat-Zahlen wurden vor 1950 entdeckt. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Vor 1950 entdeckte Primfaktoren von Fermat-Zahlen (ohne Zuhilfenahme programmgesteuerter Rechenmaschinen)</div> <div class="NavContent" style="text-align:left"> <table class="wikitable sortable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Entdecker </th> <th>Fermat- Zahl </th> <th>Dezimal- stellen von Fn </th> <th>Faktor </th> <th data-sort-type="number">Dezimal- stellen dieses Faktors </th> <th>Faktor ausgeschrieben </th></tr> <tr> <td>1732 </td> <td><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> </td> <td>F5 <style data-mw-deduplicate="TemplateStyles:r247957335">.mw-parser-output .fussnoten-marke{font-size:0.75rem;font-style:normal;font-variant:normal;font-weight:normal;unicode-bidi:isolate;white-space:nowrap}.mw-parser-output .fussnoten-marke.reference,.mw-parser-output span.fussnoten-inhalt{padding-left:0.1rem}.mw-parser-output .fussnoten-marke.reference~.fussnoten-marke.reference,.mw-parser-output span.fussnoten-inhalt~span.fussnoten-inhalt{padding-left:0.15rem}.mw-parser-output .fussnoten-block{margin-bottom:0.1rem}.mw-parser-output div.fussnoten-inhalt{display:inline-block;padding-left:0.8rem;text-indent:-0.8rem}.mw-parser-output div.fussnoten-inhalt p,.mw-parser-output div.fussnoten-inhalt dl,.mw-parser-output div.fussnoten-inhalt ol,.mw-parser-output div.fussnoten-inhalt ul{text-indent:0}.mw-parser-output div.fussnoten-inhalt.fussnoten-floatfix{display:block}.mw-parser-output .fussnoten-box{margin-top:0.5rem;padding-left:0.8rem}.mw-parser-output .fussnoten-box,.mw-parser-output div.fussnoten-inhalt{font-size:94%}.mw-parser-output .fussnoten-box div.fussnoten-inhalt,.mw-parser-output span.fussnoten-inhalt,.mw-parser-output .fussnoten-inhalt .reference-text{font-size:inherit}.mw-parser-output .fussnoten-inhalt .reference-text{display:inline}.mw-parser-output .fussnoten-linie{display:inline-block;position:relative;top:-1em;border-top:solid 1px #808080;width:8rem}.mw-parser-output .fussnoten-linie+p,.mw-parser-output .fussnoten-linie+dl,.mw-parser-output .fussnoten-linie+ol,.mw-parser-output .fussnoten-linie+ul,.mw-parser-output .fussnoten-linie+link+div{margin-top:-1em}.mw-parser-output .fussnoten-marke.reference:target,.mw-parser-output .fussnoten-inhalt:target{background-color:#eaf3ff;box-shadow:0 0 0 0.25em #eaf3ff}.mw-parser-output .fussnoten-marke.reference:target,.mw-parser-output .fussnoten-inhalt:target .fussnoten-marke{font-weight:bold}</style> (a) </td> <td style="text-align:right">10 </td> <td style="text-align:right">5 · 27 + 1 </td> <td style="text-align:center">3 </td> <td style="text-align:right">641 </td></tr> <tr> <td>1732 </td> <td>Leonhard Euler </td> <td>F5 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> (a) </td> <td style="text-align:right">10 </td> <td style="text-align:right">52347 · 27 + 1 </td> <td style="text-align:center" data-sort-value="7,9">7 </td> <td style="text-align:right">6.700.417 </td></tr> <tr> <td>1855 </td> <td><a href="/wiki/Thomas_Clausen_(Astronom)" title="Thomas Clausen (Astronom)">Thomas Clausen</a> </td> <td>F6 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> (a) </td> <td style="text-align:right">20 </td> <td style="text-align:right">1071 · 28 + 1 </td> <td style="text-align:center" data-sort-value="6,4">6 </td> <td style="text-align:right">274.177 </td></tr> <tr> <td>1855 </td> <td>Thomas Clausen </td> <td>F6 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> (a) </td> <td style="text-align:right">20 </td> <td style="text-align:right">262814145745 · 28 + 1 </td> <td style="text-align:center">14 </td> <td style="text-align:right">67.280.421.310.721 </td></tr> <tr> <td>1877 </td> <td><a href="/w/index.php?title=Iwan_Michejewitsch_Perwuschin&action=edit&redlink=1" class="new" title="Iwan Michejewitsch Perwuschin (Seite nicht vorhanden)">Iwan Perwuschin</a> </td> <td>F12 </td> <td style="text-align:right">1.234 </td> <td style="text-align:right">7 · 214 + 1 </td> <td style="text-align:center" data-sort-value="6,1">6 </td> <td style="text-align:right">114.689 </td></tr> <tr> <td>1878 </td> <td>Iwan Perwuschin </td> <td>F23 </td> <td style="text-align:right">2.525.223 </td> <td style="text-align:right">5 · 225 + 1 </td> <td style="text-align:center">9 </td> <td style="text-align:right">167.772.161 </td></tr> <tr> <td>1886 </td> <td><a href="/w/index.php?title=Paul_Peter_Heinrich_Seelhoff&action=edit&redlink=1" class="new" title="Paul Peter Heinrich Seelhoff (Seite nicht vorhanden)">Paul Peter Heinrich Seelhoff</a> </td> <td>F36 </td> <td style="text-align:right">20.686.623.784 </td> <td style="text-align:right">5 · 239 + 1 </td> <td style="text-align:center" data-sort-value="13,1">13 </td> <td style="text-align:right">2.748.779.069.441 </td></tr> <tr> <td>1899 </td> <td><a href="/wiki/Allan_Joseph_Champneys_Cunningham" title="Allan Joseph Champneys Cunningham">Allan Joseph Champneys Cunningham</a> </td> <td>F11 </td> <td style="text-align:right">617 </td> <td style="text-align:right">39 · 213 + 1 </td> <td style="text-align:center" data-sort-value="6,6">6 </td> <td style="text-align:right">319.489 </td></tr> <tr> <td>1899 </td> <td>Allan Joseph Champneys Cunningham </td> <td>F11 </td> <td style="text-align:right">617 </td> <td style="text-align:right">119 · 213 + 1 </td> <td style="text-align:center" data-sort-value="6,9">6 </td> <td style="text-align:right">974.849 </td></tr> <tr> <td>1903 </td> <td><a href="/w/index.php?title=Alfred_Edward_Western&action=edit&redlink=1" class="new" title="Alfred Edward Western (Seite nicht vorhanden)">Alfred Edward Western</a> </td> <td>F9 </td> <td style="text-align:right">155 </td> <td style="text-align:right">37 · 216 + 1 </td> <td style="text-align:center" data-sort-value="7,1">7 </td> <td style="text-align:right">2.424.833 </td></tr> <tr> <td>1903 </td> <td>Alfred Edward Western </td> <td>F12 </td> <td style="text-align:right">1.234 </td> <td style="text-align:right">397 · 216 + 1 </td> <td style="text-align:center" data-sort-value="8,5">8 </td> <td style="text-align:right">26.017.793 </td></tr> <tr> <td>1903 </td> <td>Alfred Edward Western </td> <td>F12 </td> <td style="text-align:right">1.234 </td> <td style="text-align:right">973 · 216 + 1 </td> <td style="text-align:center" data-sort-value="8,9">8 </td> <td style="text-align:right">63.766.529 </td></tr> <tr> <td>1903 </td> <td>Alfred Edward Western </td> <td>F18 </td> <td style="text-align:right">78.914 </td> <td style="text-align:right">13 · 220 + 1 </td> <td style="text-align:center" data-sort-value="8,1">8 </td> <td style="text-align:right">13.631.489 </td></tr> <tr> <td>1903 </td> <td><a href="/wiki/James_Cullen_(Mathematiker)" title="James Cullen (Mathematiker)">James Cullen</a> </td> <td>F38 </td> <td style="text-align:right">82.746.495.136 </td> <td style="text-align:right">3 · 241 + 1 </td> <td style="text-align:center" data-sort-value="13,9">13 </td> <td style="text-align:right">6.597.069.766.657 </td></tr> <tr> <td>1906 </td> <td><a href="/w/index.php?title=James_Caddall_Morehead&action=edit&redlink=1" class="new" title="James Caddall Morehead (Seite nicht vorhanden)">James Caddall Morehead</a> </td> <td>F73 </td> <td style="text-align:right">2.843.147.923.723.958.896.933 </td> <td style="text-align:right">5 · 275 + 1 </td> <td style="text-align:center">24 </td> <td style="text-align:right">188.894.659.314.785.808.547.841 </td></tr> <tr> <td>1925 </td> <td><a href="/wiki/Maurice_Kraitchik" title="Maurice Kraitchik">Maurice Borissowitsch Kraitchik</a> </td> <td>F15 </td> <td style="text-align:right">9.865 </td> <td style="text-align:right">579 · 221 + 1 </td> <td style="text-align:center">10 </td> <td style="text-align:right">1.214.251.009 </td></tr></tbody></table> <div style="text-align:left"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-block"><div class="fussnoten-inhalt references">(a) <div class="reference-text">Diese Zahlen waren damit vollständig faktorisiert.</div></div></div> </div> </div> </div> Seit 1950 wurden alle weiteren Faktoren durch Einsatz von Computern gefunden.<a href="#cite_note-Entdeckungsdaten-14">[14]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Anzahl der entdeckte Primfaktoren von Fermat-Zahlen (unter Zuhilfenahme programmgesteuerter Rechenmaschinen)</div> <div class="NavContent" style="text-align:left"> <table class="toptextcells"> <tbody><tr> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>1951</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1952</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1953</td> <td>02 </td></tr> <tr> <td>1954</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1955</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1956</td> <td>14 </td></tr> <tr> <td>1957</td> <td>06 </td></tr> <tr> <td>1958</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1959</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1960</td> <td data-sort-value="0">0– </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">22 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>1961</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1962</td> <td>02 </td></tr> <tr> <td>1963</td> <td>11 </td></tr> <tr> <td>1964</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1965</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1966</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1967</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1968</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1969</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1970</td> <td>02 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">15 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>1971</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1972</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1973</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1974</td> <td>02 </td></tr> <tr> <td>1975</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1976</td> <td>02 </td></tr> <tr> <td>1977</td> <td>04 </td></tr> <tr> <td>1978</td> <td>02 </td></tr> <tr> <td>1979</td> <td>13 </td></tr> <tr> <td>1980</td> <td>09 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">32 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>1981</td> <td>03 </td></tr> <tr> <td>1982</td> <td>02 </td></tr> <tr> <td>1983</td> <td>02 </td></tr> <tr> <td>1984</td> <td>07 </td></tr> <tr> <td>1985</td> <td>02 </td></tr> <tr> <td>1986</td> <td>12 </td></tr> <tr> <td>1987</td> <td>05 </td></tr> <tr> <td>1988</td> <td>04 </td></tr> <tr> <td>1989</td> <td data-sort-value="0">0– </td></tr> <tr> <td>1990</td> <td>08 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">45 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>1991</td> <td>12 </td></tr> <tr> <td>1992</td> <td>10 </td></tr> <tr> <td>1993</td> <td>10 </td></tr> <tr> <td>1994</td> <td>01 </td></tr> <tr> <td>1995</td> <td>08 </td></tr> <tr> <td>1996</td> <td>07 </td></tr> <tr> <td>1997</td> <td>04 </td></tr> <tr> <td>1998</td> <td>08 </td></tr> <tr> <td>1999</td> <td>09 </td></tr> <tr> <td>2000</td> <td>13 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">82 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>2001</td> <td>22 </td></tr> <tr> <td>2002</td> <td>08 </td></tr> <tr> <td>2003</td> <td>08 </td></tr> <tr> <td>2004</td> <td>02 </td></tr> <tr> <td>2005</td> <td>07 </td></tr> <tr> <td>2006</td> <td>01 </td></tr> <tr> <td>2007</td> <td>04 </td></tr> <tr> <td>2008</td> <td>06 </td></tr> <tr> <td>2009</td> <td>06 </td></tr> <tr> <td>2010</td> <td>07 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">71 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>2011</td> <td>09 </td></tr> <tr> <td>2012</td> <td>16 </td></tr> <tr> <td>2013</td> <td>07 </td></tr> <tr> <td>2014</td> <td>07 </td></tr> <tr> <td>2015</td> <td>06 </td></tr> <tr> <td>2016</td> <td>07 </td></tr> <tr> <td>2017</td> <td>05 </td></tr> <tr> <td>2018</td> <td>07 </td></tr> <tr> <td>2019</td> <td>03 </td></tr> <tr> <td>2020</td> <td>05 </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">72 </td></tr></tbody></table> </td> <td> <table class="wikitable sortable" style="text-align:center"> <tbody><tr class="hintergrundfarbe6"> <th>Jahr </th> <th>Teiler </th></tr> <tr> <td>2021</td> <td>05 </td></tr> <tr> <td>2022</td> <td data-sort-value="0">0– </td></tr> <tr> <td>2023</td> <td>08 </td></tr> <tr> <td>2024</td> <td>02 </td></tr> <tr> <td>2025</td> <td> </td></tr> <tr> <td>2026</td> <td> </td></tr> <tr> <td>2027</td> <td> </td></tr> <tr> <td>2028</td> <td> </td></tr> <tr> <td>2029</td> <td> </td></tr> <tr> <td>2030</td> <td> </td></tr> <tr class="sortbottom"> <td style="text-align:center;" bgcolor="#CCCCFF">TOTAL</td> <td bgcolor="#CCCCFF">15 </td></tr></tbody></table> </td></tr></tbody></table> Es wurden somit bisher 354 Primfaktoren von Fermat-Zahlen mit Computern gefunden (Stand: 6. Juli 2024). </div> </div> <div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=4" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <ul><li>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"> hat jeder <a href="/wiki/Teilbarkeit" title="Teilbarkeit">Teiler</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{n+2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{n+2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcf94c18df6cf27fa7453f61d554482e72adba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.375ex; height:2.843ex;" alt="{\displaystyle k\cdot 2^{n+2}+1}"> (bewiesen von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> und <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Lucas</a>, siehe auch Artikel Quadratisches Reziprozitätsgesetz, Unterabschnitt <a href="/wiki/Quadratisches_Reziprozit%C3%A4tsgesetz#Teiler_von_Fermat-_und_Mersenne-Zahlen" title="Quadratisches Reziprozitätsgesetz"> Teiler von Fermat- und Mersenne-Zahlen</a>).</li></ul> <dl><dd>Beispiele: <dl><dd>Der Teiler 641 von F5: 641 = 5 · 27 + 1 = 5 · 128 + 1</dd> <dd>Der Teiler 6700417 von F5: 6700417 = 52347 · 27 + 1 = 52347 · 128 + 1</dd></dl></dd></dl> <ul><li>Fermat-Zahlen lassen sich auf folgende Arten <a href="/wiki/Rekursion" title="Rekursion">rekursiv</a> berechnen:</li></ul> <dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ff1c52a7a38d3afcce05117270671017d1b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.494ex; height:3.176ex;" alt="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}">  für  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69207b4ff322fc6cfb07d8b4c91d07a828278f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.703ex; height:3.343ex;" alt="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}">  für  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd82a341aa74ebed6b30fd9a2469f89ced6f024f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.987ex; height:3.509ex;" alt="{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}">  für  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93bc33460189090087c28972cd1955a76885d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}">  für  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></li></ul></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der vier Behauptungen:</div> <div class="NavContent" style="text-align:left"> Zwei der vier Beweise funktionieren mittels <a href="/wiki/Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">vollständiger Induktion</a>. Man zeigt, dass die Behauptungen für den Anfang gelten (Induktionsanfang), nimmt an, dass die Behauptung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> gilt (Induktionsvoraussetzung) und beweist, dass die Behauptung dadurch auch für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> gelten muss (Induktionsschluss). Beweis der ersten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ff1c52a7a38d3afcce05117270671017d1b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.494ex; height:3.176ex;" alt="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1=2^{2\cdot 2^{n-1}}+1=(2^{2^{n-1}})^{2}+1=({\color {red}2^{2^{n-1}}+1}-1)^{2}+1=({\color {red}F_{n-1}}-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1=2^{2\cdot 2^{n-1}}+1=(2^{2^{n-1}})^{2}+1=({\color {red}2^{2^{n-1}}+1}-1)^{2}+1=({\color {red}F_{n-1}}-1)^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e525d972b32fa81f77a5194aba07291c354a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.754ex; height:3.509ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1=2^{2\cdot 2^{n-1}}+1=(2^{2^{n-1}})^{2}+1=({\color {red}2^{2^{n-1}}+1}-1)^{2}+1=({\color {red}F_{n-1}}-1)^{2}+1}">, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl></dd></dl> Beweis der zweiten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69207b4ff322fc6cfb07d8b4c91d07a828278f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.703ex; height:3.343ex;" alt="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> <dl><dd>Der Beweis funktioniert mittels vollständiger Induktion.</dd> <dd>Induktionsanfang: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17=F_{2}=F_{1}+2^{2^{2-1}}\cdot F_{0}=5+2^{2^{1}}\cdot 3=5+2^{2}\cdot 3=5+4\cdot 3=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mn>5</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17=F_{2}=F_{1}+2^{2^{2-1}}\cdot F_{0}=5+2^{2^{1}}\cdot 3=5+2^{2}\cdot 3=5+4\cdot 3=17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec48b7930fce5081b4283bc65a7da9aaf8ae523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:67.121ex; height:3.343ex;" alt="{\displaystyle 17=F_{2}=F_{1}+2^{2^{2-1}}\cdot F_{0}=5+2^{2^{1}}\cdot 3=5+2^{2}\cdot 3=5+4\cdot 3=17}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 257=F_{3}=F_{2}+2^{2^{3-1}}\cdot F_{0}\cdot F_{1}=17+2^{4}\cdot 3\cdot 5=17+16\cdot 3\cdot 5=17+240=257}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>257</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>17</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>=</mo> <mn>17</mn> <mo>+</mo> <mn>16</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>=</mo> <mn>17</mn> <mo>+</mo> <mn>240</mn> <mo>=</mo> <mn>257</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 257=F_{3}=F_{2}+2^{2^{3-1}}\cdot F_{0}\cdot F_{1}=17+2^{4}\cdot 3\cdot 5=17+16\cdot 3\cdot 5=17+240=257}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8906b031ddc6fc9d171c57fb729ada840cdfc9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:81.604ex; height:3.343ex;" alt="{\displaystyle 257=F_{3}=F_{2}+2^{2^{3-1}}\cdot F_{0}\cdot F_{1}=17+2^{4}\cdot 3\cdot 5=17+16\cdot 3\cdot 5=17+240=257}"></dd></dl></dd> <dd>Induktionsvoraussetzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69207b4ff322fc6cfb07d8b4c91d07a828278f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.703ex; height:3.343ex;" alt="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}}"> bzw. umgeformt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}={\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}={\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a4c172868fb647b8c0951cb4efacb7c85ea9cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.973ex; height:6.176ex;" alt="{\displaystyle F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-2}={\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}"></dd> <dd>Induktionsschluss: zu zeigen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}=F_{n}+2^{2^{n}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}=F_{n}+2^{2^{n}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2ac8e133842cd665c89d33be1b28f7f1c6cbd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.998ex; height:3.009ex;" alt="{\displaystyle F_{n+1}=F_{n}+2^{2^{n}}\cdot F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})^{2}-2\cdot ({\color {magenta}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{n-1}^{2}+2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}F_{n-1}+(2^{2^{n-1}})^{2}F_{0}^{2}F_{1}^{2}\ldots F_{n-2}^{2}-2F_{n-1}-2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}+2\\&=&(F_{n-1}^{2}-2F_{n-1}+1)+1+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}\cdot (2F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}-2)\\&=&{\color {magenta}(F_{n-1}-1)^{2}+1}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}F_{0}F_{1}\ldots F_{n-2}}-{\frac {2}{2^{2^{n-1}}}})\\&=&{\color {magenta}F_{n}}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}{\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}-{\frac {2}{2^{2^{n-1}}}})&{\text{(umgeformte Induktionsvoraussetzung)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2F_{n-1}+F_{n}-F_{n-1}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}F_{n-1}}+{\color {magenta}F_{n}}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}2^{2^{n-1}}+1}+{\color {magenta}2^{2^{n}}+1}-2}{2^{2^{n-1}}}}&{\text{(Definition von Fermat-Zahlen)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}+2^{2^{n}}}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}\cdot (1+2^{2^{n-1}})}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\color {red}2^{2^{n-1}}+1})\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\color {red}F_{n-1}}&{\text{(Definition von Fermat-Zahlen)}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(erste obige Behauptung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Induktionsvoraussetzung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(umgeformte Induktionsvoraussetzung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Definition von Fermat-Zahlen)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Definition von Fermat-Zahlen)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})^{2}-2\cdot ({\color {magenta}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{n-1}^{2}+2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}F_{n-1}+(2^{2^{n-1}})^{2}F_{0}^{2}F_{1}^{2}\ldots F_{n-2}^{2}-2F_{n-1}-2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}+2\\&=&(F_{n-1}^{2}-2F_{n-1}+1)+1+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}\cdot (2F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}-2)\\&=&{\color {magenta}(F_{n-1}-1)^{2}+1}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}F_{0}F_{1}\ldots F_{n-2}}-{\frac {2}{2^{2^{n-1}}}})\\&=&{\color {magenta}F_{n}}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}{\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}-{\frac {2}{2^{2^{n-1}}}})&{\text{(umgeformte Induktionsvoraussetzung)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2F_{n-1}+F_{n}-F_{n-1}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}F_{n-1}}+{\color {magenta}F_{n}}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}2^{2^{n-1}}+1}+{\color {magenta}2^{2^{n}}+1}-2}{2^{2^{n-1}}}}&{\text{(Definition von Fermat-Zahlen)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}+2^{2^{n}}}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}\cdot (1+2^{2^{n-1}})}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\color {red}2^{2^{n-1}}+1})\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\color {red}F_{n-1}}&{\text{(Definition von Fermat-Zahlen)}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024b224093618144ad629febb8a6959c3449de2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -31.671ex; width:149.028ex; height:64.509ex;" alt="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})^{2}-2\cdot ({\color {magenta}F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{n-1}^{2}+2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}F_{n-1}+(2^{2^{n-1}})^{2}F_{0}^{2}F_{1}^{2}\ldots F_{n-2}^{2}-2F_{n-1}-2\cdot 2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}+2\\&=&(F_{n-1}^{2}-2F_{n-1}+1)+1+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}\cdot (2F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\ldots F_{n-2}-2)\\&=&{\color {magenta}(F_{n-1}-1)^{2}+1}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}F_{0}F_{1}\ldots F_{n-2}}-{\frac {2}{2^{2^{n-1}}}})\\&=&{\color {magenta}F_{n}}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\frac {2}{2^{2^{n-1}}}}F_{n-1}+{\color {red}{\frac {F_{n}-F_{n-1}}{2^{2^{n-1}}}}}-{\frac {2}{2^{2^{n-1}}}})&{\text{(umgeformte Induktionsvoraussetzung)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2F_{n-1}+F_{n}-F_{n-1}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}F_{n-1}}+{\color {magenta}F_{n}}-2}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {{\color {red}2^{2^{n-1}}+1}+{\color {magenta}2^{2^{n}}+1}-2}{2^{2^{n-1}}}}&{\text{(Definition von Fermat-Zahlen)}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}+2^{2^{n}}}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\frac {2^{2^{n-1}}\cdot (1+2^{2^{n-1}})}{2^{2^{n-1}}}}\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot ({\color {red}2^{2^{n-1}}+1})\\&=&F_{n}+2^{2^{n}}F_{0}F_{1}\ldots F_{n-2}\cdot {\color {red}F_{n-1}}&{\text{(Definition von Fermat-Zahlen)}}\end{array}}}"></dd></dl></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der dritten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd82a341aa74ebed6b30fd9a2469f89ced6f024f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.987ex; height:3.509ex;" alt="{\displaystyle F_{n}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}F_{n}=2^{2^{n}}+1&=&2^{2\cdot 2^{n-1}}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2^{n-1}}\\&=&(2^{2^{n-1}})^{2}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2\cdot 2^{n-2}}\\&=&({\color {red}2^{2^{n-1}}+1})^{2}-2\cdot (2^{2^{n-2}})^{2}\\&=&{\color {red}F_{n-1}}^{2}-2\cdot (2^{2^{n-2}}+1-1)^{2}\\&=&F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}F_{n}=2^{2^{n}}+1&=&2^{2\cdot 2^{n-1}}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2^{n-1}}\\&=&(2^{2^{n-1}})^{2}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2\cdot 2^{n-2}}\\&=&({\color {red}2^{2^{n-1}}+1})^{2}-2\cdot (2^{2^{n-2}})^{2}\\&=&{\color {red}F_{n-1}}^{2}-2\cdot (2^{2^{n-2}}+1-1)^{2}\\&=&F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef376d4a6bb977bfe26696bdf681ecd5c998d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:54.371ex; height:18.843ex;" alt="{\displaystyle {\begin{array}{rcll}F_{n}=2^{2^{n}}+1&=&2^{2\cdot 2^{n-1}}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2^{n-1}}\\&=&(2^{2^{n-1}})^{2}+2\cdot 2^{2^{n-1}}+1-2\cdot 2^{2\cdot 2^{n-2}}\\&=&({\color {red}2^{2^{n-1}}+1})^{2}-2\cdot (2^{2^{n-2}})^{2}\\&=&{\color {red}F_{n-1}}^{2}-2\cdot (2^{2^{n-2}}+1-1)^{2}\\&=&F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}\end{array}}}"></dd></dl></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der vierten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93bc33460189090087c28972cd1955a76885d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> <dl><dd>Der Beweis funktioniert mittels vollständiger Induktion.</dd> <dd>Induktionsanfang: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=F_{1}=F_{0}+2=3+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=F_{1}=F_{0}+2=3+2=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afad7f03dded32836cd0cfb945845f21acf1ea16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.985ex; height:2.509ex;" alt="{\displaystyle 5=F_{1}=F_{0}+2=3+2=5}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17=F_{2}=F_{0}\cdot F_{1}+2=3\cdot 5+2=15+2=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>15</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17=F_{2}=F_{0}\cdot F_{1}+2=3\cdot 5+2=15+2=17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f55c6e29a7699c01151ab1147c02e88fc00206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.805ex; height:2.509ex;" alt="{\displaystyle 17=F_{2}=F_{0}\cdot F_{1}+2=3\cdot 5+2=15+2=17}"></dd></dl></dd> <dd>Induktionsvoraussetzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93bc33460189090087c28972cd1955a76885d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"></dd> <dd>Induktionsschluss: zu zeigen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15f4c8614ecfdf9c9bf92444dd7bd041e856781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n+1}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n}+2}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})^{2}-2({\color {magenta}F_{0}F_{1}\ldots F_{n-1}+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+4F_{0}F_{1}\ldots F_{n-1}+4-2F_{0}F_{1}\ldots F_{n-1}-4+2\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+2F_{0}F_{1}\ldots F_{n-1}+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot ({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot {\color {red}F_{n}}+2&{\text{(Induktionsvoraussetzung)}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(erste obige Behauptung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Induktionsvoraussetzung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Induktionsvoraussetzung)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})^{2}-2({\color {magenta}F_{0}F_{1}\ldots F_{n-1}+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+4F_{0}F_{1}\ldots F_{n-1}+4-2F_{0}F_{1}\ldots F_{n-1}-4+2\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+2F_{0}F_{1}\ldots F_{n-1}+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot ({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot {\color {red}F_{n}}+2&{\text{(Induktionsvoraussetzung)}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793c033d7eb01ff1b74f0d274d1c0fcd593cb122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:102.204ex; height:24.009ex;" alt="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste obige Behauptung)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})^{2}-2({\color {magenta}F_{0}F_{1}\ldots F_{n-1}+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+4F_{0}F_{1}\ldots F_{n-1}+4-2F_{0}F_{1}\ldots F_{n-1}-4+2\\&=&F_{0}^{2}F_{1}^{2}\ldots F_{n-1}^{2}+2F_{0}F_{1}\ldots F_{n-1}+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot ({\color {red}F_{0}F_{1}\ldots F_{n-1}+2})+2\\&=&F_{0}F_{1}\ldots F_{n-1}\cdot {\color {red}F_{n}}+2&{\text{(Induktionsvoraussetzung)}}\end{array}}}"></dd></dl></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li> Es gelten folgende Darstellungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}">:</li></ul> <dl><dd><ul><li>Jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> ist von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6m-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2024a283053e9edb75b758430348a5b827f0324b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 6m-1}">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> positiv ganzzahlig ist. (mit anderen Worten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv -1{\pmod {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv -1{\pmod {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030c6ceb74c212f71ca08b011c5063a5d84fd630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.629ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv -1{\pmod {6}}}">)<a href="#cite_note-pseudoprim2-15">[15]</a></li> <li>Jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> ist von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4m+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/979236eb049f13fa685f313ec9105ff33e8ebe61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 4m+1}">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> positiv ganzzahlig ist. (mit anderen Worten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 1{\pmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 1{\pmod {4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5cf3351c0b0359bdf4fcd5fc581bf42f483a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 1{\pmod {4}}}">)</li> <li>Jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> ist von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3m+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3m+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b4e06c93b929806a2fccfbd5d1b1be9e493cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 3m+2}">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> positiv ganzzahlig ist. (mit anderen Worten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb3277760e955b2a76650f843e98faa3e481f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {3}}}">)</li> <li>Jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> ist von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10m+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>m</mi> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10m+7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42cfaf2d36ae5851b6bc05abef928c4b72397f97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.368ex; height:2.343ex;" alt="{\displaystyle 10m+7}">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> positiv ganzzahlig ist. (mit anderen Worten: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 7{\pmod {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48010e145122c25af71a6eacc26d554022199dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.983ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 7{\pmod {10}}}">)</li></ul> <dl><dd>Anders formuliert: Mit Ausnahme von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf3edb177c042d1d7138b914b2826b872bc242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=3}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aed80ee739484183d13636183bfcacf82e23974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=5}"> endet jede Fermat-Zahl im <a href="/wiki/Dezimalsystem" title="Dezimalsystem">Dezimalsystem</a> mit der Ziffer 7. Die letzten beiden Ziffern sind 17, 37, 57 oder 97.<a href="#cite_note-vorRemark3.7-16">[16]</a></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der vier Behauptungen:</div> <div class="NavContent" style="text-align:left"> Beweis der ersten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv -1{\pmod {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv -1{\pmod {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030c6ceb74c212f71ca08b011c5063a5d84fd630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.629ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv -1{\pmod {6}}}"> <dl><dd><dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. Man startet mit einer bekannten richtigen Aussage und beweist das Gewünschte.</dd> <dd>Eine weiter oben angegebene Eigenschaft besagt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93bc33460189090087c28972cd1955a76885d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}">. Somit gilt aber, weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf3edb177c042d1d7138b914b2826b872bc242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=3}"> ist: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}+1=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot (F_{1}\cdot \ldots \cdot F_{n-1}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}+1=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot (F_{1}\cdot \ldots \cdot F_{n-1}+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38bff4ec44224cd3e559bc1f927541df0492e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:80.072ex; height:2.843ex;" alt="{\displaystyle F_{n}+1=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot F_{1}\cdot \ldots \cdot F_{n-1}+3=3\cdot (F_{1}\cdot \ldots \cdot F_{n-1}+1)}">.</dd></dl></dd> <dd>Der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}\cdot \ldots \cdot F_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}\cdot \ldots \cdot F_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed8caa2bf3c5c7f7fd7274b614a72f5f1a3cf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.444ex; height:2.509ex;" alt="{\displaystyle F_{1}\cdot \ldots \cdot F_{n-1}}"> ist als Produkt von ungeraden Fermat-Zahlen selber ungerade. Addiert man 1 dazu, erhält man eine gerade Zahl. Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee094df4785180804ecac93381f8870285de3494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.716ex; height:2.509ex;" alt="{\displaystyle F_{n}+1}"> ein Produkt aus 3 und einer geraden Zahl und somit durch 6 teilbar. Es gibt also ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}+1=6m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>6</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}+1=6m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2caf7141c86019d1d4beb3be01b3d91b7ffd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.017ex; height:2.509ex;" alt="{\displaystyle F_{n}+1=6m}">. Daher ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6m-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2024a283053e9edb75b758430348a5b827f0324b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 6m-1}">, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl></dd></dl> Beweis der zweiten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 1{\pmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 1{\pmod {4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5cf3351c0b0359bdf4fcd5fc581bf42f483a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 1{\pmod {4}}}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1=(2^{2})^{\frac {2^{n}}{2}}+1=4^{2^{n-1}}+1\equiv 1{\pmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1=(2^{2})^{\frac {2^{n}}{2}}+1=4^{2^{n-1}}+1\equiv 1{\pmod {4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac3fa358b1c7e9d9161c5bf20ecdbf9a7234b6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.889ex; height:4.176ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1=(2^{2})^{\frac {2^{n}}{2}}+1=4^{2^{n-1}}+1\equiv 1{\pmod {4}}}">, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der dritten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb3277760e955b2a76650f843e98faa3e481f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {3}}}"> <dl><dd><dl><dd>Der dritte Beweis funktioniert mit <a href="/wiki/Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">vollständiger Induktion</a>:</dd> <dd>Induktionsanfang: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=2^{2^{1}}+1=5=3\cdot 1+2\equiv 2{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=2^{2^{1}}+1=5=3\cdot 1+2\equiv 2{\pmod {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2590319711752e359815a9493e70bf4822881a90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.172ex; height:3.509ex;" alt="{\displaystyle F_{1}=2^{2^{1}}+1=5=3\cdot 1+2\equiv 2{\pmod {3}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=2^{2^{2}}+1=17=3\cdot 5+2\equiv 2{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>17</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}=2^{2^{2}}+1=17=3\cdot 5+2\equiv 2{\pmod {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba8d4b221df94bd16704f568330ab8088115192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.335ex; height:3.509ex;" alt="{\displaystyle F_{2}=2^{2^{2}}+1=17=3\cdot 5+2\equiv 2{\pmod {3}}}"></dd></dl></dd> <dd>Induktionsvoraussetzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=3\cdot k+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=3\cdot k+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894083a0d0a6ea977a2058bd77132590849d8149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.867ex; height:2.509ex;" alt="{\displaystyle F_{n}=3\cdot k+2}"></dd> <dd>Induktionsschluss: zu zeigen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}=3\cdot m+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}=3\cdot m+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa42b709a1925645c6f6d8f07cfb17883879146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.797ex; height:2.509ex;" alt="{\displaystyle F_{n+1}=3\cdot m+2}"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste Behauptung weiter oben)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}3k+2})^{2}-2\cdot ({\color {magenta}3k+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&9k^{2}+12k+4-6k-4+2\\&=&9k^{2}+6k+2\\&=&3\cdot (3k^{2}+2k)+2\\&=&3m+2&{\text{(mit }}m:=3k^{2}+2k{\text{)}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(erste Behauptung weiter oben)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="magenta"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Induktionsvoraussetzung)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>9</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>6</mn> <mi>k</mi> <mo>−</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>9</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(mit </mtext> </mrow> <mi>m</mi> <mo>:=</mo> <mn>3</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste Behauptung weiter oben)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}3k+2})^{2}-2\cdot ({\color {magenta}3k+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&9k^{2}+12k+4-6k-4+2\\&=&9k^{2}+6k+2\\&=&3\cdot (3k^{2}+2k)+2\\&=&3m+2&{\text{(mit }}m:=3k^{2}+2k{\text{)}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f1486d5282adb16fab0159e266efdf246fde4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.906ex; margin-bottom: -0.266ex; width:73.156ex; height:23.509ex;" alt="{\displaystyle {\begin{array}{rcll}F_{n+1}&=&(F_{n}-1)^{2}+1&{\text{(erste Behauptung weiter oben)}}\\&=&{\color {red}F_{n}}^{2}-2{\color {magenta}F_{n}}+1+1\\&=&({\color {red}3k+2})^{2}-2\cdot ({\color {magenta}3k+2})+2&{\text{(Induktionsvoraussetzung)}}\\&=&9k^{2}+12k+4-6k-4+2\\&=&9k^{2}+6k+2\\&=&3\cdot (3k^{2}+2k)+2\\&=&3m+2&{\text{(mit }}m:=3k^{2}+2k{\text{)}}\end{array}}}"></dd></dl></dd></dl></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der vierten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 7{\pmod {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48010e145122c25af71a6eacc26d554022199dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.983ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> <dl><dd><dl><dd>Der vierte Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>:</dd> <dd>Weiter oben wurde gezeigt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93bc33460189090087c28972cd1955a76885d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdot F_{1}\cdot \ldots \cdot F_{n-1}+2}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> gilt. Daraus kann man folgern, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {F_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {F_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5147c26916947f7a1efc94bb7d54340b46ffdc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.241ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {F_{k}}}}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,\ldots ,n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff2663852067d440427721aad0c33b9b6e4378d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.244ex; height:2.509ex;" alt="{\displaystyle k=0,1,\ldots ,n-1}"> gilt. Im Speziellen gilt also für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"> (also für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aed80ee739484183d13636183bfcacf82e23974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=5}">) die Kongruenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b761cf7c87cc00651fd3b2239df81a7956aee1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {5}}}"> und somit entweder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d150d889be0e9eb7cf290535217e6a5b1673891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.983ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {10}}}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 7{\pmod {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48010e145122c25af71a6eacc26d554022199dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.983ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 7{\pmod {10}}}">. Weil aber Fermat-Zahlen immer ungerade sind, kann nur die Kongruenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 7{\pmod {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48010e145122c25af71a6eacc26d554022199dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.983ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 7{\pmod {10}}}"> zutreffen, was zu zeigen war.</dd> <dd>Die Aussage, dass die letzten beiden Ziffern 17, 37, 57 oder 97 sind, kann man der Literatur<a href="#cite_note-vorRemark3.7-16">[16]</a> entnehmen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl></dd></dl> </div> </div> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-te Fermat-Zahl. Dann gilt:</li></ul> <dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> hat unendlich viele Darstellungen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-2y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-2y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1d3ab812070fd4f0c141fd518495c5f57bc97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.413ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-2y^{2}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95f98f3b1dfb08f89299edf4793e17e83fd6ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {N} }"> positiv ganzzahlig, für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"><a href="#cite_note-Proposition3.4-17">[17]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> hat mindestens eine Darstellung der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77960c30f41a5a06c673d6f29f5915c4d75a3af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.251ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-y^{2}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95f98f3b1dfb08f89299edf4793e17e83fd6ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {N} }"> positiv ganzzahlig. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> <a href="/wiki/Zusammengesetzte_Zahl" title="Zusammengesetzte Zahl">zusammengesetzt</a>, gibt es mehrere Möglichkeiten dieser Darstellung.<a href="#cite_note-Remark3.13-18">[18]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> kann niemals als Summe von zwei Primzahlen dargestellt werden, für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a1e8ec3430c6dbdd7a5bf0faff8ad4c6906769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.948ex; height:2.343ex;" alt="{\displaystyle n\geq 2:}"><a href="#cite_note-Primzahlsumme-19">[19]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\not =p_{1}+p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\not =p_{1}+p_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc900b01d83bba780f69b450100197dc8056b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.099ex; height:2.676ex;" alt="{\displaystyle F_{n}\not =p_{1}+p_{2}}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2}\in \mathbb {P} ,n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2}\in \mathbb {P} ,n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299e06562eef0457a1078091b7ac98f9a9e96c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:16.521ex; height:2.509ex;" alt="{\displaystyle p_{1},p_{2}\in \mathbb {P} ,n\geq 2}"></dd></dl></dd></dl> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> kann niemals als Differenz von zwei p-ten Potenzen geschrieben werden, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> und p ungerade Primzahlen sind:<a href="#cite_note-pseudoprim3-20">[20]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\not =a^{p}-b^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\not =a^{p}-b^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d2a05e53cf422514e58f8e38a5c3e65ce286d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.998ex; height:2.843ex;" alt="{\displaystyle F_{n}\not =a^{p}-b^{p}}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} ,\,p\not =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} ,\,p\not =2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5c9bbdf4eedcb249a8d51e035d66068786db6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.371ex; height:2.676ex;" alt="{\displaystyle p\in \mathbb {P} ,\,p\not =2}"></dd></dl></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der vier Behauptungen:</div> <div class="NavContent" style="text-align:left"> Beweis der ersten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-2y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-2y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1d3ab812070fd4f0c141fd518495c5f57bc97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.413ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-2y^{2}}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>.</dd> <dd>Die Existenz einer solchen Darstellung konnte schon weiter oben mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=F_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=F_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d844225f544ca94c0617dc45fc22c0b342c63a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.889ex; height:2.509ex;" alt="{\displaystyle x:=F_{n-1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y:=F_{n-2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>:=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y:=F_{n-2}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365ae6ee389de7d931d21e1347db4cfcf7f4f283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.717ex; height:2.509ex;" alt="{\displaystyle y:=F_{n-2}-1}"> gezeigt werden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-2y^{2}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-2y^{2}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91647648c82bb8496e2275510df559a895f10c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.687ex; height:3.509ex;" alt="{\displaystyle F_{n}=x^{2}-2y^{2}=F_{n-1}^{2}-2\cdot (F_{n-2}-1)^{2}}"></dd> <dd>Um unendlich viele solche Darstellungen zu erhalten, betrachte man folgende <a href="/wiki/Identit%C3%A4tsgleichung" title="Identitätsgleichung">Identität</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}(3x+4y)^{2}-2\cdot (2x+3y)^{2}&=&9x^{2}+24xy+16y^{2}-2\cdot (4x^{2}+12xy+9y^{2})\\&=&9x^{2}+24xy+16y^{2}-8x^{2}-24xy-18y^{2}\\&=&x^{2}-2y^{2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>16</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>9</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>16</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>24</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>18</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}(3x+4y)^{2}-2\cdot (2x+3y)^{2}&=&9x^{2}+24xy+16y^{2}-2\cdot (4x^{2}+12xy+9y^{2})\\&=&9x^{2}+24xy+16y^{2}-8x^{2}-24xy-18y^{2}\\&=&x^{2}-2y^{2}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d469a7ae61093e134da7e692215d227a8779d031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.026ex; margin-bottom: -0.312ex; width:77.397ex; height:9.843ex;" alt="{\displaystyle {\begin{array}{rcl}(3x+4y)^{2}-2\cdot (2x+3y)^{2}&=&9x^{2}+24xy+16y^{2}-2\cdot (4x^{2}+12xy+9y^{2})\\&=&9x^{2}+24xy+16y^{2}-8x^{2}-24xy-18y^{2}\\&=&x^{2}-2y^{2}\end{array}}}"></dd></dl></dd> <dd>Weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95f98f3b1dfb08f89299edf4793e17e83fd6ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {N} }"> ist, gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+4y>x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>></mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+4y>x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7966a1106cee4e98cb9563c214b2ffc03b5634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.079ex; height:2.509ex;" alt="{\displaystyle 3x+4y>x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+3y>y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>></mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+3y>y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6583e5a9197e6ed2ecb7a99fbd2ba14f34784489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.904ex; height:2.509ex;" alt="{\displaystyle 2x+3y>y}">. Somit kann man aus dem Darstellungspaar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> ein (größeres) Darstellungspaar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3x+4y,2x+3y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>,</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3x+4y,2x+3y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1f2dd434dfd5afd1893ce3ca9b4a6a27ee22cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.144ex; height:2.843ex;" alt="{\displaystyle (3x+4y,2x+3y)}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> konstruieren. Aus diesem kann man mit obiger Identität das nächste (größere) Darstellungspaar für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> konstruieren und so fort. Man erhält also unendlich viele Darstellungspaare für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> und somit auch unendlich viele Darstellungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-2y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-2y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1d3ab812070fd4f0c141fd518495c5f57bc97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.413ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-2y^{2}}">, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der zweiten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77960c30f41a5a06c673d6f29f5915c4d75a3af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.251ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-y^{2}}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>.</dd> <dd>Es gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1=2^{2^{n}}+2^{2^{n}}+1-2^{2^{n}}=2^{2^{n}}+2\cdot 2^{2^{n}-1}+1-2^{2^{n}}=\left(2^{2^{n}-1}+1\right)^{2}-\left(2^{2^{n}-1}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1=2^{2^{n}}+2^{2^{n}}+1-2^{2^{n}}=2^{2^{n}}+2\cdot 2^{2^{n}-1}+1-2^{2^{n}}=\left(2^{2^{n}-1}+1\right)^{2}-\left(2^{2^{n}-1}\right)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ccaae547e366be414ec04ce95087ac7e42d26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:90.757ex; height:5.176ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1=2^{2^{n}}+2^{2^{n}}+1-2^{2^{n}}=2^{2^{n}}+2\cdot 2^{2^{n}-1}+1-2^{2^{n}}=\left(2^{2^{n}-1}+1\right)^{2}-\left(2^{2^{n}-1}\right)^{2}}"></dd></dl></dd> <dd>Somit hat man zwei Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=2^{2^{n}-1}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=2^{2^{n}-1}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d889c2e1a2d2ec20a561f5af5a2a9d01f8606f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.36ex; height:2.843ex;" alt="{\displaystyle x:=2^{2^{n}-1}+1}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y:=2^{2^{n}-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y:=2^{2^{n}-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e63923ac58b21131e6279a43ceeda9e7a5e3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.183ex; height:3.009ex;" alt="{\displaystyle y:=2^{2^{n}-1}}"> gefunden, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=x^{2}-y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=x^{2}-y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77960c30f41a5a06c673d6f29f5915c4d75a3af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.251ex; height:3.009ex;" alt="{\displaystyle F_{n}=x^{2}-y^{2}}">, also die Differenz von zwei Quadratzahlen, ist, was zu zeigen war.</dd> <dd>Die Aussage, dass es mehrere solche Darstellungsmöglichkeiten als Differenz von zwei Quadratzahlen gibt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> zusammengesetzt ist, kann man der Literatur<a href="#cite_note-Remark3.13-18">[18]</a> entnehmen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der dritten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\not =p_{1}+p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\not =p_{1}+p_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc900b01d83bba780f69b450100197dc8056b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.099ex; height:2.676ex;" alt="{\displaystyle F_{n}\not =p_{1}+p_{2}}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">indirekt</a>. Man startet mit einer Behauptung und zeigt, dass sie falsch ist, womit die Behauptung fallengelassen werden muss und das Gegenteil gilt.</dd> <dd>Alle Fermat-Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"> sind als Summe einer geraden und einer ungeraden Zahl 1 immer ungerade Zahlen. Primzahlen sind, bis auf die erste Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}">, immer ungerade. Wenn also die ungerade Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> Summe von zwei Primzahlen sein soll, so dürfen nicht beide Primzahlen ungerade sein, weil die Summe zweier ungerader Zahlen eine gerade Zahl ergibt. Eine davon muss gerade sein. Weil es nur eine gerade Primzahl gibt, muss also 2 eine der beiden Summanden sein. Der andere prime Summand ist somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315b07c26f84166a7e93b061768e169359b3e211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.716ex; height:2.509ex;" alt="{\displaystyle F_{n}-2}"> und es gilt trivialerweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2+(F_{n}-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2+(F_{n}-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a0a60f4f820c80f2ab7588c92ed3e99340d5a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.34ex; height:2.843ex;" alt="{\displaystyle F_{n}=2+(F_{n}-2)}">. Es gilt aber: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-2=2^{2^{n}}+1-2=2^{2^{n}}-1=(2^{2^{n-1}}-1)\cdot (2^{2^{n-1}}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-2=2^{2^{n}}+1-2=2^{2^{n}}-1=(2^{2^{n-1}}-1)\cdot (2^{2^{n-1}}+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2476f01eddb6898c0ceb6187d1ad3fe90a5c808e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.46ex; height:3.509ex;" alt="{\displaystyle F_{n}-2=2^{2^{n}}+1-2=2^{2^{n}}-1=(2^{2^{n-1}}-1)\cdot (2^{2^{n-1}}+1)}"></dd></dl></dd> <dd>Somit ist aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315b07c26f84166a7e93b061768e169359b3e211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.716ex; height:2.509ex;" alt="{\displaystyle F_{n}-2}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> zusammengesetzt und keine Primzahl, weil sogar der kleinere der beiden Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n-1}}-1\geq 2^{2^{2-1}}-1=2^{2}-1=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>≥</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n-1}}-1\geq 2^{2^{2-1}}-1=2^{2}-1=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8386623fa149c9da144ef897426173f129b8933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.324ex; height:3.176ex;" alt="{\displaystyle 2^{2^{n-1}}-1\geq 2^{2^{2-1}}-1=2^{2}-1=3}"> ist und somit eine nichttriviale Faktorisierung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315b07c26f84166a7e93b061768e169359b3e211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.716ex; height:2.509ex;" alt="{\displaystyle F_{n}-2}"> existiert. Wir erhalten einen Widerspruch. Die Annahme, dass man eine Fermat-Zahl als Summe zweier Primzahlen darstellen kann, muss fallengelassen werden, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> Beweis der vierten Behauptung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\not =a^{p}-b^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\not =a^{p}-b^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d2a05e53cf422514e58f8e38a5c3e65ce286d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.998ex; height:2.843ex;" alt="{\displaystyle F_{n}\not =a^{p}-b^{p}}"> <dl><dd>Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">indirekt</a>. Man startet mit einer Behauptung und zeigt, dass sie falsch ist, womit die Behauptung fallengelassen werden muss und das Gegenteil gilt.</dd> <dd>Angenommen, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7891a37dcb8b0ee507f9ef2038a853e245d76657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }"> ist eine ungerade Primzahl und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> kann dargestellt werden als Differenz von zwei p-ten Potenzen. Es sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=a^{p}-b^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=a^{p}-b^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902007197d2fcc90357007a951d682e2f99d95cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.998ex; height:2.676ex;" alt="{\displaystyle F_{n}=a^{p}-b^{p}}">. Dann gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=a^{p}-b^{p}=(a-b)\cdot (a^{p-1}+a^{p-2}b+\ldots +ab^{p-2}+b^{p-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=a^{p}-b^{p}=(a-b)\cdot (a^{p-1}+a^{p-2}b+\ldots +ab^{p-2}+b^{p-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f621f6e74f04fde288ee67f2625d2747c5574f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.867ex; height:3.176ex;" alt="{\displaystyle F_{n}=a^{p}-b^{p}=(a-b)\cdot (a^{p-1}+a^{p-2}b+\ldots +ab^{p-2}+b^{p-1})}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></dd></dl></dd> <dd>Weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> prim ist und somit nicht zwei Teiler haben darf, muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fb72b1984dcc0dc347c53d0a86fd40ce7918a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.329ex; height:2.343ex;" alt="{\displaystyle a-b=1}"> sein. Wegen des <a href="/wiki/Kleiner_fermatscher_Satz" title="Kleiner fermatscher Satz">kleinen fermatschen Satzes</a> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{p}\equiv a{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≡</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{p}\equiv a{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff656f721894b9a50a2b1d18538463a6a4ec15f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.471ex; height:2.843ex;" alt="{\displaystyle a^{p}\equiv a{\pmod {p}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{p}\equiv b{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≡</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{p}\equiv b{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087e49abc33df8f5ea67dd725bcd721c5bb165d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.006ex; height:2.843ex;" alt="{\displaystyle b^{p}\equiv b{\pmod {p}}}"> und somit gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=a^{p}-b^{p}\equiv a-b\equiv 1{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≡</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=a^{p}-b^{p}\equiv a-b\equiv 1{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd732f59ef689995a335be220de480f5ee03c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.278ex; height:2.843ex;" alt="{\displaystyle F_{n}=a^{p}-b^{p}\equiv a-b\equiv 1{\pmod {p}}}"></dd></dl></dd> <dd>Somit muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> ein Teiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba6392151483569ffa418fb5c074b91465a9387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.282ex; height:3.009ex;" alt="{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}"> sein, was aber nicht sein kann, weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc9417763ad5d68870290ddaa2ca025ffdaf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{2^{n}}}"> nur Zweierpotenzen als Teiler hat.</dd> <dd>Die Annahme muss also fallengelassen werden, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> kann daher nicht dargestellt werden als Differenz von zwei p-ten Potenzen.</dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-te Fermat-Zahl und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c770d316f90c97bc02eb0813fe986be35b37f623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle D(n)}"> die Anzahl der Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}">. Dann gilt:<a href="#cite_note-Remark3.7-21">[21]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(n)=\lfloor \log _{10}\left(2^{2^{n}}+1\right)+1\rfloor \approx \lfloor \log _{10}2^{2^{n}}+1\rfloor =\lfloor 2^{n}\cdot \log _{10}2+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>≈</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(n)=\lfloor \log _{10}\left(2^{2^{n}}+1\right)+1\rfloor \approx \lfloor \log _{10}2^{2^{n}}+1\rfloor =\lfloor 2^{n}\cdot \log _{10}2+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722ebb8c74428a553690c909cb6896030fbe22f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.309ex; height:4.843ex;" alt="{\displaystyle D(n)=\lfloor \log _{10}\left(2^{2^{n}}+1\right)+1\rfloor \approx \lfloor \log _{10}2^{2^{n}}+1\rfloor =\lfloor 2^{n}\cdot \log _{10}2+1\rfloor }"></dd> <dd>wobei mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738c94c88678dd08a289f90a47a609ce44eedf14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rfloor }"> die <a href="/wiki/Floor-Funktion" class="mw-redirect" title="Floor-Funktion">Floor-Funktion</a> gemeint ist (also die größte ganze Zahl, die kleiner oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> ist)</dd></dl></dd></dl> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-te Fermat-Zahl mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}">. Dann gilt:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> ist eine Primzahl <a href="/wiki/Genau_dann,_wenn" class="mw-redirect" title="Genau dann, wenn">genau dann, wenn</a> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e895adb2d79055dc3671fe60a2c376718e510c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.194ex; height:4.343ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"></dd></dl></dd> <dd>Mit anderen Worten: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\in \mathbb {P} \Longleftrightarrow 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">⟺</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\in \mathbb {P} \Longleftrightarrow 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d65444667f12558ab78e868880240b98cc0a683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.775ex; height:4.343ex;" alt="{\displaystyle F_{n}\in \mathbb {P} \Longleftrightarrow 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"></dd></dl></dd> <dd>Dieser Satz nennt sich <a href="/wiki/P%C3%A9pin-Test" title="Pépin-Test">Pépin-Test</a>.</dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der Behauptung:</div> <div class="NavContent" style="text-align:left"> Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. Man startet mit dem linken Teil der Aussage und zeigt, dass daraus die rechte folgert. Danach startet man mit dem rechten Teil der Aussage und zeigt, dass daraus die linke Seite folgert. Beweis: <dl><dd>„<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">“: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840a1f2b26dadf1011fb0f1766a6c6242d7a2de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.974ex; height:2.509ex;" alt="{\displaystyle F_{n}\in \mathbb {P} }"> eine Primzahl mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}">. Man muss zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8123ad629851d8177ed64e288e04960caea05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.772ex; height:4.176ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}"> ist. <dl><dd>Es gilt nach dem Eulerschein Kriterium für das <a href="/wiki/Legendre-Symbol#Berechnung" title="Legendre-Symbol">Legendre-Symbol</a> die folgende Kongruenz: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)\mod F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)\mod F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09d68dd139be3987e9d93d915d6b0e46169b607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.772ex; height:6.176ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)\mod F_{n}}"></dd></dl></dd> <dd>Weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 1{\pmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 1{\pmod {4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5cf3351c0b0359bdf4fcd5fc581bf42f483a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 1{\pmod {4}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\equiv 2{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\equiv 2{\pmod {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb3277760e955b2a76650f843e98faa3e481f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\equiv 2{\pmod {3}}}"> gilt (wurde <a href="#Darstellungen_von_F_n">weiter oben</a> bewiesen), erhält man wegen des <a href="/wiki/Quadratisches_Reziprozit%C3%A4tsgesetz" title="Quadratisches Reziprozitätsgesetz">Quadratischen Reziprozitätsgesetzes</a> für das Legendre-Symbol: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {3}{F_{n}}}\right)=\left({\frac {F_{n}}{3}}\right)=\left({\frac {2}{3}}\right)=(-1)^{\frac {3^{2}-1}{8}}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>8</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {3}{F_{n}}}\right)=\left({\frac {F_{n}}{3}}\right)=\left({\frac {2}{3}}\right)=(-1)^{\frac {3^{2}-1}{8}}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e251f65445d2466de8dbf735b857f50d0edceef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.747ex; height:6.343ex;" alt="{\displaystyle \left({\frac {3}{F_{n}}}\right)=\left({\frac {F_{n}}{3}}\right)=\left({\frac {2}{3}}\right)=(-1)^{\frac {3^{2}-1}{8}}=-1}"></dd></dl></dd> <dd>Somit erhält man: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)=-1\mod F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)=-1\mod F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1afee60cdc86c12885ee3f87e85ca0718f7a214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.841ex; height:6.176ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right)=-1\mod F_{n}}"></dd></dl></dd> <dd>Damit ist eine Richtung des obigen Satzes gezeigt worden.</dd></dl></dd> <dd>„<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }">“: Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8123ad629851d8177ed64e288e04960caea05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.772ex; height:4.176ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1\mod F_{n}}">. Man muss zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840a1f2b26dadf1011fb0f1766a6c6242d7a2de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.974ex; height:2.509ex;" alt="{\displaystyle F_{n}\in \mathbb {P} }"> eine Primzahl ist. <dl><dd>Quadriert man diese Kongruenz, erhält man: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3^{\frac {F_{n}-1}{2}})^{2}=3^{F_{n}-1}\equiv (-1)^{2}=1\mod F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3^{\frac {F_{n}-1}{2}})^{2}=3^{F_{n}-1}\equiv (-1)^{2}=1\mod F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fa4d4c512f3020d97216fdcade4925050cb32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.376ex; height:4.343ex;" alt="{\displaystyle (3^{\frac {F_{n}-1}{2}})^{2}=3^{F_{n}-1}\equiv (-1)^{2}=1\mod F_{n}}"></dd></dl></dd> <dd>Nach dem <a href="/wiki/Lucas-Test_(Mathematik)#Erweiterungen_von_Lehmer,_Brillhart_und_Selfridge" title="Lucas-Test (Mathematik)">verbesserten Lucas-Test</a> folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> prim ist (weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba6392151483569ffa418fb5c074b91465a9387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.282ex; height:3.009ex;" alt="{\displaystyle F_{n}-1=2^{2^{n}}+1-1=2^{2^{n}}}"> nur einen einzigen Primteiler, nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"> hat und für diesen Primfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"> auch laut Voraussetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{p}}=3^{\frac {F_{n}-1}{2}}\equiv -1\not \equiv 1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mo>≢</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{p}}=3^{\frac {F_{n}-1}{2}}\equiv -1\not \equiv 1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11362c6f70fddafd81e37af6f0dff097ea54e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.283ex; height:4.343ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{p}}=3^{\frac {F_{n}-1}{2}}\equiv -1\not \equiv 1{\pmod {F_{n}}}}"> gilt).</dd></dl></dd> <dd>Damit sind beide Richtungen obiger Aussage bewiesen, sie hat sich somit als richtig herausgestellt. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> gilt:<a href="#cite_note-Theorem3.9-22">[22]</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 1{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 1{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbca4ffda9fe6fcc613a00590333c8b39d8863dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.844ex; height:5.176ex;" alt="{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 1{\pmod {F_{n+1}}}}"></dd></dl> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e065e9e67334dd15eeaba57d309d21f5151510d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.848ex; height:2.509ex;" alt="{\displaystyle F_{n+k}}"> prim. Dann gilt:<a href="#cite_note-Theorem3.9-22">[22]</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{\frac {F_{n+k}-1}{2}}\equiv 1{\pmod {F_{n+k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{\frac {F_{n+k}-1}{2}}\equiv 1{\pmod {F_{n+k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3b7723fd6cf7ea3c6ecb2e651130e5f3787999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.907ex; height:5.176ex;" alt="{\displaystyle F_{n}^{\frac {F_{n+k}-1}{2}}\equiv 1{\pmod {F_{n+k}}}}"></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der ersten Behauptung:</div> <div class="NavContent" style="text-align:left"> Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. Man startet mit einer bekannten richtigen Aussage und beweist mittels Umformungen und <a href="/wiki/Modulo" class="mw-redirect" title="Modulo">Modulo</a>-Rechnungen das Gewünschte. Beweis der ersten Behauptung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{2}=(2^{2^{n}}+1)^{2}={\color {red}2^{2^{n+1}}+1}+2\cdot 2^{2^{n}}={\color {red}F_{n+1}}+2\cdot 2^{2^{n}}\equiv 2\cdot 2^{2^{n}}{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>≡</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{2}=(2^{2^{n}}+1)^{2}={\color {red}2^{2^{n+1}}+1}+2\cdot 2^{2^{n}}={\color {red}F_{n+1}}+2\cdot 2^{2^{n}}\equiv 2\cdot 2^{2^{n}}{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59701ec5649a2aa0bb10b8391e357996ba8bc55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:77.262ex; height:3.509ex;" alt="{\displaystyle F_{n}^{2}=(2^{2^{n}}+1)^{2}={\color {red}2^{2^{n+1}}+1}+2\cdot 2^{2^{n}}={\color {red}F_{n+1}}+2\cdot 2^{2^{n}}\equiv 2\cdot 2^{2^{n}}{\pmod {F_{n+1}}}}"></dd> <dd>Somit gilt:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{2^{2}}\equiv (2\cdot 2^{2^{n}})^{2}=4\cdot 2^{2^{n+1}}=4\cdot (F_{n+1}-1)=4\cdot F_{n+1}-4\equiv -4\equiv -2^{2}{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msubsup> <mo>≡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>4</mn> <mo>≡</mo> <mo>−</mo> <mn>4</mn> <mo>≡</mo> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{2^{2}}\equiv (2\cdot 2^{2^{n}})^{2}=4\cdot 2^{2^{n+1}}=4\cdot (F_{n+1}-1)=4\cdot F_{n+1}-4\equiv -4\equiv -2^{2}{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de7429b4bbcae27f71f15bb534b352e4696b278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.525ex; height:3.509ex;" alt="{\displaystyle F_{n}^{2^{2}}\equiv (2\cdot 2^{2^{n}})^{2}=4\cdot 2^{2^{n+1}}=4\cdot (F_{n+1}-1)=4\cdot F_{n+1}-4\equiv -4\equiv -2^{2}{\pmod {F_{n+1}}}}"></dd> <dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0b1582d6f884e01e27786508ef410fae3de5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 3}"> erhält man:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{2^{k}}=(F_{n}^{2^{2}})^{2^{k-2}}\equiv (-2^{2})^{2^{k-2}}\equiv 2^{2^{k-1}}{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>≡</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{2^{k}}=(F_{n}^{2^{2}})^{2^{k-2}}\equiv (-2^{2})^{2^{k-2}}\equiv 2^{2^{k-1}}{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde3e5785adb3ce540e06111e0b33114af89ab7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.885ex; height:3.509ex;" alt="{\displaystyle F_{n}^{2^{k}}=(F_{n}^{2^{2}})^{2^{k-2}}\equiv (-2^{2})^{2^{k-2}}\equiv 2^{2^{k-1}}{\pmod {F_{n+1}}}}"></dd> <dd>Setzt man nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2^{n+1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2^{n+1}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2dff50a8a583159ae3ce92c4d08d8c0e7591fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.794ex; height:2.843ex;" alt="{\displaystyle k=2^{n+1}-1}"> in obiges Ergebnis ein, dann erhält man:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{2^{2^{n+1}-1}}=F_{n}^{\frac {2^{2^{n+1}}}{2}}=F_{n}^{\frac {{\color {red}2^{2^{n+1}}+1}-1}{2}}=F_{n}^{\frac {{\color {red}F_{n+1}}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>≡</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{2^{2^{n+1}-1}}=F_{n}^{\frac {2^{2^{n+1}}}{2}}=F_{n}^{\frac {{\color {red}2^{2^{n+1}}+1}-1}{2}}=F_{n}^{\frac {{\color {red}F_{n+1}}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be554b9b66851253760b789a7cf74107733e3ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.807ex; height:5.676ex;" alt="{\displaystyle F_{n}^{2^{2^{n+1}-1}}=F_{n}^{\frac {2^{2^{n+1}}}{2}}=F_{n}^{\frac {{\color {red}2^{2^{n+1}}+1}-1}{2}}=F_{n}^{\frac {{\color {red}F_{n+1}}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}{\pmod {F_{n+1}}}}"></dd> <dd>Die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n+1}-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n+1}-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f963af995cfa2196239e331a002f0feefabdb9fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.987ex; height:3.009ex;" alt="{\displaystyle 2^{2^{n+1}-2}}"> ist als Potenz von 2 durch jede kleinere Potenz von 2 teilbar, somit für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> auch durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec7da881b9da391c503b68a69a46ba3c43e189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n+1}}">. Es existiert also eine positive ganze Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b985ba501f78cb9890f3ecda3e2e315cbd5cb26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.582ex; height:2.176ex;" alt="{\displaystyle N\in \mathbb {N} }"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n+1}-2}=2^{n+1}\cdot 2N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mn>2</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n+1}-2}=2^{n+1}\cdot 2N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87c3f33cb9d7004dc9c0e02c72f56d11b953597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.473ex; height:3.009ex;" alt="{\displaystyle 2^{2^{n+1}-2}=2^{n+1}\cdot 2N}">. Wenn man dies in obiges Ergebnis einsetzt, erhält man:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}=(2^{2^{n+1}})^{2N}=({\color {red}2^{2^{n+1}}+1}-1)^{2N}=({\color {red}F_{n+1}}-1)^{2N}\equiv (-1)^{2N}\equiv 1{\pmod {F_{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>≡</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}=(2^{2^{n+1}})^{2N}=({\color {red}2^{2^{n+1}}+1}-1)^{2N}=({\color {red}F_{n+1}}-1)^{2N}\equiv (-1)^{2N}\equiv 1{\pmod {F_{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/604373d81f9afdb8b232436abb97b50adbfa90fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:96.587ex; height:5.176ex;" alt="{\displaystyle F_{n}^{\frac {F_{n+1}-1}{2}}\equiv 2^{2^{2^{n+1}-2}}=(2^{2^{n+1}})^{2N}=({\color {red}2^{2^{n+1}}+1}-1)^{2N}=({\color {red}F_{n+1}}-1)^{2N}\equiv (-1)^{2N}\equiv 1{\pmod {F_{n+1}}}}"></dd> <dd>Womit die erste Behauptung bewiesen ist. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> eine Primzahl und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175ebd03e0644b0967a63d648c2843a5e883257b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.621ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {Z} }"> eine ganze Zahl. Dann gilt für jede prime Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}\leq F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}\leq F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6378e7032240356f6e6ed492e80336207c5e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.395ex; height:2.509ex;" alt="{\displaystyle F_{k}\leq F_{n}}">:<a href="#cite_note-Theorem3.11-23">[23]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{n}}-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{n}}-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e630c32805aa5b80b903d8f1026fdb22474130b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.554ex; height:2.843ex;" alt="{\displaystyle a^{F_{n}}-a}"></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der Behauptung:</div> <div class="NavContent" style="text-align:left"> Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. Man startet mit einer bekannten richtigen Aussage und beweist das Gewünschte. Beweis: <dl><dd>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> eine prime Fermat-Zahl mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq k\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b429c3c44b3dc10332285272eee6f754dbf985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq k\leq n}">. <dl><dd>Sei weiters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> ein Teiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> auch ein Teiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb55a50daf3f282d64552ed4e9de917469c1a073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.484ex; height:2.676ex;" alt="{\displaystyle a^{F_{n}}}"> und somit auch Teiler der Differenz. Also gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{n}}-a=a\cdot (a^{F_{n}-1}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{n}}-a=a\cdot (a^{F_{n}-1}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873e3349a833de8b47b175c64e6a2825e38e6b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.958ex; height:3.176ex;" alt="{\displaystyle a^{F_{n}}-a=a\cdot (a^{F_{n}-1}-1)}"></dd></dl></dd> <dd>Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> kein Teiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">. Dann gilt wegen des <a href="/wiki/Kleiner_fermatscher_Satz" title="Kleiner fermatscher Satz">kleinen fermatschen Satzes</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{k}-1}\equiv 1{\pmod {F_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{k}-1}\equiv 1{\pmod {F_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a8abac0775398d3b1ad1fb5c385b590b05c6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.007ex; height:3.176ex;" alt="{\displaystyle a^{F_{k}-1}\equiv 1{\pmod {F_{k}}}}"> und somit: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{k}-1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{k}-1}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b776ffd5508c2920d08f93045f1ce93d276c87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle a^{F_{k}-1}-1}"></dd> <dd>Weil aber jede kleine Zweierpotenz jede größere Zweierpotenz teilt, gilt auch:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}-1=2^{2^{k}}+1-1=2^{2^{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}-1=2^{2^{k}}+1-1=2^{2^{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb450deeed0871b34175468893ddfa2b0c5d18c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.941ex; height:3.343ex;" alt="{\displaystyle F_{k}-1=2^{2^{k}}+1-1=2^{2^{k}}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}=2^{2^{n}}+1-1=F_{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}=2^{2^{n}}+1-1=F_{n}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137b69716f4e0e4c89bbb2739668083bf776f7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.282ex; height:3.009ex;" alt="{\displaystyle 2^{2^{n}}=2^{2^{n}}+1-1=F_{n}-1}"></dd> <dd>Weiters gilt bei mehrfacher Anwendung der dritten <a href="/wiki/Binomische_Formeln#Formeln" title="Binomische Formeln">binomischen Formel</a>:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{k}-1}-1=a^{2^{2^{k}}+1-1}-1=a^{2^{2^{k}}}-1^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>−</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{k}-1}-1=a^{2^{2^{k}}+1-1}-1=a^{2^{2^{k}}}-1^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68414467a057d785cb44080670a4abe717f3eeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.828ex; height:3.676ex;" alt="{\displaystyle a^{F_{k}-1}-1=a^{2^{2^{k}}+1-1}-1=a^{2^{2^{k}}}-1^{2}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{2^{n}}}-1^{2}=a^{2^{2^{n}}+1-1}-1=a^{F_{n}-1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>−</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{2^{n}}}-1^{2}=a^{2^{2^{n}}+1-1}-1=a^{F_{n}-1}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d56b8dbee2d32cc46554cd240649487e9a44e633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:37.144ex; height:3.343ex;" alt="{\displaystyle a^{2^{2^{n}}}-1^{2}=a^{2^{2^{n}}+1-1}-1=a^{F_{n}-1}-1}"></dd> <dd>Obige Ergebnisse zusammengefasst ergibt:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{k}-1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{k}-1}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b776ffd5508c2920d08f93045f1ce93d276c87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle a^{F_{k}-1}-1}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{F_{n}-1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{F_{n}-1}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384bf574dca5a713e32a033f60f6de5bd1cd72b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.587ex; height:2.843ex;" alt="{\displaystyle a^{F_{n}-1}-1}"> teilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (a^{F_{n}-1}-1)=a^{F_{n}}-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (a^{F_{n}-1}-1)=a^{F_{n}}-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cce575fa3ba91f9618900c853bc29ddbcca4a5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.958ex; height:3.176ex;" alt="{\displaystyle a\cdot (a^{F_{n}-1}-1)=a^{F_{n}}-a}"></dd></dl></dd></dl></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}:=2^{2^{n+1}}+2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}:=2^{2^{n+1}}+2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f385fcda4533282237ccdda7909745da0ecb96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.807ex; height:3.343ex;" alt="{\displaystyle H_{n}:=2^{2^{n+1}}+2^{2^{n}}+1}">. Dann gilt:<a href="#cite_note-Proposition3.5-24">[24]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}=(F_{n-1}^{2}-3F_{n-1}+3)\cdot H_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}=(F_{n-1}^{2}-3F_{n-1}+3)\cdot H_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc744d3fded279b1099b0f5282cfe27b8487b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.62ex; height:3.343ex;" alt="{\displaystyle H_{n}=(F_{n-1}^{2}-3F_{n-1}+3)\cdot H_{n-1}}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der Behauptung:</div> <div class="NavContent" style="text-align:left"> Der Beweis funktioniert <a href="/wiki/Beweis_(Mathematik)#Beweismethoden" title="Beweis (Mathematik)">direkt</a>. Beweis: <dl><dd>Man betrachte die folgende <a href="/wiki/Identit%C3%A4tsgleichung" title="Identitätsgleichung">Identität</a> unter Verwendung der <a href="/wiki/Binomische_Formeln#Formeln" title="Binomische Formeln">dritten binomischen Formel</a>:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)&=&(x^{2}+2x+1-3x-3+3)\cdot (x^{2}+x+1)&\\&=&(x^{2}-x+1)\cdot (x^{2}+x+1)&\\&=&((x^{2}+1)-x)\cdot ((x^{2}+1)+x)&{\text{(dritte binomische Formel)}}\\&=&(x^{2}+1)^{2}-x^{2}&\\&=&x^{4}+2x^{2}+1-x^{2}&\\&=&x^{4}+x^{2}+1&\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(dritte binomische Formel)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)&=&(x^{2}+2x+1-3x-3+3)\cdot (x^{2}+x+1)&\\&=&(x^{2}-x+1)\cdot (x^{2}+x+1)&\\&=&((x^{2}+1)-x)\cdot ((x^{2}+1)+x)&{\text{(dritte binomische Formel)}}\\&=&(x^{2}+1)^{2}-x^{2}&\\&=&x^{4}+2x^{2}+1-x^{2}&\\&=&x^{4}+x^{2}+1&\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036ade0a7aa7c3aacb5b2c059ecbeab1245fb25f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:116.889ex; height:20.509ex;" alt="{\displaystyle {\begin{array}{rcll}\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)&=&(x^{2}+2x+1-3x-3+3)\cdot (x^{2}+x+1)&\\&=&(x^{2}-x+1)\cdot (x^{2}+x+1)&\\&=&((x^{2}+1)-x)\cdot ((x^{2}+1)+x)&{\text{(dritte binomische Formel)}}\\&=&(x^{2}+1)^{2}-x^{2}&\\&=&x^{4}+2x^{2}+1-x^{2}&\\&=&x^{4}+x^{2}+1&\end{array}}}"></dd> <dd>Wenn man nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=2^{2^{n-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=2^{2^{n-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e68f06913ce4cbdaa5d57d194f68254fa56ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.962ex; height:3.009ex;" alt="{\displaystyle x:=2^{2^{n-1}}}"> <a href="/wiki/Substitution_(Mathematik)" title="Substitution (Mathematik)">substituiert</a>, sich ins Gedächtnis zurückruft, dass Fermatzahlen die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-1}=2^{2^{n-1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-1}=2^{2^{n-1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e239e8d412ce4bae4cd89476fa91f0a686ddbcab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.802ex; height:3.343ex;" alt="{\displaystyle F_{n-1}=2^{2^{n-1}}+1}"> haben und dass laut Definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n-1}:=2^{2^{n}}+2^{2^{n-1}}+1=(2^{2^{n-1}})^{2}+2^{2^{n-1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n-1}:=2^{2^{n}}+2^{2^{n-1}}+1=(2^{2^{n-1}})^{2}+2^{2^{n-1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5bc3f06a535850fb95475d831175d8c121b1be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.486ex; height:3.509ex;" alt="{\displaystyle H_{n-1}:=2^{2^{n}}+2^{2^{n-1}}+1=(2^{2^{n-1}})^{2}+2^{2^{n-1}}+1}"> ist, erhält man das gewünschte Ergebnis:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}({\color {red}F_{n-1}}^{2}-3{\color {red}F_{n-1}}+3)\cdot {\color {blue}H_{n-1}}&=&\left[({\color {red}2^{2^{n-1}}+1})^{2}-3({\color {red}2^{2^{n-1}}+1})+3\right]\cdot ({\color {blue}({2^{2^{n-1}}})^{2}+2^{2^{n-1}}+1})\\&=&\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)\\&=&x^{4}+x^{2}+1\\&=&(2^{2^{n-1}})^{4}+(2^{2^{n-1}})^{2}+1\\&=&2^{2^{2}\cdot 2^{n-1}}+2^{2\cdot 2^{n-1}}+1\\&=&2^{2^{n+1}}+2^{2^{n}}+1\\&=&H_{n}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}({\color {red}F_{n-1}}^{2}-3{\color {red}F_{n-1}}+3)\cdot {\color {blue}H_{n-1}}&=&\left[({\color {red}2^{2^{n-1}}+1})^{2}-3({\color {red}2^{2^{n-1}}+1})+3\right]\cdot ({\color {blue}({2^{2^{n-1}}})^{2}+2^{2^{n-1}}+1})\\&=&\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)\\&=&x^{4}+x^{2}+1\\&=&(2^{2^{n-1}})^{4}+(2^{2^{n-1}})^{2}+1\\&=&2^{2^{2}\cdot 2^{n-1}}+2^{2\cdot 2^{n-1}}+1\\&=&2^{2^{n+1}}+2^{2^{n}}+1\\&=&H_{n}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53cc074527be4899b3ea8cae0becf56402dcf5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:90.253ex; height:26.509ex;" alt="{\displaystyle {\begin{array}{rcl}({\color {red}F_{n-1}}^{2}-3{\color {red}F_{n-1}}+3)\cdot {\color {blue}H_{n-1}}&=&\left[({\color {red}2^{2^{n-1}}+1})^{2}-3({\color {red}2^{2^{n-1}}+1})+3\right]\cdot ({\color {blue}({2^{2^{n-1}}})^{2}+2^{2^{n-1}}+1})\\&=&\left[(x+1)^{2}-3(x+1)+3\right]\cdot (x^{2}+x+1)\\&=&x^{4}+x^{2}+1\\&=&(2^{2^{n-1}})^{4}+(2^{2^{n-1}})^{2}+1\\&=&2^{2^{2}\cdot 2^{n-1}}+2^{2\cdot 2^{n-1}}+1\\&=&2^{2^{n+1}}+2^{2^{n}}+1\\&=&H_{n}\end{array}}}"></dd> <dd>Was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> eine Primzahl. Dann gilt:<a href="#cite_note-Nielsen-25">[25]</a><a href="#cite_note-Sierpinski-26">[26]</a></li></ul> <dl><dd><dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=F_{m}-1=2^{2^{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=F_{m}-1=2^{2^{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f7387e4648103dce6090f01c6c7ad1b17ccd5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.316ex; height:3.009ex;" alt="{\displaystyle n=F_{m}-1=2^{2^{m}}}"> mit einer positiven ganzen Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{m}+m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{m}+m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c8f7a63ec361b238108fc0cb8881bf0c0213f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.32ex; height:2.843ex;" alt="{\displaystyle n^{n}+1=F_{2^{m}+m}}"></li></ul></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame" style="margin:0.5em auto;width:90%;;"> <div class="NavHead" style="text-align:left">Beweis der Behauptung:</div> <div class="NavContent" style="text-align:left"> Beweis von Teil 1 durch <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">Widerspruch</a>: Man führt die Annahme, dass das zu Beweisende falsch sei, zu einem Widerspruch (analog zum Beweis <a href="#Fermatsche_Primzahlen">weiter oben</a>). <dl><dd>Annahme: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> ist prim und die Hochzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> hat einen ungeraden Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d6b5c226641ad3def8e65631f0eb463d9c67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}">.</dd> <dd>Dann gilt</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=n^{{\frac {n}{c}}\cdot c}+1=(n^{\frac {n}{c}})^{c}+1^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>c</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=n^{{\frac {n}{c}}\cdot c}+1=(n^{\frac {n}{c}})^{c}+1^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1968c6b07d28bdfbea4d41d636a64e44b7eed2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.213ex; height:3.509ex;" alt="{\displaystyle n^{n}+1=n^{{\frac {n}{c}}\cdot c}+1=(n^{\frac {n}{c}})^{c}+1^{c}}"></dd> <dd>mit einer ganzen Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/698512ab6efc76df2690703a166731a6c57d0f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{c}}}">. Nach Annahme ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> ungerade, also ist diese Summe <a href="/wiki/Binomische_Formeln#Der_binomische_Lehrsatz" title="Binomische Formeln">bekanntlich</a> durch die Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\frac {n}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\frac {n}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdecdb38619e7a3f5a52d4188323b2cb738364f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.267ex; height:3.176ex;" alt="{\displaystyle n^{\frac {n}{c}}+1}"> der beiden Basen teilbar:</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=(n^{\frac {n}{c}})^{c}+1^{c}=(n^{\frac {n}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (n^{\frac {n}{c}})^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=(n^{\frac {n}{c}})^{c}+1^{c}=(n^{\frac {n}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (n^{\frac {n}{c}})^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa9906d32235fbaf8788bbc80c94400fdeeaf6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.239ex; height:7.676ex;" alt="{\displaystyle n^{n}+1=(n^{\frac {n}{c}})^{c}+1^{c}=(n^{\frac {n}{c}}+1)\cdot \sum _{j=0}^{c-1}(-1)^{j}\cdot (n^{\frac {n}{c}})^{j}}"></dd> <dd>Weil die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> prim ist, muss ihr Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\frac {n}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\frac {n}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdecdb38619e7a3f5a52d4188323b2cb738364f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.267ex; height:3.176ex;" alt="{\displaystyle n^{\frac {n}{c}}+1}"> gleich 1 oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> sein. Aber im Widerspruch dazu ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\frac {n}{c}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\frac {n}{c}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdecdb38619e7a3f5a52d4188323b2cb738364f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.267ex; height:3.176ex;" alt="{\displaystyle n^{\frac {n}{c}}+1}"> (wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\frac {n}{c}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\frac {n}{c}}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f2ed8a00ddb3faab77ca9827acbb3c6b7247db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.525ex; height:3.009ex;" alt="{\displaystyle n^{\frac {n}{c}}>0}">) größer als 1 und (wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{c}}<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>c</mi> </mfrac> </mrow> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{c}}<n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6536240cfd9fe3e9e2d6f31ba817a87646dd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.724ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{c}}<n}">) kleiner als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}">. Die Annahme, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> prim ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> einen ungeraden Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d6b5c226641ad3def8e65631f0eb463d9c67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}"> hat, muss daher fallengelassen werden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> kann nur prim sein, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> eine Zweierpotenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d82641ae2702b0db07dd11830af27b9ee0cd196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79214ef55efadfb1d9c9b02252eb8a71cf6f8b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 0}"> ist.</dd> <dd>Es ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=(2^{k})^{2^{k}}+1=2^{k\cdot 2^{k}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=(2^{k})^{2^{k}}+1=2^{k\cdot 2^{k}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c8475032cb458cc66d7fc8817672698ef69ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.183ex; height:3.509ex;" alt="{\displaystyle n^{n}+1=(2^{k})^{2^{k}}+1=2^{k\cdot 2^{k}}+1}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ce30228c74c7fb8b0d262d7d9363f87d30d42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.613ex; height:2.343ex;" alt="{\displaystyle n^{n}}"> ist somit eine Zweierpotenz.</dd> <dd>Es wurde aber <a href="#Fermatsche_Primzahlen">weiter oben</a> gezeigt, dass eine Zahl der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{t}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{t}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f259150d74e1f2a41eab90309e9e126774feab78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.991ex; height:2.676ex;" alt="{\displaystyle 2^{t}+1}"> nur dann eine Primzahl ist, wenn die Hochzahl (also der <a href="/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik)">Exponent</a>) selbst eine Zweierpotenz ist. Es gibt also ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {N} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50e2cd8430e9a081c4ffabe25b8857b515bad6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.413ex; height:2.509ex;" alt="{\displaystyle t\in \mathbb {N} _{0}}">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{k}=2^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{k}=2^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7a9d9716a0d88a648e61c19906d70718c44f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.228ex; height:2.676ex;" alt="{\displaystyle k\cdot 2^{k}=2^{t}}"> ist. Somit muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> selbst eine Zweierpotenz (also ohne ungerade Teiler) sein, daher gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd16ef926b19751e63d1b03b6090f09a492af88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {N} _{0}}">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bee44bac58afc263bdf838d40583408573b5940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.147ex; height:2.343ex;" alt="{\displaystyle k=2^{m}}"> ist. Es ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{k}=2^{2^{m}}=2^{2^{m}}+1-1=F_{m}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{k}=2^{2^{m}}=2^{2^{m}}+1-1=F_{m}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62dfdd27728f3120650752be060a34b83465045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.322ex; height:3.009ex;" alt="{\displaystyle n=2^{k}=2^{2^{m}}=2^{2^{m}}+1-1=F_{m}-1}">, was als Erstes zu zeigen war.</dd> <dd>Weiters gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=(2^{k})^{(2^{k})}+1=(2^{2^{m}})^{(2^{2^{m}})}+1=2^{2^{m}\cdot 2^{2^{m}}}+1=2^{2^{m+2^{m}}}+1=2^{2^{2^{m}+m}}+1=F_{2^{m}+m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=(2^{k})^{(2^{k})}+1=(2^{2^{m}})^{(2^{2^{m}})}+1=2^{2^{m}\cdot 2^{2^{m}}}+1=2^{2^{m+2^{m}}}+1=2^{2^{2^{m}+m}}+1=F_{2^{m}+m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99f18e409f0fce07feababbcf801c6d13302740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:89.004ex; height:3.676ex;" alt="{\displaystyle n^{n}+1=(2^{k})^{(2^{k})}+1=(2^{2^{m}})^{(2^{2^{m}})}+1=2^{2^{m}\cdot 2^{2^{m}}}+1=2^{2^{m+2^{m}}}+1=2^{2^{2^{m}+m}}+1=F_{2^{m}+m}}">, was zu zeigen war. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></dd></dl> </div> </div> <dl><dd><dl><dd>Beispiele: <dl><dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{0}+0}=F_{1}=5\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{0}+0}=F_{1}=5\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea5b898b980c5d49aae36327f160eae47a3bc59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.365ex; height:3.009ex;" alt="{\displaystyle n^{n}+1=F_{2^{0}+0}=F_{1}=5\in \mathbb {P} }"></dd> <dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{1}+1}=F_{3}=257\in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>257</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{1}+1}=F_{3}=257\in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7ee795186f1adfa782fb0de475eb94dee4acf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.689ex; height:3.009ex;" alt="{\displaystyle n^{n}+1=F_{2^{1}+1}=F_{3}=257\in \mathbb {P} }"></dd> <dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{2}+2}=F_{6}=18.446.744.073.709.551.617\not \in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>18.446.744.073.709.551.617</mn> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{2}+2}=F_{6}=18.446.744.073.709.551.617\not \in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/472bf792cc235481e09aadd7faf5dc059b22826c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.332ex; height:3.009ex;" alt="{\displaystyle n^{n}+1=F_{2^{2}+2}=F_{6}=18.446.744.073.709.551.617\not \in \mathbb {P} }"> (eine 20-stellige Zahl)</dd> <dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{3}+3}=F_{11}\not \in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{3}+3}=F_{11}\not \in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f3d949e2ffbab8a9f5383fae8ce48728f3a21f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.926ex; height:3.009ex;" alt="{\displaystyle n^{n}+1=F_{2^{3}+3}=F_{11}\not \in \mathbb {P} }"> (eine 617-stellige Zahl)</dd> <dd>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0002ab187a5f0920f4c5eff6741f9964cbe2abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=4}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1=F_{2^{4}+4}=F_{20}\not \in \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1=F_{2^{4}+4}=F_{20}\not \in \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6c1c80a93918617c3f3afdbc296a26078c6f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.926ex; height:3.009ex;" alt="{\displaystyle n^{n}+1=F_{2^{4}+4}=F_{20}\not \in \mathbb {P} }"> (eine 315653-stellige Zahl)</dd> <dd>Auch für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed436becdbbbc62c94b057f6922d53e6df39d67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=5}"> (eine 41373247568-stellige Zahl) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=11}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcfdb4444c11806dc1fc6bef70d5db4520d6a280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.464ex; height:2.176ex;" alt="{\displaystyle m=11}"> (die Anzahl der Stellen dieser Zahl hat 620 Stellen) erhält man keine Primzahlen. Für alle anderen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> ist noch nicht bekannt, ob es sich um Primzahlen handelt oder nicht.</dd> <dd>Könnte man zeigen, dass es keine weiteren Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ebbaa162bae928654d45446265b1c23597665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle n^{n}+1}"> gibt, so wäre gleichzeitig auch bewiesen, dass es unendlich viele zusammengesetzte Fermat-Zahlen gibt.</dd></dl></dd></dl></dd></dl> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a757a9ed218d8a300ee354d01171f0d5609536a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.581ex; height:2.843ex;" alt="{\displaystyle n^{n^{n}}+1}"> eine Primzahl. Dann gilt:<a href="#cite_note-Sierpinski-26">[26]</a></li></ul> <dl><dd><dl><dd><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=F_{m}-1=2^{2^{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=F_{m}-1=2^{2^{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f7387e4648103dce6090f01c6c7ad1b17ccd5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.316ex; height:3.009ex;" alt="{\displaystyle n=F_{m}-1=2^{2^{m}}}"> mit einer positiven ganzen Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n^{n}}+1=F_{2^{2^{m}+m}+m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n^{n}}+1=F_{2^{2^{m}+m}+m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeaf0a7af56792536a5e3f59804bdf6da86bbd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.295ex; height:3.509ex;" alt="{\displaystyle n^{n^{n}}+1=F_{2^{2^{m}+m}+m}}"></li></ul></dd></dl></dd></dl> <ul><li>Die Menge aller <a href="/wiki/Quadratischer_Rest" title="Quadratischer Rest">quadratischen Nichtreste</a> einer primen Fermat-Zahl ist gleich der Menge aller ihrer <a href="/wiki/Primitivwurzel" title="Primitivwurzel">Primitivwurzeln</a>.<a href="#cite_note-Theorem3.10-27">[27]</a></li></ul> <ul><li>Zwei Fermat-Zahlen sind gleich oder <a href="/wiki/Teilerfremd" class="mw-redirect" title="Teilerfremd">teilerfremd</a>, wie aus der letzten Aussage folgt (Goldbachs Theorem, nach <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a>, 1730). Daraus lässt sich folgern, dass es unendlich viele Primzahlen gibt (siehe auch <a href="https://de.wikibooks.org/wiki/Beweisarchiv:_Zahlentheorie:_Elementare_Zahlentheorie:_Satz_von_Euklid" class="extiw" title="b:Beweisarchiv: Zahlentheorie: Elementare Zahlentheorie: Satz von Euklid">Beweisarchiv</a>).</li></ul> <ul><li>Die Summe der <a href="/wiki/Kehrwert" title="Kehrwert">Kehrwerte</a> aller Fermat-Zahlen ist eine <a href="/wiki/Irrationale_Zahl" title="Irrationale Zahl">irrationale Zahl</a> (bewiesen von <a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a> im Jahr 1963).<a href="#cite_note-Golomb-28">[28]</a> Es gilt:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{F_{n}}}=\sum _{n=0}^{\infty }{\frac {1}{2^{2^{n}}+1}}\approx 0{,}59606317211782167942379392586279}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0,596</mn> <mn>06317211782167942379392586279</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{F_{n}}}=\sum _{n=0}^{\infty }{\frac {1}{2^{2^{n}}+1}}\approx 0{,}59606317211782167942379392586279}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62dcba759c76be2a195e2a27e6d3e97c24bf5dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.259ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{F_{n}}}=\sum _{n=0}^{\infty }{\frac {1}{2^{2^{n}}+1}}\approx 0{,}59606317211782167942379392586279}"> (Folge <a href="//oeis.org/A051158" class="extiw" title="oeis:A051158">A051158</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl></dd></dl> <ul><li>Keine Fermat-Zahl ist eine <a href="/wiki/Perfekte_Zahl" class="mw-redirect" title="Perfekte Zahl">perfekte Zahl</a>. Keine Fermat-Zahl ist Teil eines Paares <a href="/wiki/Befreundete_Zahlen" title="Befreundete Zahlen">befreundeter Zahlen</a> (bewiesen von <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a> im Jahr 2000).<a href="#cite_note-Luca-29">[29]</a></li></ul> <ul><li>Die Summe der Kehrwerte aller Primteiler von Fermat-Zahlen ist <a href="/wiki/Grenzwert_(Folge)" title="Grenzwert (Folge)">konvergent</a> (bewiesen von <a href="/w/index.php?title=Michal_K%C5%99%C3%AD%C5%BEek&action=edit&redlink=1" class="new" title="Michal Křížek (Seite nicht vorhanden)">Michal Křížek</a>, Florian Luca und <a href="/w/index.php?title=Lawrence_Somer&action=edit&redlink=1" class="new" title="Lawrence Somer (Seite nicht vorhanden)">Lawrence Somer</a> im Jahr 2002).<a href="#cite_note-Somer-30">[30]</a> Mit anderen Worten:</li></ul> <dl><dd>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{F}\subset \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{F}\subset \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf1b817b7667f646d0b9f8498975f1e5e005af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.474ex; height:2.509ex;" alt="{\displaystyle P_{F}\subset \mathbb {P} }"> die Menge aller Primzahlen, die irgendeine Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> teilen. Dann gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{p\in P_{F}}{\frac {1}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∈</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p\in P_{F}}{\frac {1}{p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f9766d15a6af4cd8d7ab74ce54184b3a843152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:6.598ex; height:6.676ex;" alt="{\displaystyle \sum _{p\in P_{F}}{\frac {1}{p}}}"> ist konvergent.</dd></dl></dd></dl> <ul><li>Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(F_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(F_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79ccb4e6f0f135df01e89983318169568d49839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.781ex; height:2.843ex;" alt="{\displaystyle p(F_{n})}"> der größte Primteiler der Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}">. Dann gilt:<a href="#cite_note-Wojtowicz-31">[31]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(F_{n})\geq 2^{n+2}\cdot (4n+9)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(F_{n})\geq 2^{n+2}\cdot (4n+9)+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fa536bc24b3b4fd918e0cf7c13889a2c1cdb63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:27.412ex; height:3.176ex;" alt="{\displaystyle p(F_{n})\geq 2^{n+2}\cdot (4n+9)+1}"></dd></dl></dd> <dd>für alle  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25010fec4b0f68f1b46f49d14917d962acca0b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 4}"> (bewiesen von <a href="/w/index.php?title=Aleksander_Grytczuk&action=edit&redlink=1" class="new" title="Aleksander Grytczuk (Seite nicht vorhanden)">Aleksander Grytczuk</a>, Florian Luca und <a href="/w/index.php?title=Marek_W%C3%B3jtowicz&action=edit&redlink=1" class="new" title="Marek Wójtowicz (Seite nicht vorhanden)">Marek Wójtowicz</a> im Jahr 2001).</dd></dl> <ul><li>Jede zusammengesetzte Fermat-Zahl ist eine <a href="/wiki/Starke_Pseudoprimzahl" title="Starke Pseudoprimzahl">starke Pseudoprimzahl</a> zur Basis 2, weil für alle Fermat-Zahlen gilt:<a href="#cite_note-pseudoprim-32">[32]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{r}}\equiv -1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{r}}\equiv -1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69551e369f95f261afa3b4606d972a35116729d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.449ex; height:3.176ex;" alt="{\displaystyle 2^{2^{r}}\equiv -1{\pmod {F_{n}}}}"></dd></dl></dd> <dd>für mindestens ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r<2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r<2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34aa671f16b3fe74855a724587eae955cc9b2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.789ex; height:2.509ex;" alt="{\displaystyle 0\leq r<2^{n}}"> (im Speziellen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9e24b8d7d089b120f8b65983bbfa97de38a071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.542ex; height:1.676ex;" alt="{\displaystyle r=n}">).</dd></dl> <ul><li>Jede zusammengesetzte Fermat-Zahl ist eine <a href="/wiki/Eulersche_Pseudoprimzahl" title="Eulersche Pseudoprimzahl">eulersche Pseudoprimzahl</a> zur Basis 2, weil für alle Fermat-Zahlen gilt:<a href="#cite_note-pseudoprim-32">[32]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {F_{n}-1}{2}}=2^{2^{n}-1}\equiv \pm 1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {F_{n}-1}{2}}=2^{2^{n}-1}\equiv \pm 1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c879da453f66f641fce35b9cb266fc7d557fb0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.575ex; height:4.343ex;" alt="{\displaystyle 2^{\frac {F_{n}-1}{2}}=2^{2^{n}-1}\equiv \pm 1{\pmod {F_{n}}}}"></dd></dl></dd></dl> <ul><li>Jede zusammengesetzte Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> ist eine <a href="/wiki/Fermatsche_Pseudoprimzahl" title="Fermatsche Pseudoprimzahl">fermatsche Pseudoprimzahl</a> zur Basis 2. Das heißt, für alle Fermat-Zahlen gilt:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa84381b4d951079e94a544ef8dcb5f67f97b437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.175ex; height:3.176ex;" alt="{\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}"></dd></dl></dd></dl> <ul><li>Eine prime Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> ist niemals eine <a href="/wiki/Wieferich-Primzahl" title="Wieferich-Primzahl">Wieferich-Primzahl</a>.<a href="#cite_note-Kennard-33">[33]</a> Das heißt, für alle primen Fermat-Zahlen gilt:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{F_{n}-1}\not \equiv 1{\pmod {F_{n}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≢</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{F_{n}-1}\not \equiv 1{\pmod {F_{n}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10376796a059d1c59e1dbdadde20899dbfcb3b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.332ex; height:3.176ex;" alt="{\displaystyle 2^{F_{n}-1}\not \equiv 1{\pmod {F_{n}^{2}}}}"></dd></dl></dd></dl> <ul><li>Ein Produkt</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a}\cdot F_{b}\cdot \ldots \cdot F_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a}\cdot F_{b}\cdot \ldots \cdot F_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66971afdae2ba2d605e5e99ac801f26d721913f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.287ex; height:2.509ex;" alt="{\displaystyle F_{a}\cdot F_{b}\cdot \ldots \cdot F_{s}}"></dd></dl></dd> <dd>von Fermat-Zahlen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b>\ldots >s>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>></mo> <mo>…</mo> <mo>></mo> <mi>s</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b>\ldots >s>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df9d4a2fe8192f34f5c076448fdeba24ed058a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.597ex; height:2.176ex;" alt="{\displaystyle a>b>\ldots >s>1}"> ist eine fermatsche Pseudoprimzahl zur Basis 2 <a href="/wiki/Logische_%C3%84quivalenz" title="Logische Äquivalenz">genau dann, wenn</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{s}>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{s}>a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999f69741b2ff85d9d73bbc6df58e912a12bbdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle 2^{s}>a}"> (bewiesen von <a href="/wiki/Michele_Cipolla" title="Michele Cipolla">Michele Cipolla</a> im Jahr 1904).<a href="#cite_note-Somer2-34">[34]</a></dd></dl> <ul><li>Jede Fermat-Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> hat im <a href="/wiki/Dualsystem" title="Dualsystem">Binärsystem</a> die Form</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mn>000</mn> <mo>…</mo> <mn>000</mn> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351955ff0b9064f9dcaaf8ceecaaf61da02ca9c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.125ex; height:2.509ex;" alt="{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}"></dd></dl></dd> <dd>mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"> Nullen zwischen den beiden Einsen am Anfang und Ende.<a href="#cite_note-pseudoprim4-35">[35]</a></dd> <dd>Jede Fermat-Zahl ab <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd17e0779153d765b40ebef91533489b87b2e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{2}}"> hat im <a href="/wiki/Hexadezimalsystem" title="Hexadezimalsystem">Hexadezimalsystem</a> die Form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mn>000</mn> <mo>…</mo> <mn>000</mn> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351955ff0b9064f9dcaaf8ceecaaf61da02ca9c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.125ex; height:2.509ex;" alt="{\displaystyle F_{n}=1\;\!000\ldots 000\;\!1}"></dd></dl></dd> <dd>mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n-2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-2}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c3757a47df325a5e9f9c26708025348e05680b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.484ex; height:2.843ex;" alt="{\displaystyle 2^{n-2}-1}"> Nullen zwischen den beiden Einsen am Anfang und Ende.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ungelöste_Probleme">Ungelöste Probleme</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=5" title="Abschnitt bearbeiten: Ungelöste Probleme" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Ungelöste Probleme">Quelltext bearbeiten</a>]</div> <ul><li>Ist Fn eine <a href="/wiki/Zusammengesetzte_Zahl" title="Zusammengesetzte Zahl">zusammengesetzte Zahl</a> für alle n ≥ 5?</li> <li>Gibt es unendlich viele zusammengesetzte Fermatsche Zahlen? (Diese Behauptung ist etwas schwächer als die vorherige.)</li> <li>Gibt es unendlich viele Fermatsche Primzahlen? (Diese Behauptung steht nicht im Widerspruch zur vorherigen; es könnten beide Behauptungen gelten. Es ist allerdings äußerst unwahrscheinlich, wie der <a class="mw-selflink-fragment" href="#Warum_es_wahrscheinlich_keine_weiteren_Fermat-Primzahlen_gibt">untere Abschnitt</a> zeigt.)</li> <li>Gibt es Fermatsche Zahlen, die nicht <a href="/wiki/Quadratfreie_Zahl" title="Quadratfreie Zahl">quadratfrei</a> sind?</li></ul> <div class="mw-heading mw-heading2"><h2 id="Warum_es_wahrscheinlich_keine_weiteren_Fermat-Primzahlen_gibt">Warum es wahrscheinlich keine weiteren Fermat-Primzahlen gibt</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=6" title="Abschnitt bearbeiten: Warum es wahrscheinlich keine weiteren Fermat-Primzahlen gibt" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Warum es wahrscheinlich keine weiteren Fermat-Primzahlen gibt">Quelltext bearbeiten</a>]</div> Man kann <a href="/wiki/Heuristik" title="Heuristik">heuristisch</a> annehmen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}=65537}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>65537</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}=65537}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbe6707f5e992f9fd5e1a63c211263808854fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.46ex; height:2.509ex;" alt="{\displaystyle F_{4}=65537}"> die letzte (und somit auch die größte) Fermat-Primzahl ist. Die Überlegungen dafür sind die folgenden: Der <a href="/wiki/Primzahlsatz" title="Primzahlsatz">Primzahlsatz</a> gibt an, dass eine zufällige ganze Zahl in einem geeigneten Intervall um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> mit einer Wahrscheinlichkeit von etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\ln n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\ln n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47f822d9eff6d95cbedee9f1414b355b8c3d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.581ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{\ln n}}}"> eine Primzahl ist. Wenn man nun heuristisch davon ausgeht, dass diese Aussage auch für Fermat-Primzahlen gilt, gepaart mit der Tatsache, dass die Fermat-Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5},\ldots F_{32}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5},\ldots F_{32}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951d38b7381c217939e8b8d687a0916e64080ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.064ex; height:2.509ex;" alt="{\displaystyle F_{5},\ldots F_{32}}"> alle zusammengesetzt sind, kommt man für größere Fermat-Primzahlen zu folgendem Ergebnis:<a href="#cite_note-36">[36]</a> <dl><dd>Die Wahrscheinlichkeit, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> eine Fermat-Primzahl ist, beträgt höchstens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{2^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{2^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdbb6b5a2ba3a09719dea38d5ceeb1d3a8e4089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.623ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{2^{n}}}}">.</dd></dl> Für eine neue, noch unbekannte Fermat-Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 33}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>33</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 33}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f2d70ce739e1d74d512af3f873d9f9640e2dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.818ex; height:2.343ex;" alt="{\displaystyle n\geq 33}"> sein. Somit beträgt die erwartete Anzahl an neuen, noch unbekannten Fermat-Primzahlen höchstens <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{2^{33}}}+{\frac {4}{2^{34}}}+{\frac {4}{2^{35}}}+\ldots ={\frac {4}{2^{32}}}={\frac {1}{2^{30}}}<{\frac {1}{10^{9}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>…</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msup> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{2^{33}}}+{\frac {4}{2^{34}}}+{\frac {4}{2^{35}}}+\ldots ={\frac {4}{2^{32}}}={\frac {1}{2^{30}}}<{\frac {1}{10^{9}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d124f2f63e4b51cc800716bb0ab6d4f218c99938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.129ex; height:5.676ex;" alt="{\displaystyle {\frac {4}{2^{33}}}+{\frac {4}{2^{34}}}+{\frac {4}{2^{35}}}+\ldots ={\frac {4}{2^{32}}}={\frac {1}{2^{30}}}<{\frac {1}{10^{9}}}}"></dd></dl> Die Wahrscheinlichkeit, dass es noch eine weitere Fermat-Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}>F_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}>F_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139ca906e7648ba26e053b5a82f8889a43267e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.36ex; height:2.509ex;" alt="{\displaystyle F_{n}>F_{4}}"> gibt, beträgt also weniger als 1 zu einer <a href="/wiki/Milliarde" title="Milliarde">Milliarde</a>, weswegen man davon ausgehen kann, dass es wahrscheinlich keine weiteren gibt. <div class="mw-heading mw-heading2"><h2 id="Geometrische_Anwendung_der_Fermatschen_Primzahlen">Geometrische Anwendung der Fermatschen Primzahlen</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=7" title="Abschnitt bearbeiten: Geometrische Anwendung der Fermatschen Primzahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Geometrische Anwendung der Fermatschen Primzahlen">Quelltext bearbeiten</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg/350px-Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg.png" decoding="async" width="350" height="303" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg/525px-Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg/700px-Polygonkonstruktion_nach_Gau%C3%9F_und_Wantzel.svg.png 2x" data-file-width="748" data-file-height="648" /></a><figcaption>Anzahl der Seiten bekannter konstruierbarer Polygone. Rot: Seitenzahlen der 31 bekannten regulären Polygone mit ungerader Seitenzahl (Lesart von oben nach unten: <a href="/wiki/Gleichseitiges_Dreieck" title="Gleichseitiges Dreieck">Gleichseitiges Dreieck</a> – regelmäßiges <a href="/wiki/F%C3%BCnfeck" title="Fünfeck">Fünfeck</a> – regelmäßiges <a href="/wiki/F%C3%BCnfzehneck" title="Fünfzehneck">Fünfzehneck</a> - … – <a href="/wiki/4294967295-Eck" title="4294967295-Eck">4294967295-Eck</a>) Schwarz: Seitenzahlen der (unendlich vielen) bekannten Polygone mit gerader Seitenzahl</figcaption></figure> <a href="/wiki/Carl_Friedrich_Gau%C3%9F" title="Carl Friedrich Gauß">Carl Friedrich Gauß</a> zeigte (in seinem Lehrbuch <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a>), dass es einen Zusammenhang zwischen der <a href="/wiki/Konstruierbares_Polygon" title="Konstruierbares Polygon">Konstruktion von regelmäßigen Polygonen</a> und den Fermatschen Primzahlen gibt: <dl><dd>Ein <a href="/wiki/Regelm%C3%A4%C3%9Figes_Polygon" title="Regelmäßiges Polygon">regelmäßiges Polygon</a> mit n Seiten kann dann und nur dann mit <a href="/wiki/Konstruktion_mit_Zirkel_und_Lineal" title="Konstruktion mit Zirkel und Lineal">Zirkel und Lineal konstruiert</a> werden, wenn n <ul><li>eine <a href="/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenz</a> von 2 oder</li> <li>eine Potenz von 2 multipliziert mit paarweise verschiedenen Fermatschen Primzahlen ist.<a href="#cite_note-EmilArtin-37">[37]</a></li></ul></dd></dl> Mit anderen Worten: <dl><dd>Ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-seitiges regelmäßiges Polygon kann mit Zirkel und Lineal konstruiert werden<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \Longleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mo stretchy="false">⟺</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \Longleftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc2c4122cd29a61327e359e47b5d72b3d288ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.962ex; height:1.843ex;" alt="{\displaystyle \qquad \Longleftrightarrow }"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{k}\cdot p_{1}\cdot p_{2}\cdot \dotso \cdot p_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{k}\cdot p_{1}\cdot p_{2}\cdot \dotso \cdot p_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed957fcb283d9b401ba8493c0fb8d7167d45a7bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.804ex; height:3.009ex;" alt="{\displaystyle n=2^{k}\cdot p_{1}\cdot p_{2}\cdot \dotso \cdot p_{s}}">  mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bceb13f72e37bcd50b60e5fb2fa05bcf15c265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.784ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {N} _{0}}"> und paarweise verschiedenen Fermatschen Primzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},\dotsc ,p_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},\dotsc ,p_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53bfb7738a4cb85885921a16d0bc645de76cd5ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.922ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},\dotsc ,p_{s}}"></dd></dl></dd></dl> Konkret zeigte Gauß die Konstruierbarkeit des regelmäßigen <a href="/wiki/Siebzehneck" title="Siebzehneck">Siebzehnecks</a>. Die nach der obigen Formel konstruierbaren regelmäßigen Polygone lassen sich in zwei Gruppen unterteilen: solche mit ungerader Seitenzahl und solche mit gerader Seitenzahl. Alle Polygone, in denen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"> ist, sind offensichtlich solche mit gerader Seitenzahl (durch 2 teilbar). Alle Polygone mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"> sind solche mit ungerader Seitenzahl (ein Produkt von Primzahlen größer als 2 ist immer eine ungerade Zahl). Da nur endlich viele Fermatsche Primzahlen bekannt sind, ist auch die Anzahl der bekannten, mit Zirkel und Lineal konstruierbaren, regulären Polygone mit ungerader Seitenzahl begrenzt. Unter diesen ist das <a href="/wiki/4294967295-Eck" title="4294967295-Eck">4294967295-Eck</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \prod _{i=1}^{5}p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \prod _{i=1}^{5}p_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce4d01d8855e8ed5084be779090bb87d3a6a033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.72ex; height:2.676ex;" alt="{\displaystyle \scriptstyle \prod _{i=1}^{5}p_{i}}">) dasjenige mit der größten Eckenzahl. <div class="mw-heading mw-heading2"><h2 id="Verallgemeinerte_Fermatsche_Zahlen">Verallgemeinerte Fermatsche Zahlen</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=8" title="Abschnitt bearbeiten: Verallgemeinerte Fermatsche Zahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerte Fermatsche Zahlen">Quelltext bearbeiten</a>]</div> Eine Zahl der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2^{n}}+a^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2^{n}}+a^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38a36d6558f890699a4b723ba1a6b5ca27be4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.106ex; height:2.843ex;" alt="{\displaystyle b^{2^{n}}+a^{2^{n}}}"> mit zwei <a href="/wiki/Teilerfremdheit" title="Teilerfremdheit">teilerfremden</a> natürlichen Zahlen a > 0 und b > 0 heißt verallgemeinerte Fermatsche Zahl. Ist eine solche Zahl prim, dann heißt sie verallgemeinerte Fermatsche Primzahl. Insgesamt sind schon über 11719 Faktoren von verallgemeinerten zusammengesetzten Fermat-Zahlen bekannt (Stand: 13. August 2018).<a href="#cite_note-prothsearch-38">[38]</a><a href="#cite_note-prothsearch2-39">[39]</a> Davon wurden alleine über 5100 von <a href="/w/index.php?title=Anders_Bj%C3%B6rn&action=edit&redlink=1" class="new" title="Anders Björn (Seite nicht vorhanden)">Anders Björn</a> und <a href="/wiki/Hans_Riesel" title="Hans Riesel">Hans Riesel</a> vor 1998 entdeckt. Ist a = 1, so werden die so erhaltenen verallgemeinerten Fermatschen Zahlen üblicherweise mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)=b^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155b271d27e4d7b9dd02718fdb9c20273ee64a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:3.176ex;" alt="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"></dd></dl> bezeichnet. Die Zahl b nennt man Basis. Ist a = 1 und b = 2, so handelt es sich um die schon weiter oben erwähnten Fermat-Zahlen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(2)=F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(2)=F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59181e6cfad5da108945c40cd69a218395fe1590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.779ex; height:3.176ex;" alt="{\displaystyle F_{n}(2)=F_{n}=2^{2^{n}}+1}">.</dd></dl> Es folgt eine Auflistung der ersten verallgemeinerten Fermatschen Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mi>ggT</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc0613a9c2c576ad2b122dc4c7a7ca138a8a7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.795ex; height:6.509ex;" alt="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}">. Die beiden Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> müssen, damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> prim sein kann, teilerfremd sein. Außerdem ist es auch notwendig, dass man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> durch den <a href="/wiki/GgT" class="mw-redirect" title="GgT">größten gemeinsamen Teiler</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\operatorname {ggT} (b+a,2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ggT</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\operatorname {ggT} (b+a,2)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c49e381281dd0160d550ca761b098131af1707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.076ex; height:2.843ex;" alt="{\displaystyle {\operatorname {ggT} (b+a,2)}}"> dividiert, da die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2^{n}}+a^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2^{n}}+a^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38a36d6558f890699a4b723ba1a6b5ca27be4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.106ex; height:2.843ex;" alt="{\displaystyle b^{2^{n}}+a^{2^{n}}}"> bei ungeradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> immer eine gerade Zahl wäre und somit niemals eine Primzahl sein könnte. Weiters kann man <a href="/wiki/Ohne_Beschr%C3%A4nkung_der_Allgemeinheit" title="Ohne Beschränkung der Allgemeinheit">ohne Einschränkung</a> annehmen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"> sein muss, da man bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> bedenkenlos mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> vertauschen kann und somit zum Beispiel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(5,9)=F_{n}(9,5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>9</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(5,9)=F_{n}(9,5)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc8adb9d25648b8707e52c3a3c13c4e2bf8f412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.861ex; height:2.843ex;" alt="{\displaystyle F_{n}(5,9)=F_{n}(9,5)}"> ist. Der Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"> führt niemals zu Primzahlen, da dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)=F_{n}(b,b)={\frac {b^{2^{n}}+b^{2^{n}}}{\operatorname {ggT} (b+b,2)}}={\frac {2b^{2^{n}}}{2}}=b^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mi>ggT</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)=F_{n}(b,b)={\frac {b^{2^{n}}+b^{2^{n}}}{\operatorname {ggT} (b+b,2)}}={\frac {2b^{2^{n}}}{2}}=b^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1dcf6667668c639b7a8eef1b9b35b72050d35fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.441ex; height:6.509ex;" alt="{\displaystyle F_{n}(b,a)=F_{n}(b,b)={\frac {b^{2^{n}}+b^{2^{n}}}{\operatorname {ggT} (b+b,2)}}={\frac {2b^{2^{n}}}{2}}=b^{2^{n}}}"> wäre und sicher nicht prim ist (es wären in diesem Fall auch die beiden Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> nicht wie vorausgesetzt teilerfremd). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der verallgemeinerten Fermatschen Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mi>ggT</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc0613a9c2c576ad2b122dc4c7a7ca138a8a7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.795ex; height:6.509ex;" alt="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq 16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq 16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87ea4b6d9127a628a53c7e0084358579bfba513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.421ex; height:2.343ex;" alt="{\displaystyle b\leq 16}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></div> <div class="NavContent"> <table class="toptextcells"> <tbody><tr> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th>a</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"></th> <th style="max-width:0; line-height:120%"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> prim ist </th></tr> <tr> <td>2 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/626b45a94acac04a3344f10f26eeeb96c4fdb51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 2^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, 3, 4, … </td></tr> <tr> <td>3 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875827d3f14bcac012abeb993592a3367d34e6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {3^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 2, 4, 5, 6, … </td></tr> <tr> <td>3 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8623fd7db5efacdc2c2666ff49ad882b6ffc1aa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 3^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>4 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6063351d1e95ae7274ff1ecba99e54b22ffc6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 4^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, 3, … </td></tr> <tr> <td>4 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{n}}+3^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{n}}+3^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167676c2ca92a9e269ba9731f2e94a65f67fdb5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 4^{2^{n}}+3^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 4, … </td></tr> <tr> <td>5 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3959f8eed2b47aa3bbe11f6e0fc3715f0c561f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {5^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 2 … </td></tr> <tr> <td>5 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cd8adce7219506ceffe2e969827021de19aa1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 5^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>5 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{n}}+3^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{n}}+3^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d057c16eb9e598625e2cfbed7b0d6f79a96746ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {5^{2^{n}}+3^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 2, 3, … </td></tr> <tr> <td>5 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d38a80e75266e588d87ebd9aa0125dbc957e270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 5^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>6 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50bb8c888d61ae61ca21ea9760ff8b76521ee704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 6^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>6 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6^{2^{n}}+5^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6^{2^{n}}+5^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccc552b1e7123ee6a0c041b69310fae1fc3cb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 6^{2^{n}}+5^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 3, 4, … </td></tr> <tr> <td>7 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0542ae0c982ac1634c87438e2a65b5a8f3f203f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {7^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>7 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80870a4aec26df588268b3566bc337bbb62ffccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 7^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>7 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7^{2^{n}}+3^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7^{2^{n}}+3^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a79d7b10ea16dbe27b5e6f237e51a3d5b253e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {7^{2^{n}}+3^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 8, … </td></tr> <tr> <td>7 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5393581cdfd55cbef1ed9f08fde5322ae9ab689a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 7^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>7 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7^{2^{n}}+5^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7^{2^{n}}+5^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c94b84a8f8b28504ee8b7aed6fcd741fc1b9300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {7^{2^{n}}+5^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 4, … </td></tr> <tr> <td>7 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7^{2^{n}}+6^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7^{2^{n}}+6^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936e299d639dc4dd45b8e5f5dca2326d78e9eebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 7^{2^{n}}+6^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 4, … </td></tr> <tr> <td>8 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc18cf28fa4fc9a4800ceb7b0462f5f1707887d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 8^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left; background:#E8EAEC; max-width:120px">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>8 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{2^{n}}+3^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2^{n}}+3^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e8be2d77448e96fa641e4b228cf62e17d9ec92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 8^{2^{n}}+3^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>8 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{2^{n}}+5^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2^{n}}+5^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2553183484befe899ae0ed95991fbd0563293f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 8^{2^{n}}+5^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th>a</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"></th> <th style="max-width:0; line-height:120%"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> prim ist </th></tr> <tr> <td>8 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{2^{n}}+7^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2^{n}}+7^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ee6a015f4f9592da06f4488acb35840cadabfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 8^{2^{n}}+7^{2^{n}}}"> </td> <td style="text-align:left">1, 4, … </td></tr> <tr> <td>9 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {9^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576595b66ebbcbbb06179fa008544f75e59e7ec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {9^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 3, 4, 5, … </td></tr> <tr> <td>9 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0476d827485c2d19f21b84db2ceca43f3c64a1eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 9^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>9 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d982423c52d227cc8587121eeb036511563580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 9^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>9 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {9^{2^{n}}+5^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9^{2^{n}}+5^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb718d5426efad245c33f745b8f14f78f0e69489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {9^{2^{n}}+5^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>9 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {9^{2^{n}}+7^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9^{2^{n}}+7^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e6657911ba0e2fb3f65846f3f2507f295a367a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.04ex; height:5.843ex;" alt="{\displaystyle {\frac {9^{2^{n}}+7^{2^{n}}}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>9 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9^{2^{n}}+8^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9^{2^{n}}+8^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f814ef0d7c159817962e939be91ea6d5f3e08d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle 9^{2^{n}}+8^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 5, … </td></tr> <tr> <td>10 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a963a78592485a53a9bd00f562661b3e460fb136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 10^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>10 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2^{n}}+3^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2^{n}}+3^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd121c029a877f84e2698781c0bc54811c97f82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 10^{2^{n}}+3^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 3, … </td></tr> <tr> <td>10 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2^{n}}+7^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2^{n}}+7^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038d2f2e66ee5e1232c021cb3115bbf84220a0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 10^{2^{n}}+7^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>10 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2^{n}}+9^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2^{n}}+9^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6561e209c4a6b936f0e7b329bc4e3f60701d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 10^{2^{n}}+9^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>11 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097f372bed1ed581dfaf045bf7e638a4eceb6daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>11 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76fdbec09431f19cf29e7d3963ee59e1b4771921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 11^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>11 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+3^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+3^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d89f605cfd1fe21efefa44c9e52b6722cf0db3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+3^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 3, … </td></tr> <tr> <td>11 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f0f24573221b2cd8918d2b0264b9aff9453cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 11^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>11 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+5^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+5^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b8c76e670f9753aaee4b68cf7cf6b5d643fb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+5^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>11 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11^{2^{n}}+6^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11^{2^{n}}+6^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a8be06c9891fb0391f7f96d9facaea888b366d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 11^{2^{n}}+6^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>11 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+7^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+7^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c56baeca2cbd2f781cc501844edfd9383a24ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+7^{2^{n}}}{2}}}"> </td> <td style="text-align:left">2, 4, 5, … </td></tr> <tr> <td>11 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11^{2^{n}}+8^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11^{2^{n}}+8^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/603972af61031c8836ebefde7c0b1f63d26dbf08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 11^{2^{n}}+8^{2^{n}}}"> </td> <td style="text-align:left">0, 6, … </td></tr> <tr> <td>11 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+9^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+9^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245ca78fd21f65cc5307d002c516f1cdc36228d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+9^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th>a</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"></th> <th style="max-width:0; line-height:120%"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> prim ist </th></tr> <tr> <td>11 </td> <td>10 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11^{2^{n}}+10^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11^{2^{n}}+10^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bbb33e4a947a2c1962b638b3c3c554527c1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 11^{2^{n}}+10^{2^{n}}}"> </td> <td style="text-align:left">5, … </td></tr> <tr> <td>12 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73165376957d61095373f9be19579232367ffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 12^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>12 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{n}}+5^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{n}}+5^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da42743c2707c7e52b51ebc68d01ba565d6104a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 12^{2^{n}}+5^{2^{n}}}"> </td> <td style="text-align:left">0, 4, … </td></tr> <tr> <td>12 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{n}}+7^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{n}}+7^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c4b7ab3cf0a6a32949ab9e795339374746dc7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 12^{2^{n}}+7^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 3, … </td></tr> <tr> <td>12 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{n}}+11^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{n}}+11^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b47b8cbb7b18301065166130d70bf140fb3d4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 12^{2^{n}}+11^{2^{n}}}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>13 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c829cba6a7225c10136c8736bc688fde5f965226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 2, 3, … </td></tr> <tr> <td>13 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c13d25d4c9d834e90ba995e9eca0595bdcdba32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">1, 3, 9, … </td></tr> <tr> <td>13 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+3^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+3^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b62343a0aeb2f616108a04ba6bdbeb5b8d3bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+3^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>13 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0e17fe033ef54e8bb6428be61bf69de2cfd7ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>13 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+5^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+5^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034a85c33b905bd6d4a73c3bce079f577c01e840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+5^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 2, 4, … </td></tr> <tr> <td>13 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+6^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+6^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7da47b52a1e1910ed6cafb1b5110a4e4c6ab469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+6^{2^{n}}}"> </td> <td style="text-align:left">0, 6, … </td></tr> <tr> <td>13 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+7^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+7^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3ddddfbe4dce178d0fed50277591d027c60969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+7^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>13 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+8^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+8^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc011cbcdbfe771560bf11f266bfbbbac27efd57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+8^{2^{n}}}"> </td> <td style="text-align:left">1, 3, 4, … </td></tr> <tr> <td>13 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+9^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+9^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd9f341e3ffbcfb845c31762c85019378784d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+9^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 3, … </td></tr> <tr> <td>13 </td> <td>10 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+10^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+10^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f784d5747dba6e7657c7b1ff28ddf04b7865b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+10^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, 4, … </td></tr> <tr> <td>13 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+11^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+11^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3ce698d9c3ba50bc83a0871f1ea7ad1f5f5c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.365ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+11^{2^{n}}}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>13 </td> <td>12 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13^{2^{n}}+12^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13^{2^{n}}+12^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0951953d3a9b1c4954f59a6566e80f5bb6d9b3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 13^{2^{n}}+12^{2^{n}}}"> </td> <td style="text-align:left">1, 2, 5, … </td></tr> <tr> <td>14 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0cb44802d2e866f0c66a9dd106e871d0e12117b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>14 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+3^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+3^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a670b257cdc6a98d966e6ad12eb655ad10e4a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+3^{2^{n}}}"> </td> <td style="text-align:left">0, 3, … </td></tr> <tr> <td>14 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+5^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+5^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c3eea1b68df3ed2c66edab3915bb554c0b3b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+5^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 4, 8, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th>a</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"></th> <th style="max-width:0; line-height:120%"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> prim ist </th></tr> <tr> <td>14 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+9^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+9^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c8f2675305d554d30a04ddedb50541263ade91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+9^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 8, … </td></tr> <tr> <td>14 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+11^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+11^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73963d022f8b0d36ea89f8f8910b04d587f77342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+11^{2^{n}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>14 </td> <td>13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+13^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+13^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75a45e4822467a5d16b2fa8a33533e1284e1c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+13^{2^{n}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>15 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {15^{2^{n}}+1^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {15^{2^{n}}+1^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce354bc36482585e8a9732e9e0c94d200d1d7e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {15^{2^{n}}+1^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>15 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{2^{n}}+2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{2^{n}}+2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9142a6237e4011d4c01480af9cf8fbf86d154a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 15^{2^{n}}+2^{2^{n}}}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>15 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{2^{n}}+4^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{2^{n}}+4^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9ea3c5de741952977b42f495615d73315084cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 15^{2^{n}}+4^{2^{n}}}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>15 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {15^{2^{n}}+7^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {15^{2^{n}}+7^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5201a84ffb7287ce299aeb9e73dce06bee17fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.202ex; height:5.843ex;" alt="{\displaystyle {\frac {15^{2^{n}}+7^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>15 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{2^{n}}+8^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{2^{n}}+8^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d46207e8bc10193359c3c573208d61a87d65bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 15^{2^{n}}+8^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 3, … </td></tr> <tr> <td>15 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {15^{2^{n}}+11^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {15^{2^{n}}+11^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf83a693d8f41798a37bf6d61fcc8065f40bdda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.365ex; height:5.843ex;" alt="{\displaystyle {\frac {15^{2^{n}}+11^{2^{n}}}{2}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>15 </td> <td>13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {15^{2^{n}}+13^{2^{n}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {15^{2^{n}}+13^{2^{n}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9daaae33ed22ceaeb4c093a270a6824f9b9e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.365ex; height:5.843ex;" alt="{\displaystyle {\frac {15^{2^{n}}+13^{2^{n}}}{2}}}"> </td> <td style="text-align:left">1, 4, … </td></tr> <tr> <td>15 </td> <td>14 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{2^{n}}+14^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{2^{n}}+14^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a2750978f83543995a23f089c0fc333144432c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 15^{2^{n}}+14^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, 4, … </td></tr> <tr> <td>16 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+1^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+1^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f051fdf762c33c911833d4bbebb612e488904f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+1^{2^{n}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>16 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+3^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+3^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd75e5b0be1d22426d50d0ff739422831c2d5603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+3^{2^{n}}}"> </td> <td style="text-align:left">0, 2, 8, … </td></tr> <tr> <td>16 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+5^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+5^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee150f9db2dbda4bc418f7ed076eb8beb597563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+5^{2^{n}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>16 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+7^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+7^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0db6cf671fc9a7152006ec4356253af7c30acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+7^{2^{n}}}"> </td> <td style="text-align:left">0, 6, … </td></tr> <tr> <td>16 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+9^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+9^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd8c8388b72179d6e85ee1b983b81cebd84713f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.366ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+9^{2^{n}}}"> </td> <td style="text-align:left">1, 3, … </td></tr> <tr> <td>16 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+11^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+11^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266f29046990d691b0a604a24ecf7d7176d8f7d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+11^{2^{n}}}"> </td> <td style="text-align:left">2, 4, … </td></tr> <tr> <td>16 </td> <td>13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+13^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+13^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cef0a0505efb223753da238dad349fff9f31e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+13^{2^{n}}}"> </td> <td style="text-align:left">0, 3, … </td></tr> <tr> <td>16 </td> <td>15 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+15^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+15^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646b1db7bba49f7f214feeabcc92d409c6706e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+15^{2^{n}}}"> </td> <td style="text-align:left">0, … </td></tr></tbody></table> </td></tr></tbody></table> </div> </div> <a href="/wiki/Fast_alle" title="Fast alle">Fast alle</a> verallgemeinerten Fermatschen Zahlen sind wahrscheinlich zusammengesetzt. Bewiesen ist diese Aussage aber nicht, denn schon für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32584049ed5f72969777f89d69b74ee462875e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=2}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6104442ed30596ef4d7795d3186273f68d796ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}"> (das sind die ursprünglichen Fermat-Zahlen) wurde weiter oben im Kapitel <a href="#Ungelöste_Probleme">Ungelöste Probleme</a> erwähnt, dass man noch nicht weiß, ob ab <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a752e15bfe1dac8d617d014a77c275bfd4af0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 5}"> alle weiteren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> zusammengesetzt sind oder nicht. Ähnlich verhält es sich mit anderen Basen und Hochzahlen. Und obwohl schon über 11000 Faktoren von verallgemeinerten Fermatschen Zahlen bekannt sind (siehe weiter oben), ist es schwierig, solche Faktoren zu finden, zumal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"> sehr schnell sehr groß wird. Zum Teil weiß man zwar, dass diese Zahlen zusammengesetzt sein müssen, aber Primteiler kennt man von den wenigsten. Bekannt ist, dass solche Primteiler die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{m}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{m}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e596004fee10e811ca95f0b45d0801080e1aa077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.731ex; height:2.509ex;" alt="{\displaystyle k\cdot 2^{m}+1}"> haben müssen. Es folgt eine Auflistung von Primfaktoren kleinerer verallgemeinerter Fermatschen Zahlen inklusive zweier etwas höherer Zahlenbeispiele, anhand derer man erkennen kann, wie schnell die Zahlen sehr hoch werden. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste von ausgewählten Primfaktoren von verallgemeinerten Fermatschen Zahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mi>ggT</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc0613a9c2c576ad2b122dc4c7a7ca138a8a7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.795ex; height:6.509ex;" alt="{\displaystyle F_{n}(b,a)={\frac {b^{2^{n}}+a^{2^{n}}}{\operatorname {ggT} (b+a,2)}}}"></div> <div class="NavContent"> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th colspan="5">verallgemeinerte zusammengesetzte Fermatsche Zahl</th> <th colspan="4">Primteiler </th></tr> <tr class="hintergrundfarbe6"> <th>b</th> <th>a</th> <th>n</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b,a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd20adb752da868c0fe06bc33744921980fb1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(b,a)}"></th> <th>Dezimalschreibweise</th> <th>k</th> <th>m</th> <th>Primteiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{m}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{m}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e596004fee10e811ca95f0b45d0801080e1aa077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.731ex; height:2.509ex;" alt="{\displaystyle k\cdot 2^{m}+1}"></th> <th>Dezimalschreibweise </th></tr> <tr> <th rowspan="2">2 </th> <th rowspan="2">1 </th> <th rowspan="2">5 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{5}}+1^{2^{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{5}}+1^{2^{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3359b2c616d10f51e08c021904300f08513bf5e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 2^{2^{5}}+1^{2^{5}}}"> </th> <th rowspan="2">4.294.967.297 (=<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64647603e8358bf2b07099963d5ac2d8b75ee9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{5}}">) </th> <td>5 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d3e7e954a01c1d2f1b8e5a117b566434f54ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{7}+1}"> </td> <td>641 </td></tr> <tr> <td>52347 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 52347\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>52347</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 52347\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e28bcad1647d7ddca3c12b69d033dfc51a28d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.711ex; height:2.843ex;" alt="{\displaystyle 52347\cdot 2^{7}+1}"> </td> <td>6.700.417 </td></tr> <tr> <th rowspan="2">2 </th> <th rowspan="2">1 </th> <th rowspan="2">6 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{6}}+1^{2^{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{6}}+1^{2^{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26697f0c63b7b0052ee4000c91dc002d26ba879c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 2^{2^{6}}+1^{2^{6}}}"> </th> <th rowspan="2">18.446.744.073.709.551.617 (=<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f760637659fa525945b3fdb906c673089642ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{6}}">) </th> <td>1071 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1071\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1071</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1071\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1a689078f22c1298e81ad2aff61c1858170dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle 1071\cdot 2^{8}+1}"> </td> <td>274.177 </td></tr> <tr> <td>262814145745 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 262814145745\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>262814145745</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 262814145745\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e867c688a29ba92d2c6bc5377132cf19acb21d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.848ex; height:2.843ex;" alt="{\displaystyle 262814145745\cdot 2^{8}+1}"> </td> <td>67.280.421.310.721 </td></tr> <tr> <th rowspan="2">3 </th> <th rowspan="2">1 </th> <th rowspan="2">3 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{2^{3}}+1^{2^{3}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{2^{3}}+1^{2^{3}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1822525f79c767482880c21b0cc59547f74e52a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.773ex; height:6.009ex;" alt="{\displaystyle {\frac {3^{2^{3}}+1^{2^{3}}}{2}}}"> </th> <th rowspan="2">3.281 </th> <td>1 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64df7df94ba2284768ae7025aef8ecaf48847ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{4}+1}"> </td> <td>17 </td></tr> <tr> <td>3 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2^{6}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{6}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cca20310c8e36b3662f26b64d1d0f29f7be726b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{6}+1}"> </td> <td>193 </td></tr> <tr> <th rowspan="2">3 </th> <th rowspan="2">2 </th> <th rowspan="2">3 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2^{3}}+2^{2^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2^{3}}+2^{2^{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3ad72579c1f59c5c9434efe1d08f8e3f6851b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 3^{2^{3}}+2^{2^{3}}}"> </th> <th rowspan="2">6.817 </th> <td>1 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64df7df94ba2284768ae7025aef8ecaf48847ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{4}+1}"> </td> <td>17 </td></tr> <tr> <td>25 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d85e5a302110801f244a1f77602b2d0e827f37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.224ex; height:2.843ex;" alt="{\displaystyle 25\cdot 2^{4}+1}"> </td> <td>401 </td></tr> <tr> <th rowspan="2">3 </th> <th rowspan="2">2 </th> <th rowspan="2">4 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2^{4}}+2^{2^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2^{4}}+2^{2^{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5705ff0119bd120a9e69335114c3ad60cfbf35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 3^{2^{4}}+2^{2^{4}}}"> </th> <th rowspan="2">43.112.257 </th> <td>95 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 95\cdot 2^{5}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>95</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 95\cdot 2^{5}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d1acf190fc45314deb62c26815f95684983db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.224ex; height:2.843ex;" alt="{\displaystyle 95\cdot 2^{5}+1}"> </td> <td>3.041 </td></tr> <tr> <td>443 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 443\cdot 2^{5}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>443</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 443\cdot 2^{5}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8622eba38f0b23cc458648c2f8bd226e7abdc3f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.386ex; height:2.843ex;" alt="{\displaystyle 443\cdot 2^{5}+1}"> </td> <td>14.177 </td></tr> <tr> <th rowspan="2">3 </th> <th rowspan="2">2 </th> <th rowspan="2">5 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2^{5}}+2^{2^{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2^{5}}+2^{2^{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c9c4acdc0e9b3dc401115f4f961bf7cc214cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 3^{2^{5}}+2^{2^{5}}}"> </th> <th rowspan="2">1.853.024.483.819.137 </th> <td>9 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6307d373fd56ced16e72ff5aeb87b0f0804af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 9\cdot 2^{7}+1}"> </td> <td>1.153 </td></tr> <tr> <td>3138931869 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3138931869\cdot 2^{9}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3138931869</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3138931869\cdot 2^{9}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8203ee90f4bb29193f738dffd048d33884360ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.523ex; height:2.843ex;" alt="{\displaystyle 3138931869\cdot 2^{9}+1}"> </td> <td>1.607.133.116.929 </td></tr> <tr> <th rowspan="3">3 </th> <th rowspan="3">2 </th> <th rowspan="3">6 </th> <th rowspan="3"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2^{6}}+2^{2^{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2^{6}}+2^{2^{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6ecb4cee1431026da49590de789b252350224c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 3^{2^{6}}+2^{2^{6}}}"> </th> <th rowspan="3">3.433.683.820.310.959.228.731.558.640.897 </th> <td>3 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d476681784d41d63691d3ee3c12149375ef7babc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{8}+1}"> </td> <td>769 </td></tr> <tr> <td>952341149 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 952341149\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>952341149</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 952341149\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3cf3b3bc2de5b216291667a342cc527cde661a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.361ex; height:2.843ex;" alt="{\displaystyle 952341149\cdot 2^{7}+1}"> </td> <td>121.899.667.073 </td></tr> <tr> <td>286168266760535 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 286168266760535\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>286168266760535</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 286168266760535\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f259ff25ff162d62bab6efe00054b3b72d0d35e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.335ex; height:2.843ex;" alt="{\displaystyle 286168266760535\cdot 2^{7}+1}"> </td> <td>36.629.538.145.348.481 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">1 </th> <th rowspan="2">4 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{4}}+1^{2^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{4}}+1^{2^{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c2628a094d09f0392f7fa84f2bedeee6e2c0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{4}}+1^{2^{4}}}"> </th> <th rowspan="2">4.294.967.297 </th> <td>5 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d3e7e954a01c1d2f1b8e5a117b566434f54ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{7}+1}"> </td> <td>641 </td></tr> <tr> <td>52347 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 52347\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>52347</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 52347\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e28bcad1647d7ddca3c12b69d033dfc51a28d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.711ex; height:2.843ex;" alt="{\displaystyle 52347\cdot 2^{7}+1}"> </td> <td>6.700.417 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">1 </th> <th rowspan="2">5 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{5}}+1^{2^{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{5}}+1^{2^{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f61851ce1d1eb65f303acff75f71f81fcac0762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{5}}+1^{2^{5}}}"> </th> <th rowspan="2">18.446.744.073.709.551.617 </th> <td>1071 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1071\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1071</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1071\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1a689078f22c1298e81ad2aff61c1858170dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle 1071\cdot 2^{8}+1}"> </td> <td>274.177 </td></tr> <tr> <td>262814145745 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 262814145745\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>262814145745</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 262814145745\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b452e79196ab8e7de8d2bbad92104f07a73937ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.848ex; height:2.843ex;" alt="{\displaystyle 262814145745\cdot 2^{7}+1}"> </td> <td>67.280.421.310.721 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">1 </th> <th rowspan="2">6 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{6}}+1^{2^{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{6}}+1^{2^{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e27123a32b05da18f03ebdc69dba0c04663ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{6}}+1^{2^{6}}}"> </th> <th rowspan="2">340.282.366.920.938.463.463.374.607.431.768.211.457 </th> <td>116503103764643 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 116503103764643\cdot 2^{9}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>116503103764643</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 116503103764643\cdot 2^{9}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7aec9b43ab150fa4701c4bee4a424b21838430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.335ex; height:2.843ex;" alt="{\displaystyle 116503103764643\cdot 2^{9}+1}"> </td> <td>59.649.589.127.497.217 </td></tr> <tr> <td>11141971095088142685 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11141971095088142685\cdot 2^{9}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>11141971095088142685</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11141971095088142685\cdot 2^{9}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1e66831b4602dd45c7768a18b448a3c972ae2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.148ex; height:2.843ex;" alt="{\displaystyle 11141971095088142685\cdot 2^{9}+1}"> </td> <td>5.704.689.200.685.129.054.721 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">3 </th> <th rowspan="2">1 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{1}}+3^{2^{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{1}}+3^{2^{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c42d8a5bf0e924ae27b95067f33c4fede81db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{1}}+3^{2^{1}}}"> </th> <th rowspan="2">25 </th> <td>1 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c358ec384245ecebbed71a745d6e1a2b3636a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{2}+1}"> </td> <td>5 </td></tr> <tr> <td>1 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c358ec384245ecebbed71a745d6e1a2b3636a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{2}+1}"> </td> <td>5 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">3 </th> <th rowspan="2">3 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{3}}+3^{2^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{3}}+3^{2^{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef720717fd48af67d097ff3b1374940701c58e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{3}}+3^{2^{3}}}"> </th> <th rowspan="2">72.097 </th> <td>1 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64df7df94ba2284768ae7025aef8ecaf48847ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{4}+1}"> </td> <td>17 </td></tr> <tr> <td>265 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 265\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>265</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 265\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353213e80da7aa5b4bc4d28ddf8ba7beba1b0d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.386ex; height:2.843ex;" alt="{\displaystyle 265\cdot 2^{4}+1}"> </td> <td>4.241 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">3 </th> <th rowspan="2">5 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{5}}+3^{2^{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{5}}+3^{2^{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a924c65eaaef86e51658bb8bc1733dd9c73956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{5}}+3^{2^{5}}}"> </th> <th rowspan="2">18.448.597.093.898.403.457 </th> <td>187 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 187\cdot 2^{6}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>187</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 187\cdot 2^{6}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd60f814b922c47d83784a7f41c80449efbafe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.386ex; height:2.843ex;" alt="{\displaystyle 187\cdot 2^{6}+1}"> </td> <td>11.969 </td></tr> <tr> <td>24083827353343 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 24083827353343\cdot 2^{6}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24083827353343</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24083827353343\cdot 2^{6}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/964696b2a5f097565b18a8e0a321db0b7417df28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.173ex; height:2.843ex;" alt="{\displaystyle 24083827353343\cdot 2^{6}+1}"> </td> <td>1.541.364.950.613.953 </td></tr> <tr> <th rowspan="2">4 </th> <th rowspan="2">3 </th> <th rowspan="2">6 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{6}}+3^{2^{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{6}}+3^{2^{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74b997f22b24e100de5670a5ed7d148c7331d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle 4^{2^{6}}+3^{2^{6}}}"> </th> <th rowspan="2">340.282.370.354.622.283.755.887.092.089.617.300.737 </th> <td>1317 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1317\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1317</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1317\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7484ddba8692806546268df19af6907d194d65d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle 1317\cdot 2^{8}+1}"> </td> <td>337.153 </td></tr> <tr> <td>492813355211781926870528348211 </td> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 492813355211781926870528348211\cdot 2^{11}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>492813355211781926870528348211</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 492813355211781926870528348211\cdot 2^{11}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7729ad301e118497995a1207710679545d19069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:43.594ex; height:2.843ex;" alt="{\displaystyle 492813355211781926870528348211\cdot 2^{11}+1}"> </td> <td>1.009.281.751.473.729.386.230.842.057.136.129 </td></tr> <tr> <th rowspan="2">5 </th> <th rowspan="2">1 </th> <th rowspan="2">3 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{3}}+1^{2^{3}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{3}}+1^{2^{3}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a6f12869c00eaffe707c1a5b95ee5e9c3d16cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.773ex; height:6.009ex;" alt="{\displaystyle {\frac {5^{2^{3}}+1^{2^{3}}}{2}}}"> </th> <th rowspan="2">195.313 </th> <td>1 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64df7df94ba2284768ae7025aef8ecaf48847ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{4}+1}"> </td> <td>17 </td></tr> <tr> <td>359 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 359\cdot 2^{5}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>359</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 359\cdot 2^{5}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f708598a5ed302dd50a0647c25055c6fa7a8c368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.386ex; height:2.843ex;" alt="{\displaystyle 359\cdot 2^{5}+1}"> </td> <td>11.489 </td></tr> <tr> <th rowspan="2">5 </th> <th rowspan="2">1 </th> <th rowspan="2">4 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{4}}+1^{2^{4}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{4}}+1^{2^{4}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d2f50c44d0ab17c9ffb91e0c7a53ebe21bc24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.773ex; height:6.009ex;" alt="{\displaystyle {\frac {5^{2^{4}}+1^{2^{4}}}{2}}}"> </th> <th rowspan="2">76.293.945.313 </th> <td>81 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 81\cdot 2^{5}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>81</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 81\cdot 2^{5}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b346109be81014af5ddb380d52f04e6c2ba82fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.224ex; height:2.843ex;" alt="{\displaystyle 81\cdot 2^{5}+1}"> </td> <td>2.593 </td></tr> <tr> <td>459735 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 459735\cdot 2^{6}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>459735</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 459735\cdot 2^{6}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7975d72aa4bc384d61de67915bec6c66266e78c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.873ex; height:2.843ex;" alt="{\displaystyle 459735\cdot 2^{6}+1}"> </td> <td>29.423.041 </td></tr> <tr> <th rowspan="3">5 </th> <th rowspan="3">1 </th> <th rowspan="3">5 </th> <th rowspan="3"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{5}}+1^{2^{5}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{5}}+1^{2^{5}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654d27ca51b958e4e606fbc6144ee47abfce884c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.773ex; height:6.009ex;" alt="{\displaystyle {\frac {5^{2^{5}}+1^{2^{5}}}{2}}}"> </th> <th rowspan="3">11.641.532.182.693.481.445.313 </th> <td>5 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d3e7e954a01c1d2f1b8e5a117b566434f54ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{7}+1}"> </td> <td>641 </td></tr> <tr> <td>1172953 </td> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1172953\cdot 2^{6}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1172953</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1172953\cdot 2^{6}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e6af88443aa0e8a9167fee981001e26099f6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.036ex; height:2.843ex;" alt="{\displaystyle 1172953\cdot 2^{6}+1}"> </td> <td>75.068.993 </td></tr> <tr> <td>945042975 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 945042975\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>945042975</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 945042975\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39cbf3f638fcdf30e23f86b75bbaaaf52a9c0737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.361ex; height:2.843ex;" alt="{\displaystyle 945042975\cdot 2^{8}+1}"> </td> <td>241.931.001.601 </td></tr> <tr> <th rowspan="3">5 </th> <th rowspan="3">1 </th> <th rowspan="3">6 </th> <th rowspan="3"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{6}}+1^{2^{6}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{6}}+1^{2^{6}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9601d7253193e30b50053ba6654a9c19d6d22d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.773ex; height:6.009ex;" alt="{\displaystyle {\frac {5^{2^{6}}+1^{2^{6}}}{2}}}"> </th> <th rowspan="3">271.050.543.121.376.108.501.863.200.217.485.427.856.445.313 (Zahl hat 45 (also abgerundet etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f89eb50e31c21a232c81e0c880681945a550fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{1}}">) Stellen) </th> <td>3 </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d476681784d41d63691d3ee3c12149375ef7babc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{8}+1}"> </td> <td>769 </td></tr> <tr> <td>28644528117 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 28644528117\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28644528117</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28644528117\cdot 2^{7}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b690616187a2a91f63e2a78f2dfd431010f64071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.686ex; height:2.843ex;" alt="{\displaystyle 28644528117\cdot 2^{7}+1}"> </td> <td>3.666.499.598.977 </td></tr> <tr> <td>187759681216101058498487625 </td> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 187759681216101058498487625\cdot 2^{9}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>187759681216101058498487625</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 187759681216101058498487625\cdot 2^{9}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/080e32d2b5a60934a57cc26bc776f86c25c44cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:39.285ex; height:2.843ex;" alt="{\displaystyle 187759681216101058498487625\cdot 2^{9}+1}"> </td> <td>96.132.956.782.643.741.951.225.664.001 </td></tr> <tr> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <td>… </td> <td>… </td> <td>… </td> <td>… </td></tr> <tr> <td>… </td> <td>… </td> <td>… </td> <td>… </td></tr> <tr> <th rowspan="2">12 </th> <th rowspan="2">11 </th> <th rowspan="2">37 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{37}}+11^{2^{37}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>37</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>37</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{37}}+11^{2^{37}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74cecff64247d29327572d3d665ac1da2360c76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.596ex; height:3.176ex;" alt="{\displaystyle 12^{2^{37}}+11^{2^{37}}}"> </th> <th rowspan="2">Zahl hat 148.321.541.064 (also etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{11}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ebdf600787afffb64c1bc390b75a982329a4e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.201ex; height:2.676ex;" alt="{\displaystyle 10^{11}}">) Stellen </th> <td>1776222707793 </td> <td>38 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1776222707793\cdot 2^{38}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1776222707793</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>38</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1776222707793\cdot 2^{38}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214ae9fd3022794ccb845b7fa57d547e5d58b52b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.833ex; height:2.843ex;" alt="{\displaystyle 1776222707793\cdot 2^{38}+1}"> </td> <td>488.244.380.184.543.957.614.593 </td></tr> <tr> <th colspan="4">und noch ein Faktor, von dem man nicht weiß, ob er <a href="/wiki/Zusammengesetzte_Zahl" title="Zusammengesetzte Zahl">zusammengesetzt</a> ist oder nicht </th></tr> <tr> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <th rowspan="2">… </th> <td>… </td> <td>… </td> <td>… </td> <td>… </td></tr> <tr> <td>… </td> <td>… </td> <td>… </td> <td>… </td></tr> <tr> <th rowspan="2">12 </th> <th rowspan="2">11 </th> <th rowspan="2">7033640 </th> <th rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{7033640}}+11^{2^{7033640}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7033640</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7033640</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{7033640}}+11^{2^{7033640}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa346d0905ae2351d5c9dd272a5ccc589ce69e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.27ex; height:3.176ex;" alt="{\displaystyle 12^{2^{7033640}}+11^{2^{7033640}}}"> </th> <th rowspan="2">Zahl hat etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2117337}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2117337</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2117337}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8165b37482d70283d5f3da2d631a3382a83bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.311ex; height:2.676ex;" alt="{\displaystyle 10^{2117337}}"> Stellen </th> <td>3 </td> <td>7033641 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2^{7033641}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7033641</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{7033641}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a42ecdcc43838b4c85b41ab217d369ce2aeda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.993ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{7033641}+1}"> </td> <td>Primteiler hat 2117338 Stellen </td></tr> <tr> <th colspan="4">und noch ein Faktor, von dem man nicht weiß, ob er <a href="/wiki/Zusammengesetzte_Zahl" title="Zusammengesetzte Zahl">zusammengesetzt</a> ist oder nicht </th></tr></tbody></table> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Verallgemeinerte_Fermatsche_Zahlen_der_Form_Fn(b)">Verallgemeinerte Fermatsche Zahlen der Form Fn(b)</h3>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=9" title="Abschnitt bearbeiten: Verallgemeinerte Fermatsche Zahlen der Form Fn(b)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerte Fermatsche Zahlen der Form Fn(b)">Quelltext bearbeiten</a>]</div> Ist b eine <a href="/wiki/Parit%C3%A4t_(Mathematik)" title="Parität (Mathematik)">gerade</a> Zahl, so kann Fn(b) sowohl zusammengesetzt als auch prim sein. Beispiel 1: <dl><dd>b = 8, n = 3 ergibt die zusammengesetzte Zahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}(8)=8^{2^{3}}+1=8^{8}+1=16.777.217=97\cdot 257\cdot 673}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>16.777.217</mn> <mo>=</mo> <mn>97</mn> <mo>⋅</mo> <mn>257</mn> <mo>⋅</mo> <mn>673</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}(8)=8^{2^{3}}+1=8^{8}+1=16.777.217=97\cdot 257\cdot 673}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eedb2bcafa578a4ebf3f86f3e30db57bc8469b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.436ex; height:3.509ex;" alt="{\displaystyle F_{3}(8)=8^{2^{3}}+1=8^{8}+1=16.777.217=97\cdot 257\cdot 673}">.</dd></dl></dd></dl> Beispiel 2: <dl><dd>b = 6, n = 2 ergibt die Primzahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}(6)=6^{2^{2}}+1=6^{4}+1=1297}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1297</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}(6)=6^{2^{2}}+1=6^{4}+1=1297}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108508bbd8ce5a001ba326341760157f183fa4cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.736ex; height:3.509ex;" alt="{\displaystyle F_{2}(6)=6^{2^{2}}+1=6^{4}+1=1297}">.</dd></dl></dd></dl> Beispiel 3: <dl><dd>b = 30, n = 5 ergibt die 48-stellige Primzahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}(30)=30^{2^{5}}+1=30^{32}+1=185.302.018.885.184.100.000.000.000.000.000.000.000.000.000.001}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>30</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>185.302.018.885.184.100.000.000.000.000.000.000.000.000.000.001</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}(30)=30^{2^{5}}+1=30^{32}+1=185.302.018.885.184.100.000.000.000.000.000.000.000.000.000.001}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2115e0cbd8bc37c20828e6f4a2b8d7cc691b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:97.896ex; height:3.509ex;" alt="{\displaystyle F_{5}(30)=30^{2^{5}}+1=30^{32}+1=185.302.018.885.184.100.000.000.000.000.000.000.000.000.000.001}"></dd></dl></dd> <dd>und ist gleichzeitig die kleinste verallgemeinerte Fermatsche Primzahl mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b13dc8b113121cdaf76a723a61aa4f8be1468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>4}">.</dd></dl> Ist b eine <a href="/wiki/Parit%C3%A4t_(Mathematik)" title="Parität (Mathematik)">ungerade</a> Zahl, so ist Fn(b) als Summe einer Potenz einer ungeraden Zahl (die selbst wieder ungerade ist) und 1 immer eine gerade Zahl, somit durch 2 teilbar und deshalb für b > 1 keine Primzahl, sondern zusammengesetzt. In diesem Fall wird häufig die Zahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{n}(b)}{2}}={\frac {b^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{n}(b)}{2}}={\frac {b^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc80e48305c6e6d59b6039e5ab095b4ff8f3416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.31ex; height:5.676ex;" alt="{\displaystyle {\frac {F_{n}(b)}{2}}={\frac {b^{2^{n}}+1}{2}}}"></dd></dl> auf ihre <a href="/wiki/Primzahl" title="Primzahl">Primalität</a> untersucht. Diese Zahlen werden auch halbe verallgemeinerte Fermatsche Zahlen genannt. Beispiel 4: <dl><dd>b = 3, n = 2 ergibt die gerade und somit zusammengesetzte Zahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}(3)=3^{2^{2}}+1=3^{4}+1=81+1=82=2\cdot 41}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>81</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>82</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>41</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}(3)=3^{2^{2}}+1=3^{4}+1=81+1=82=2\cdot 41}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be4c8fe8251befd918541c38ec23e1e800422b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.103ex; height:3.509ex;" alt="{\displaystyle F_{2}(3)=3^{2^{2}}+1=3^{4}+1=81+1=82=2\cdot 41}">.</dd></dl></dd> <dd>Es ist aber <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{2}(3)}{2}}={\frac {3^{2^{2}}+1}{2}}={\frac {82}{2}}=41}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>82</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>41</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{2}(3)}{2}}={\frac {3^{2^{2}}+1}{2}}={\frac {82}{2}}=41}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/809883890aed2b4254f1687ca31165aefbc3947d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.025ex; height:6.009ex;" alt="{\displaystyle {\frac {F_{2}(3)}{2}}={\frac {3^{2^{2}}+1}{2}}={\frac {82}{2}}=41}"></dd></dl></dd> <dd>eine Primzahl.</dd></dl> Beispiel 5: <dl><dd>b = 5, n = 3 ergibt die gerade und somit zusammengesetzte Zahl <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}(5)=5^{2^{3}}+1=5^{8}+1=390625+1=390626=2\cdot 17\cdot 11489.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>390625</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>390626</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>17</mn> <mo>⋅</mo> <mn>11489.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}(5)=5^{2^{3}}+1=5^{8}+1=390625+1=390626=2\cdot 17\cdot 11489.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e27759fb348566b57955e25984a1ed4bc7348b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.541ex; height:3.509ex;" alt="{\displaystyle F_{3}(5)=5^{2^{3}}+1=5^{8}+1=390625+1=390626=2\cdot 17\cdot 11489.}"></dd></dl></dd> <dd>Es ist aber <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{3}(5)}{2}}={\frac {5^{2^{3}}+1}{2}}={\frac {390626}{2}}=195313=17\cdot 11489}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>390626</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>195313</mn> <mo>=</mo> <mn>17</mn> <mo>⋅</mo> <mn>11489</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{3}(5)}{2}}={\frac {5^{2^{3}}+1}{2}}={\frac {390626}{2}}=195313=17\cdot 11489}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc26591630a2a413f3d74930d2e59e362c0a2fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.24ex; height:6.009ex;" alt="{\displaystyle {\frac {F_{3}(5)}{2}}={\frac {5^{2^{3}}+1}{2}}={\frac {390626}{2}}=195313=17\cdot 11489}"></dd></dl></dd> <dd>eine zusammengesetzte Zahl.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Liste_der_Primzahlen_der_Form_Fn(b)">Liste der Primzahlen der Form Fn(b)</h3>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=10" title="Abschnitt bearbeiten: Liste der Primzahlen der Form Fn(b)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Liste der Primzahlen der Form Fn(b)">Quelltext bearbeiten</a>]</div> Verallgemeinerte Fermatsche Zahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)=b^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155b271d27e4d7b9dd02718fdb9c20273ee64a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:3.176ex;" alt="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> (für gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">) bzw. der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d383969cf9781f4ae185207991540af6ca4706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.349ex; height:4.343ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}"> (für ungerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">) sind in den meisten Fällen zusammengesetzt. Weil diese Zahlen sehr schnell sehr groß werden, sind nicht besonders viele Primzahlen dieser Art bekannt. Es folgt eine Auflistung von Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq 60}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mn>60</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq 60}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e16156d79617517d01e9aac8b2da759ff8f842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.421ex; height:2.343ex;" alt="{\displaystyle b\leq 60}">: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der verallgemeinerten Fermatschen Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)=b^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155b271d27e4d7b9dd02718fdb9c20273ee64a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:3.176ex;" alt="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> bzw. der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d383969cf9781f4ae185207991540af6ca4706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.349ex; height:4.343ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}={\tfrac {b^{2^{n}}+1}{2}}}"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq 60}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mn>60</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq 60}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e16156d79617517d01e9aac8b2da759ff8f842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.421ex; height:2.343ex;" alt="{\displaystyle b\leq 60}"></div> <div class="NavContent"> <table class="toptextcells"> <tbody><tr> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> prim ist </th></tr> <tr> <td>0 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{2^{n}}+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{2^{n}}+1=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9afd6b789d835c6fee7c128b2ad9ebbcbd7cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.445ex; height:2.843ex;" alt="{\displaystyle 0^{2^{n}}+1=1}"> </td> <td style="text-align:left; background:lightgrey">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1^{2^{n}}+1}{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1^{2^{n}}+1}{2}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13924b0b1eddafc9d49c9413fd9b96d756548b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.281ex; height:5.843ex;" alt="{\displaystyle {\frac {1^{2^{n}}+1}{2}}=1}"> </td> <td style="text-align:left; background:lightgrey">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27f57a4191be088259902a790ef2fb093ffb812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 2^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, 2, 3, 4, … </td></tr> <tr> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99291771a49020860e009467ee22780bc572f4d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.021ex; height:5.843ex;" alt="{\displaystyle {\frac {3^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 1, 2, 4, 5, 6, … </td></tr> <tr> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5207c6d921850cd8f590b9e799eeafa110f303c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 4^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, 2, 3, … </td></tr> <tr> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0372c358f1556bc5d5b33087304b9634dad4170f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.021ex; height:5.843ex;" alt="{\displaystyle {\frac {5^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f382e5927cfff43a8792fc1e366b75e734b2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 6^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4e0ab918322a6230f744d891f658d43350d475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.021ex; height:5.843ex;" alt="{\displaystyle {\frac {7^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ed284106656b81a36c5c5ee814da4875ed7b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 8^{2^{n}}+1}"> </td> <td style="text-align:left; background:lightgrey">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {9^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e896ee4260c8436af94a748df0ae1255c2f093f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.021ex; height:5.843ex;" alt="{\displaystyle {\frac {9^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 1, 3, 4, 5, … </td></tr> <tr> <td>10 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b776e9ec49df29bf8a28214303b2ff0380bc26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 10^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20cfd911587c6ea675f49b8465a51aeea6abf1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {11^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>12 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4d195fa7d029accd4571ba77b779340ec7b360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 12^{2^{n}}+1}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5b680dd4093f8f37dbd2a66c988aa5c2d35d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {13^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 2, 3, … </td></tr> <tr> <td>14 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc0d284c1c6f369fddd40fc5a61628cc1829b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 14^{2^{n}}+1}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>15 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {15^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {15^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98fe93128e75a14344cf1d94209a7d3113bc1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {15^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9436a086720db7d2a1b424cb1d1e00317b0dd678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.356ex; height:5.676ex;" alt="{\displaystyle {\frac {F_{n}(b)}{2}}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9436a086720db7d2a1b424cb1d1e00317b0dd678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.356ex; height:5.676ex;" alt="{\displaystyle {\frac {F_{n}(b)}{2}}}"> prim ist </th></tr> <tr> <td>16 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f06de704a7724a762a921fc132e20592fe807e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 16^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, 2, … </td></tr> <tr> <td>17 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {17^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>17</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {17^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ba030834fd77940ac97a09bebacdf080d80620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {17^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>18 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eade296523270c69ddc53ff93b480ed0d8ab06c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 18^{2^{n}}+1}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>19 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {19^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>19</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {19^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bc02b0221878df3176033fb50eb7a1807446ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {19^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>20 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/151ffb141ce8779b6304dd496a54d39a3bb5c99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 20^{2^{n}}+1}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>21 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {21^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>21</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {21^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129f0539a30382abbd24c3570865f1dedab18fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {21^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 2, 5, … </td></tr> <tr> <td>22 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 22^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>22</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 22^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7ecbac9f0b0d3dad62fcf4bddfc25282f93d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 22^{2^{n}}+1}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>23 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {23^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>23</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {23^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bef01ecbb660dded3008891f57aef20ff04a4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {23^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>24 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 24^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>24</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d867deb744f1c08af8bcac097e4717f9c9df781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 24^{2^{n}}+1}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>25 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {25^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>25</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {25^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c349812bbc56247f2ec3c0e28cf7c195d41526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {25^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>26 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 26^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>26</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 26^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23ba61b5bbe48278732e554ab1c54854e1b8780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 26^{2^{n}}+1}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>27 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {27^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {27^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34811f53dbba7240db115fc59d5a6499fe9598c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {27^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left; background:lightgrey">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>28 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 28^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>28</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828bc39747226eefe2650583e4d62d86c1d46e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 28^{2^{n}}+1}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>29 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {29^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>29</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {29^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2fce1eb69fefce7aa99c6e4caf81f737fd9618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {29^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, 4, … </td></tr> <tr> <td>30 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 30^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1112633e449ef3f7c31892fad4d03475d18609b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 30^{2^{n}}+1}"> </td> <td style="text-align:left">0, 5, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> prim ist </th></tr> <tr> <td>31 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {31^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>31</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {31^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d907ad7dd114922e8ea0cf2f99aaf033d5c67fac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {31^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left; background: yellow">noch keine bekannt </td></tr> <tr> <td>32 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 32^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07ffef567e350697e1eb09b9a8437d580323f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 32^{2^{n}}+1}"> </td> <td style="text-align:left; background:lightgrey">es gibt keine Primzahlen dieser Form </td></tr> <tr> <td>33 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {33^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>33</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {33^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877d7445a6de5bb1425b1effb4199da625b27d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {33^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 3, … </td></tr> <tr> <td>34 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 34^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>34</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 34^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824c7bb59939722424d5af67bccf3622c8fa32f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 34^{2^{n}}+1}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>35 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {35^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>35</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {35^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6bb4f9e89d53c61da26eb441567b87a39421e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {35^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, 6, … </td></tr> <tr> <td>36 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>36</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66970f79c169cce43d370f1db0e147a548bd30e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 36^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>37 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {37^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>37</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {37^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a18a007e17a686863df5bd57ab68502f21a5f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {37^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>38 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 38^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>38</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 38^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d38a0967b860e58dd56eba4909508f07e79b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 38^{2^{n}}+1}"> </td> <td style="text-align:left; background: yellow">noch keine bekannt </td></tr> <tr> <td>39 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {39^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>39</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {39^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a4f2f902dc72f72bc4b5b9297beb0c39aa9067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {39^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>40 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 40^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>40</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 40^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62c11c448101d5fec714333dfb3ccb5263b3081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 40^{2^{n}}+1}"> </td> <td style="text-align:left">0, 1, … </td></tr> <tr> <td>41 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {41^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>41</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {41^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be0291233a365992a32ec60e0aa0ab6bc0cefbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {41^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">4, … </td></tr> <tr> <td>42 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 42^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>42</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 42^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4566392c1982bfc069c42e3057e942127dee85e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 42^{2^{n}}+1}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>43 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {43^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>43</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {43^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298b6b29105c3faf4db3f54f73a35478d853959f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {43^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">3, … </td></tr> <tr> <td>44 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 44^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>44</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 44^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e73200054237907a72262a2a9819deb76a242a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 44^{2^{n}}+1}"> </td> <td style="text-align:left">4, … </td></tr> <tr> <td>45 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {45^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {45^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadf799cbfb5a0ae3e0c7bf639fa07b18a8732b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {45^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 1, … </td></tr></tbody></table> </td> <td> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>b</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> prim ist </th></tr> <tr> <td>46 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {46^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>46</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {46^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38178eb4dde31caa087486e4b98595c1a4f3091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {46^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, 2, 9, … </td></tr> <tr> <td>47 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 47^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>47</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 47^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91cf9037f71adabd1fcd48b4c3e4524eb55f004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 47^{2^{n}}+1}"> </td> <td style="text-align:left">3, … </td></tr> <tr> <td>48 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {48^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>48</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {48^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8a6b62b1f1bab7889528b846e5bd2d8a59fb9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {48^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">2, … </td></tr> <tr> <td>49 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 49^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 49^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655fe235957146d22c35af62e4756f4e82cf9245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 49^{2^{n}}+1}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>50 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {50^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {50^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/487e3a87fec995c9a13429efa7862f74611f80bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {50^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left; background: yellow">noch keine bekannt </td></tr> <tr> <td>51 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 51^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>51</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 51^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf862b118a2e6a2ba12f9bd6ec230ec5f2c94b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 51^{2^{n}}+1}"> </td> <td style="text-align:left">1, 3, 6, … </td></tr> <tr> <td>52 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {52^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>52</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {52^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78557dd4ce901c6afba875e6c127aca7aa7585d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {52^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>53 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 53^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>53</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 53^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f225eef60fece9d201f5237d845b03d30aceaeb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 53^{2^{n}}+1}"> </td> <td style="text-align:left">3, … </td></tr> <tr> <td>54 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {54^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>54</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {54^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3541fe6d8616c3425a37daa638b7689d4717da89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {54^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, 5, … </td></tr> <tr> <td>55 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 55^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>55</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 55^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ed9c6ab2e96430f573b217d42aeb65a396abf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 55^{2^{n}}+1}"> </td> <td style="text-align:left; background: yellow">noch keine bekannt </td></tr> <tr> <td>56 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {56^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>56</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {56^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93da18ebba79d00dcb7de402af27f5493d6bdf11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {56^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">1, 2, … </td></tr> <tr> <td>57 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 57^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>57</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 57^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3156deeffb7caedf5447582dbdeec8bec81af5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 57^{2^{n}}+1}"> </td> <td style="text-align:left">0, 2, … </td></tr> <tr> <td>58 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {58^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>58</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {58^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203c4d6b5edf5898d998c51da5b440ec7e41117d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {58^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, … </td></tr> <tr> <td>59 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 59^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>59</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 59^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef24355b16c2e3c9c31ada32e4fa8f41bbf3191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.347ex; height:2.843ex;" alt="{\displaystyle 59^{2^{n}}+1}"> </td> <td style="text-align:left">1, … </td></tr> <tr> <td>60 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {60^{2^{n}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {60^{2^{n}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa790657802f683c87311003a391fd4ebdfff320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.183ex; height:5.843ex;" alt="{\displaystyle {\frac {60^{2^{n}}+1}{2}}}"> </td> <td style="text-align:left">0, … </td></tr></tbody></table> </td></tr></tbody></table> </div> </div> Die kleinsten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> (ab <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32584049ed5f72969777f89d69b74ee462875e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=2}">), für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {F_{n}(b)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc9f70bce6c80fd8f01300506ba2570c4f0aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:4.176ex;" alt="{\displaystyle {\tfrac {F_{n}(b)}{2}}}"> erstmals eine Primzahl ergibt, kann man der obigen Tabelle entnehmen, was für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq 1500}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mn>1500</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq 1500}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3344b3405df9d6fcfc1ee78d2bb553d2cc351d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.746ex; height:2.343ex;" alt="{\displaystyle b\leq 1500}"> die folgende Liste ergibt (der Wert −1 bedeutet „nicht existent“ bzw. „noch keine bekannt“): <dl><dd>0, 0, 0, 0, 0, 2, −1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, −1, 0, 1, 0, −1, −1, 0, 2, 1, 0, 0, −1, 1, 0, 4, 0, 3, 4, 0, 0, 3, 2, 1, −1, 1, 0, 3, 1, −1, 1, 0, 0, 1, 0, … (Folge <a href="//oeis.org/A253242" class="extiw" title="oeis:A253242">A253242</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> Mehr Informationen für gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> bis zur Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1000}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9176f70403c87186d46f8c739d36fe9db038fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.746ex; height:2.176ex;" alt="{\displaystyle b=1000}"> findet man im Internet.<a href="#cite_note-40">[40]</a> Nun folgt eine Auflistung von Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Liste der verallgemeinerten Fermatschen Primzahlen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)=b^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155b271d27e4d7b9dd02718fdb9c20273ee64a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:3.176ex;" alt="{\displaystyle F_{n}(b)=b^{2^{n}}+1}"> mit konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></div> <div class="NavContent"> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>n</th> <th>Fn(b)</th> <th>b, für die Fn(b) prim ist</th> <th><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>-Folge </th></tr> <tr> <td>0 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85850bdef4531128cb1e0f6847923948f33bbf6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5ex; height:2.343ex;" alt="{\displaystyle b+1}"> </td> <td style="text-align:left">1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, … (alle Primzahlen minus 1) </td> <td style="text-align:center">(Folge <a href="//oeis.org/A006093" class="extiw" title="oeis:A006093">A006093</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40535a246808369904baa0fc3231a9e454899c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.055ex; height:2.843ex;" alt="{\displaystyle b^{2}+1}"> </td> <td style="text-align:left">1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A005574" class="extiw" title="oeis:A005574">A005574</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{4}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{4}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4dd12dec5513fbf888d365383e20aef9ff063ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.055ex; height:2.843ex;" alt="{\displaystyle b^{4}+1}"> </td> <td style="text-align:left">1, 2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, 238, 242, 248, 254, 266, 272, 276, 278, 288, 296, 312, 320, 328, 334, 340, 352, 364, 374, 414, 430, 436, 442, 466, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A000068" class="extiw" title="oeis:A000068">A000068</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{8}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ad5671680d321f95eaea8250f314b2def99cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.055ex; height:2.843ex;" alt="{\displaystyle b^{8}+1}"> </td> <td style="text-align:left">1, 2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, 800, 808, 866, 876, 884, 892, 916, 918, 934, 956, 990, 1022, 1028, 1054, 1106, 1120, 1174, 1224, 1232, 1256, 1284, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A006314" class="extiw" title="oeis:A006314">A006314</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{16}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{16}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771404c4efc99e71c7535d3861c1d7ddd963e82d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle b^{16}+1}"> </td> <td style="text-align:left">1, 2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, 686, 688, 690, 736, 774, 776, 778, 790, 830, 832, 834, 846, 900, 916, 946, 956, 972, 982, 984, 1018, 1044, 1078, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A006313" class="extiw" title="oeis:A006313">A006313</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{32}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{32}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f81de59d6c226cf41d854446af7271c97908cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle b^{32}+1}"> </td> <td style="text-align:left">1, 30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, 1596, 1678, 1714, 1754, 1812, 1818, 1878, 1906, 1960, 1962, 2046, 2098, 2118, 2142, 2330, 2418, 2434, 2654, 2668, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A006315" class="extiw" title="oeis:A006315">A006315</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>6 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{64}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>64</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{64}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd5a6696b55412b527c85f4f3fb7d489db035fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle b^{64}+1}"> </td> <td style="text-align:left">1, 102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, 2420, 2494, 2524, 2614, 2784, 3024, 3104, 3140, 3164, 3254, 3278, 3628, 3694, 3738, 3750, 4000, 4030, 4058, 4166, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A006316" class="extiw" title="oeis:A006316">A006316</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{128}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>128</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{128}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ad4146bfe6e22916146d900af6ea907003cfe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.699ex; height:2.843ex;" alt="{\displaystyle b^{128}+1}"> </td> <td style="text-align:left">1, 120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, 9706, 10238, 10994, 11338, 11432, 11466, 11554, 11778, 12704, 12766, 13082, 13478, 13700, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A056994" class="extiw" title="oeis:A056994">A056994</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{256}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>256</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{256}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1885d5247432f38e094e6e49f556781d961aef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.699ex; height:2.843ex;" alt="{\displaystyle b^{256}+1}"> </td> <td style="text-align:left">1, 278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, 8524, 8644, 8762, 8808, 9024, 9142, 9412, 10892, 12206, 13220, 13222, 13246, 13370, 13738, 14114, 14930, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A056995" class="extiw" title="oeis:A056995">A056995</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>9 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{512}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>512</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{512}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f31fc9d35d11d9d42d2d19901eedadef77414782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.699ex; height:2.843ex;" alt="{\displaystyle b^{512}+1}"> </td> <td style="text-align:left">1, 46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, 8678, 8792, 9448, 9452, 9972, 10086, 10448, 10926, 11468, 12754, 13198, 13776, 14734, 16826, 16914, 18334, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A057465" class="extiw" title="oeis:A057465">A057465</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>10 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{1024}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1024</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{1024}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176969621de699884854374f12acd1c8b9d6d4bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle b^{1024}+1}"> </td> <td style="text-align:left">1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A057002" class="extiw" title="oeis:A057002">A057002</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>11 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2048}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2048</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2048}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89fd65c22a7a94a4244c5f36f538a421358ae57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle b^{2048}+1}"> </td> <td style="text-align:left">1, 150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, 46502, 47348, 49190, 49204, 49544, 54514, 57210, 59770, 61184, 66894, 68194, 70574, 72446, 82642, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A088361" class="extiw" title="oeis:A088361">A088361</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>12 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{4096}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4096</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{4096}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7822e78cec1dc627aaead627a4dd2a2fe0dec6da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle b^{4096}+1}"> </td> <td style="text-align:left">1, 1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, 110540, 114690, 125440, 125442, 127596, 138068, 144362, 154908, 157310, 161822, 161900, 166224, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A088362" class="extiw" title="oeis:A088362">A088362</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{8192}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8192</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{8192}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee6339619bd9cd72c1c0ebadc1ffa59b2f8bae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle b^{8192}+1}"> </td> <td style="text-align:left">1, 30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, 241860, 248744, 268032, 270674, 302368, 316970, 326260, 347962, 350830, 397468, 410938, 416010, 441238, 443718, 458520, 462678, 463012, 475158, 481750, …, 352666770, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A226528" class="extiw" title="oeis:A226528">A226528</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>14 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{16384}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16384</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{16384}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6676a8fe14b06daebca0be8064dd05bf84648a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.343ex; height:2.843ex;" alt="{\displaystyle b^{16384}+1}"> </td> <td style="text-align:left">1, 67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, 509622, 528614, 572934, 581424, 638980, 641762, 656210, 698480, 704930, 730352, 795810, 840796, 908086, 975248, 976914, 990908, 1007874, 1037748, 1039970, 1067896, 1082054, 1097352, 1102754, 1132526, 1162996, 1171010, 1177808, 1181388, …, 10841645805132531666786792405311319418846637043199917731311876, … </td> <td style="text-align:center">(Folge <a href="//oeis.org/A226529" class="extiw" title="oeis:A226529">A226529</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>15 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{32768}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32768</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{32768}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1459135002066ece4919876b388d1d28c618c1ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.343ex; height:2.843ex;" alt="{\displaystyle b^{32768}+1}"> </td> <td style="text-align:left">1, 70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, 1074542, 1096382, 1113768, 1161054, 1167528, 1169486, 1171824, 1210354, 1217284, 1277444, 1519380, 1755378, 1909372, 1922592, 1986700, 2034902, 2147196, 2167350, …, 3292665455999520712131951642528, …<a href="#cite_note-F15Prime-41">[41]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A226530" class="extiw" title="oeis:A226530">A226530</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>16 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{65536}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>65536</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{65536}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5607e5ed784335d2f8cb6481006d6ca380640460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.343ex; height:2.843ex;" alt="{\displaystyle b^{65536}+1}"> </td> <td style="text-align:left">1, 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, 1266062, 1361846, 1374038, 1478036, 1483076, 1540550, 1828502, 1874512, 1927034, 1966374, 2019300, 2041898, 2056292, 2162068, 2177038, 2187182, 2251082, 2313394, …, 1814570322984178, …<a href="#cite_note-F16Prime-42">[42]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A251597" class="extiw" title="oeis:A251597">A251597</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>17 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{131072}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>131072</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{131072}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d3bde1a4a79aebe97ed30765e1af09601ff39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.165ex; height:2.843ex;" alt="{\displaystyle b^{131072}+1}"> </td> <td style="text-align:left">1, 62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, 1955556, 2194180, 2280466, 2639850, 3450080, 3615210, 3814944, 4085818, 4329134, 4893072, 4974408, 5326454, 5400728, 5471814, 5586416, 5734100, 5877582, 6391936, 6403134, 6705932, 7379442, 7832704, 7858180, 7926326, 8150484, 8704114, 8770526, 9240606, 9419976, 9785844, 9907326, 10037266, 10368632, 10453790, 10765720, 10921162, 10962066, 10994460, 11036888, 11195602, 11267296, 11292782, 11778792, 11876066, 12343130, 12357518, 12512992, 12661786, 12687374, 12851074, 12961862, 12978952, 13083418, 13433028, 13613070, 13800346, 14020004, 14217182, 14613898, 14790404, 15091270, 15147290, 15237960, 15342502, 15567144, 15667716, 15731520, 16329572, 16741226, 16985784, 17025822, 17052490, 17119936, 17138628, 17141888, 17643330, 17814792, 17958952, 18298534, 18309468, 18501600, 18509226, 18608780, 18813106, 18968126, 19216648, 19464034, 20185276, 20227142, 20234282, 20674450, 20968936, 21517658, 21582550, 21757066, 21869554, 21917442, 22007146, 22705306, 22718284, 22808110, 22901508, 22980158, 23011666, 23045178, 23363426, 24297936, 24486806, 24522386, 24641166, 24642712, 24644826, 24734116, 25124378, 25128150, 26172278, 26500832, 26599558, 26757382, 26896670, 27022768, 27408050, 27544748, 27557876, 27758510, 27822108, 28175634, 28294666, 28398204, 28497098, 29169314, 29505368, 29607314, 29959190, 30022816, 30059800, 30225714, 30300414, 30315072, 30318724, 30819256, 30844300, 31044982, 31145080, 31469984, 31768014, 31821360, 32055422, 32096608, 32137342, 32200644, 32286660, 32348894, 32417420, 32430486, 32608738, 32704348, 32792696, 32869172, 33191418, 33395198, 33474284, 33732746, 34087952, 34530386, 34585314, 34763644, 34871942, 34957136, 35047222, 35139782, 35141602, 35282096, 35327718, 35372304, 35391288, 35957420, 35997532, 36038176, 36416848, 36422846, 36531196, 37909914, 38152876, 38196496, 38310998, 38734748, 38824296, 39100746, 39324372, 39502358, 39597790, 39746366, 40151896, 40463598, 40550398, 41001148, 41007562, 41102236, 41237116, 41364744, 41688706, 42168978, 42230406, 42243204, 42254832, 42414020, 42550702, 42654182, 43163894, 43165206, 44049878, 44085096, 44330870, 44438760, 44919410, 45315256, 45570624, 46077492, 46371508, 46385310, 46413358, 46730280, 46736070, 46776558, 47090246, 47179704, 48273828, 48370248, 48643706, 49038514, 49090656, 49225986, 49243622, 49331672, 49397682, 49530004, 49817700, 50055102, 50110436, 50217306, 50495632, 50844724, 50963598, 51269192, 51567684, 51570250, 51580416, 51872628, 51954384, 52043532, 52412612, 52712138, 53078434, 53161266, 53659976, 54032538, 54161106, 54206254, 54212352, 54334044, 54361742, 54548788, 55015050, 55184170, 55268442, 55579418, 55645700, 56307420, 56383242, 56459558, 56584816, 56735576, 56917336, 57438404, 57594734, 57694224, 57704312, 58247118, 58447642, 58447816, 58523466, 58589880, 59161754, 59210784, 59305348, 59362002, 59405420, 59515830, 59692546, 59720358, 60133106, 60455792, 60540024, 60642326, 61030988, 61267078, 61837354, 62146946, 62276102, 63112418, 63165756, 63168480, 63823568, 64024604, 64476916, 64506894, 64568930, 64791668, 64911056, 65200798, 65305572, 65569854, 65791182, 66131722, 66272848, 66901180, 66982940, 67371416, 67725850, 67894288, 68275006, 68372810, 68536972, 68811158, 68918852, 68924112, 68999820, 69534788, 69565722, 69622572, 69689592, 69742382, 69915032, 70022042, 70050828, 70421038, 70658696, 70893680, 70934282, 70948704, 70960658, 71450224, 71679108, 71732900, 72070092, 72602370, 73099962, 73132228, 73160610, 73404316, 73690464, 73839292, 74325990, 74363146, 74381296, 74396818, 74817490, 74833516, 75521414, 75647276, 75861530, 76018874, 76026988, 76416048, 77281404, 77469882, 77918854, 77924964, 78089172, 78240016, 78439440, 78714954, 78851276, 78880690, 78910032, 79201682, 79383608, 79428414, 79485098, 79789806, 79801426, 79912550, 80146408, 80284312, 81096098, 81444036, 81477176, 81976506, 82003030, 82008736, 83003850, 83328182, 83364886, 84149050, 84384358, 84445014, 84679936, 84715930, 84723284, 84757790, 84765338, 84817722, 84924212, 85115888, 86060696, 86295564, 86347638, 86413544, 86829162, 87039658, 87116452, 87192538, 87268788, 87352356, 87370574, 87454694, 87547832, 87920992, 88068088, 88166868, 88243020, 88760062, 89113896, 89285798, 89790434, 89977312, 90006846, 90382348, 90857490, 90938686, 90942952, 91033554, 91049202, 91069366, 91655310, 91685784, 91689894, 91707732, 91767880, 92198216, 92460588, 93035888, 93514592, 93773904, 93886318, 93950924, 94978760, 95308284, 95596816, 95635202, 95940796, 96111850, 96475576, 96734274, 96821302, 97046574, 97512766, 98137862, 98200338, 98240694, 98518362, 98557818, 98652282, 98922946, 98978354, 99189780, 99351950, 99557826, 99650934, 99665972, 100010426, 100324226, 100369508, 100382228, 100441116, 100520930, 100534258, 100719472, 100865034, 101270816, 101328382, 101607438, 101856256, 101915106, 102021074, 102050324, 102257714, 102397132, 102469684, 102507732, 103013294, 103094212, 103209792, 103280694, 103289324, 103605376, 103828182, 108584736, 108581414, 108195632, 108161744, 108080390, 107979316, 107922308, 107732730, 107627678, 107492880, 107420312, 107404768, 107222132, 107126228, 106901434, 106508704, 106440698, 106019242, 105937832, 105861526, 105850338, 105534478, 105058710, 104907548, 104808996, 104641854, …, 271643232, 314187728, 399866798, …<a href="#cite_note-F17Prime-43">[43]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A253854" class="extiw" title="oeis:A253854">A253854</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>18 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{262144}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>262144</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{262144}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a504f90bf769a4deb932f5ad5eb889031e69c3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.165ex; height:2.843ex;" alt="{\displaystyle b^{262144}+1}"> </td> <td style="text-align:left">1, 24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, 3547726, 3596074, 3673932, 3853792, 3933508, 4246258, 4489246, 5152128, 5205422, 5828034, 6287774, 6291332, 8521794, 8883864, 9125820, 9450844, 9750938, 9812766, 10578478, 10578478, 10578478, 10627360, 10793312, 10829576, 10979776, 11081688, 12189878, 12304152, 12529818, 12582496, 12959788, 13039868, 13553882, 13640376, 13911580, 14103144, 14399216, 14741470, …<a href="#cite_note-F18Prime-44">[44]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A244150" class="extiw" title="oeis:A244150">A244150</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>19 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{524288}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>524288</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{524288}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80edba9884cbe909199c0c8a0db8bdb2edbc59f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.165ex; height:2.843ex;" alt="{\displaystyle b^{524288}+1}"> </td> <td style="text-align:left">1, 75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, 6339004, …<a href="#cite_note-F19Prime-45">[45]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A243959" class="extiw" title="oeis:A243959">A243959</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr> <tr> <td>20 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{1048576}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1048576</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{1048576}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e612e64b4e95f70d85652878fd5b223618e13ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.987ex; height:2.843ex;" alt="{\displaystyle b^{1048576}+1}"> </td> <td style="text-align:left">1, 919444, 1059094, 1951734, 1963736, …<a href="#cite_note-F20Prime-46">[46]</a> </td> <td style="text-align:center">(Folge <a href="//oeis.org/A321323" class="extiw" title="oeis:A321323">A321323</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </td></tr></tbody></table> <div style="text-align:left"> Stand: 18. Juli 2023 </div> </div> </div> Die kleinsten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3370d3e4bd3efea4b3691ba395ffcb5011b038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.258ex; height:2.343ex;" alt="{\displaystyle b\geq 2}">), für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdb787c6eecda75a33d0125c93a6056a944f3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle F_{n}(b)}"> erstmals eine Primzahl ergibt, kann man der obigen Tabelle entnehmen, was die folgende Liste ergibt: <dl><dd>2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, … (Folge <a href="//oeis.org/A056993" class="extiw" title="oeis:A056993"> A056993</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Die_10_größten_bekannten_verallgemeinerten_Fermatschen_Primzahlen">Die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen</h3>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=11" title="Abschnitt bearbeiten: Die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen">Quelltext bearbeiten</a>]</div> Der folgenden Liste kann man die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen entnehmen. Sämtliche Entdecker dieser Primzahlen sind Teilnehmer des <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a>-Projektes. In der zweiten Spalte steht, die wievieltgrößte bekannte Primzahl diese Fermatsche Primzahl im Moment ist. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r248673343"><div class="NavFrame"> <div class="NavHead" style="text-align:left">Die 10 größten bekannten verallgemeinerten Fermatschen Primzahlen</div> <div class="NavContent"> <table class="wikitable" style="margin-left:2em"> <tbody><tr class="hintergrundfarbe6"> <th>Rang</th> <th>wievieltgrößte bekannte Primzahl<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> a<a href="#cite_note-primesutmedu1-47">[47]</a><a href="#cite_note-primesutmedu2-48">[48]</a><a href="#cite_note-listefermat-49">[49]</a></th> <th>Primzahl</th> <th>Fn(b)</th> <th>Dezimalstellen von Fn(b)</th> <th>Entdeckungsdatum</th> <th>Entdecker</th> <th>Quelle </th></tr> <tr> <td style="text-align:right">1 </td> <td style="text-align:center">14 </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot 5^{11786358}+1=(2\cdot 5^{5893179})^{2^{1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11786358</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5893179</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 5^{11786358}+1=(2\cdot 5^{5893179})^{2^{1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354ff1b48997181e87776096e8eab1c8d3e79510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.602ex; height:3.509ex;" alt="{\displaystyle 4\cdot 5^{11786358}+1=(2\cdot 5^{5893179})^{2^{1}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}(5893179)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>5893179</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}(5893179)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8022d6f741d34ef57ed31f70d4fcbd256af4047f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.495ex; height:2.843ex;" alt="{\displaystyle F_{1}(5893179)}"> </td> <td style="text-align:center">8.238.312 </td> <td style="text-align:right">1. Oktober 2024 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid11786358-50">[50]</a> </td></tr> <tr> <td style="text-align:right">2 </td> <td style="text-align:center">20. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1963736^{1048576}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1963736</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1048576</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1963736^{1048576}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12744d499a39df631b6be806af8d1d337be00fd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.126ex; height:2.843ex;" alt="{\displaystyle 1963736^{1048576}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{20}(1963736)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1963736</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{20}(1963736)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8310f2922a15fdfb020d4439861a9820e5ac56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.317ex; height:2.843ex;" alt="{\displaystyle F_{20}(1963736)}"> </td> <td style="text-align:center">6.598.776 </td> <td style="text-align:right">26. September 2022 </td> <td>Tom Greer (<a href="/wiki/Vereinigte_Staaten" title="Vereinigte Staaten">USA</a>) </td> <td style="text-align:center"><a href="#cite_note-primegrid1963736-51">[51]</a> </td></tr> <tr> <td style="text-align:right">3 </td> <td style="text-align:center">21. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1951734^{1048576}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1951734</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1048576</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1951734^{1048576}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24bbcfe2142634b939f2a01d67579a2d3eea64f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.126ex; height:2.843ex;" alt="{\displaystyle 1951734^{1048576}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{20}(1951734)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1951734</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{20}(1951734)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83785ff7cffe0f7536c0ad2f5d1e76c254eccba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.317ex; height:2.843ex;" alt="{\displaystyle F_{20}(1951734)}"> </td> <td style="text-align:center">6.595.985 </td> <td style="text-align:right">9. August 2022 </td> <td>Kazuya Tanaka (<a href="/wiki/Japan" title="Japan">JAP</a>) </td> <td style="text-align:center"><a href="#cite_note-primegrid1951734-52">[52]</a> </td></tr> <tr> <td style="text-align:right">4 </td> <td style="text-align:center">24. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1059094^{1048576}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1059094</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1048576</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1059094^{1048576}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f6a6308c7d17afe15cae9ec3d9e563c6d6bb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.126ex; height:2.843ex;" alt="{\displaystyle 1059094^{1048576}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{20}(1059094)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1059094</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{20}(1059094)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1850aa25794da7a52bd5df87e6226202761de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.317ex; height:2.843ex;" alt="{\displaystyle F_{20}(1059094)}"> </td> <td style="text-align:center">6.317.602 </td> <td style="text-align:right">31. Oktober 2018 </td> <td>Rob Gahan (<a href="/wiki/Irland" title="Irland">IRL</a>) </td> <td style="text-align:center"><a href="#cite_note-primegrid1059094-53">[53]</a> </td></tr> <tr> <td style="text-align:right">5 </td> <td style="text-align:center">26. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 919444^{1048576}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>919444</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1048576</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 919444^{1048576}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27f80e3e99449aa9e4a99d98dd49f3f4c4c36d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.964ex; height:2.843ex;" alt="{\displaystyle 919444^{1048576}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{20}(919444)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>919444</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{20}(919444)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe1919dda52ea5332e41cbe70670893a1c2077d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.155ex; height:2.843ex;" alt="{\displaystyle F_{20}(919444)}"> </td> <td style="text-align:center">6.253.210 </td> <td style="text-align:right">29. August 2017 </td> <td>Sylvanus A. Zimmerman (USA) </td> <td style="text-align:center"><a href="#cite_note-primegrid919444-54">[54]</a> </td></tr> <tr> <td style="text-align:right">6 </td> <td style="text-align:center">27. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 81\cdot 2^{20498148}+1=(3\cdot 2^{5124537})^{2^{2}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>81</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>20498148</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5124537</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 81\cdot 2^{20498148}+1=(3\cdot 2^{5124537})^{2^{2}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f19f8192b52a1346d92f6f9077c0ee45487bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.764ex; height:3.509ex;" alt="{\displaystyle 81\cdot 2^{20498148}+1=(3\cdot 2^{5124537})^{2^{2}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}(3\cdot 2^{20498148})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>20498148</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}(3\cdot 2^{20498148})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae617f313bdc1b62ca3f038b736a99c3ff4d48c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.17ex; height:3.176ex;" alt="{\displaystyle F_{2}(3\cdot 2^{20498148})}"> </td> <td style="text-align:center">6.170.560 </td> <td style="text-align:right">13. Juni 2023 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid20498148-55">[55]</a> </td></tr> <tr> <td style="text-align:right">7 </td> <td style="text-align:center">29. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot 5^{8431178}+1=(2\cdot 5^{4215589})^{2^{1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8431178</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4215589</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 5^{8431178}+1=(2\cdot 5^{4215589})^{2^{1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36d38d730ba0903849501c735d4c110da701d7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.78ex; height:3.509ex;" alt="{\displaystyle 4\cdot 5^{8431178}+1=(2\cdot 5^{4215589})^{2^{1}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}(2\cdot 2^{4215589})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4215589</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}(2\cdot 2^{4215589})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60cc733ef1d40fcf4f6b63b4473ba1be4ae14fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.348ex; height:3.176ex;" alt="{\displaystyle F_{1}(2\cdot 2^{4215589})}"> </td> <td style="text-align:center">5.893.142 </td> <td style="text-align:right">2. Januar 2024 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid4215589-56">[56]</a> </td></tr> <tr> <td style="text-align:right">8 </td> <td style="text-align:center">38. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot 3^{11279466}+1=(2\cdot 3^{5639733})^{2^{1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11279466</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5639733</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 3^{11279466}+1=(2\cdot 3^{5639733})^{2^{1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d668e484cba84968b48adfeea2346ff1bc9a4bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.602ex; height:3.509ex;" alt="{\displaystyle 4\cdot 3^{11279466}+1=(2\cdot 3^{5639733})^{2^{1}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}(2\cdot 2^{5639733})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5639733</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}(2\cdot 2^{5639733})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d7f44b8829d84fbadcf8ebea7d4e48d0c763d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.348ex; height:3.176ex;" alt="{\displaystyle F_{1}(2\cdot 2^{5639733})}"> </td> <td style="text-align:center">5.381.674 </td> <td style="text-align:right">10. September 2024 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid11279466-57">[57]</a> </td></tr> <tr> <td style="text-align:right">9 </td> <td style="text-align:center">73. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25\cdot 2^{13719266}+1=(5\cdot 2^{6859633})^{2^{1}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13719266</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6859633</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25\cdot 2^{13719266}+1=(5\cdot 2^{6859633})^{2^{1}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722946f135e5ac76a4bfcded734b25d6d58b8040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.764ex; height:3.509ex;" alt="{\displaystyle 25\cdot 2^{13719266}+1=(5\cdot 2^{6859633})^{2^{1}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}(5\cdot 2^{6859633})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6859633</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}(5\cdot 2^{6859633})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2950a7b2bd70b3a73da2ba4e1f59e2b40c7d1257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.348ex; height:3.176ex;" alt="{\displaystyle F_{1}(5\cdot 2^{6859633})}"> </td> <td style="text-align:center">4.129.912 </td> <td style="text-align:right">21. September 2022 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid2^13719266-58">[58]</a> </td></tr> <tr> <td style="text-align:right">10 </td> <td style="text-align:center">74. </td> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 81\cdot 2^{13708272}+1=(3\cdot 2^{3427068})^{2^{2}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>81</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13708272</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3427068</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 81\cdot 2^{13708272}+1=(3\cdot 2^{3427068})^{2^{2}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a487bb5e4b047da31800663dd55e1c1c2349a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.764ex; height:3.509ex;" alt="{\displaystyle 81\cdot 2^{13708272}+1=(3\cdot 2^{3427068})^{2^{2}}+1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}(3\cdot 2^{3427068})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3427068</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}(3\cdot 2^{3427068})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d45c24531f6c2187fa01db7e36c33f0023101b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.348ex; height:3.176ex;" alt="{\displaystyle F_{2}(3\cdot 2^{3427068})}"> </td> <td style="text-align:center">4.126.603 </td> <td style="text-align:right">11. Oktober 2022 </td> <td>Ryan Propper </td> <td style="text-align:center"><a href="#cite_note-primegrid2^13708272-59">[59]</a> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-box"> <div class="fussnoten-linie" aria-hidden="true" role="presentation"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-block"><div class="fussnoten-inhalt references">a <div class="reference-text">Stand: 26. Oktober 2024</div></div></div> </div> </div> </div> Die meisten der oben genannten Ergebnisse konnten natürlich nur mit Hilfe von Computern gefunden werden. <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=12" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Mersenne-Primzahl" class="mw-redirect" title="Mersenne-Primzahl">Mersenne-Primzahl</a></li> <li><a href="/wiki/Prothsche_Primzahl" title="Prothsche Primzahl">Prothsche Primzahl</a></li> <li><a href="/wiki/257-Eck" title="257-Eck">257-Eck</a></li> <li><a href="/wiki/4294967295-Eck" title="4294967295-Eck">4294967295-Eck</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=13" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a>: On the sum of the reciprocals of the Fermat numbers and related irrationalities. In: <a href="/wiki/Canadian_Mathematical_Society" title="Canadian Mathematical Society">Canad. J. Math.</a>, Vol. 15, 1963, S. 475–478.</li> <li><a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>: The Anti-Social Fermat Number. In: <a href="/wiki/American_Mathematical_Monthly" title="American Mathematical Monthly">American Mathematical Monthly</a>, Vol. 107, Nr. 2, Februar 2000, S. 171–173.</li> <li><a href="/w/index.php?title=Michal_K%C5%99%C3%AD%C5%BEek&action=edit&redlink=1" class="new" title="Michal Křížek (Seite nicht vorhanden)">Michal Křížek</a>, <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>, <a href="/w/index.php?title=Lawrence_Somer&action=edit&redlink=1" class="new" title="Lawrence Somer (Seite nicht vorhanden)">Lawrence Somer</a>: On the Convergence of Series of Reciprocals of Primes Related to the Fermat Numbers. In: <a href="/wiki/Journal_of_Number_Theory" title="Journal of Number Theory">Journal of Number Theory</a>, Vol. 97, Nr. 1 (Nov. 2002), S. 95–112.</li> <li><a href="/w/index.php?title=Aleksander_Grytczuk&action=edit&redlink=1" class="new" title="Aleksander Grytczuk (Seite nicht vorhanden)">Aleksander Grytczuk</a>, <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>, <a href="/w/index.php?title=Marek_W%C3%B3jtowicz&action=edit&redlink=1" class="new" title="Marek Wójtowicz (Seite nicht vorhanden)">Marek Wójtowicz</a>: Another note on the greatest prime factors of Fermat numbers. In: <a href="/wiki/Southeast_Asian_Bulletin_of_Mathematics" title="Southeast Asian Bulletin of Mathematics">Southeast Asian Bulletin of Mathematics</a>, Vol. 25, Nr. 1 (Juli 2001), S. 111–115.</li> <li>Michal Křížek, Florian Luca, Lawrence Somer: 17 Lectures on Fermat Numbers: From Number Theory to Geometry. In: <a href="/wiki/Canadian_Mathematical_Society" title="Canadian Mathematical Society">Canad. J. Math.</a>, S. 132–138.</li> <li><a href="/w/index.php?title=Fredrick_Kennard&action=edit&redlink=1" class="new" title="Fredrick Kennard (Seite nicht vorhanden)">Fredrick Kennard</a>: Unsolved Problems in Mathematics. <a href="/wiki/Lulu.com" title="Lulu.com">Lulu.com</a>, Morrisville (NC) 2015. <a href="/wiki/Spezial:ISBN-Suche/9781312938113" class="internal mw-magiclink-isbn">ISBN 978-1312938113</a>. S. 56.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=14" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.fermatsearch.org/">Distributed Search for Fermat Number Divisors.</a></li> <li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number.</a> In: <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (englisch).</li> <li><a rel="nofollow" class="external text" href="http://www.prothsearch.com/JFougGFNf.html">Factors of generalized Fermat numbers found by Björn & Riesel.</a></li> <li><a rel="nofollow" class="external text" href="http://www.prothsearch.com/GFNfacs.html">Factors of generalized Fermat numbers found after Björn & Riesel.</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Fermat-Zahl&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Fermat-Zahl&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-Fermat1640-1"><a href="#cite_ref-Fermat1640_1-0">↑</a> W. Narkiewicz: <cite style="font-style:italic">The Development of Prime Number Theory – From Euclid to Hardy and Littlewood</cite>. Springer-Verlag, 2000, S. 24 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=2gf9CAAAQBAJ&pg=PA24&lpg=PA24&dq=fermat-number+1640&source=bl&ots=6AmvB_YOi4&sig=ACfU3U0d4YIoPPoTRFf2bbKpyCZlIT3zuw&hl=de&sa=X&ved=2ahUKEwiG--XVuIbnAhUkl4sKHUvBAjEQ6AEwBXoECAkQAQ#v=onepage&q=fermat-number%201640&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=W.+Narkiewicz&rft.btitle=The+Development+of+Prime+Number+Theory+-+From+Euclid+to+Hardy+and+Littlewood&rft.date=2000&rft.genre=book&rft.pages=24&rft.pub=Springer-Verlag" style="display:none">  </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> Edward Sandifer: <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/hedi/HEDI-2007-03.pdf">How Euler Did It – Factoring F5.</a> <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a> Online, März 2007, S. 1–4, abgerufen am 23. März 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=How+Euler+Did+It+%E2%80%93+Factoring+F%3Csub%3E5%3C%2Fsub%3E&rft.description=How+Euler+Did+It+%E2%80%93+Factoring+F%3Csub%3E5%3C%2Fsub%3E&rft.identifier=http%3A%2F%2Feulerarchive.maa.org%2Fhedi%2FHEDI-2007-03.pdf&rft.creator=Edward+Sandifer&rft.publisher=%5B%5BMathematical+Association+of+America%7CMAA%5D%5D+Online">  </li> <li id="cite_note-OEISA000215-3"><a href="#cite_ref-OEISA000215_3-0">↑</a> Folge <a href="//oeis.org/A000215" class="extiw" title="oeis:A000215">A000215</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>. </li> <li id="cite_note-euler-4"><a href="#cite_ref-euler_4-0">↑</a> Leonhard Euler: <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/docs/originals/E026.pdf">Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus.</a> (PDF; 399 kB). [E26]. In: Commentarii academiae scientiarum Petropolitanae. 6 (1732/33), St. Petersburg 1738, S. 103–107, hier S. 104. Nachdruck in Opera Omnia, Band 1/2, S. 1–5. Englische Übersetzung von Ian Bruce: <a rel="nofollow" class="external text" href="http://www.17centurymaths.com/contents/euler/e026&54tr.pdf">Observations concerning a certain theorem of Fermat and other considerations regarding prime numbers.</a> (PDF; 100 kB) bzw. von David Zhao: <a rel="nofollow" class="external text" href="http://www.math.dartmouth.edu/~euler/docs/translations/E026tr.pdf">Oberservations on a certain theorem of Fermat and on others regarding prime numbers.</a> (PDF; 101 kB). </li> <li id="cite_note-Status-5">↑ <a href="#cite_ref-Status_5-0">a</a> <a href="#cite_ref-Status_5-1">b</a> <a rel="nofollow" class="external text" href="http://www.prothsearch.com/fermat.html">Faktorisierungsstatus aller Fermatzahlen.</a> Stand: 29. Juli 2018 (englisch). </li> <li id="cite_note-morrison-6"><a href="#cite_ref-morrison_6-0">↑</a> Siehe <a href="/wiki/Kettenbruchmethode#Algorithmus_nach_Morrison_und_Brillhart" title="Kettenbruchmethode">Algorithmus nach Morrison und Brillhart</a>. </li> <li id="cite_note-YoungBuell-7"><a href="#cite_ref-YoungBuell_7-0">↑</a> <a href="/w/index.php?title=Jeff_Young&action=edit&redlink=1" class="new" title="Jeff Young (Seite nicht vorhanden)">Jeff Young</a>, <a href="/w/index.php?title=Duncan_A._Buell&action=edit&redlink=1" class="new" title="Duncan A. Buell (Seite nicht vorhanden)">Duncan A. Buell</a>: <cite style="font-style:italic">The Twentieth Fermat Number is Composite</cite>. In: <cite style="font-style:italic"><a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a></cite>. Vol. 50, Nr. 181, Januar 1988, S. 261–263 (<a rel="nofollow" class="external text" href="http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf">ams.org</a> [PDF; abgerufen am 14. August 2016]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=The+Twentieth+Fermat+Number+is+Composite&rft.au=Jeff+Young%2C+Duncan+A.+Buell&rft.date=1988-01&rft.genre=journal&rft.issue=181&rft.jtitle=Mathematics+of+Computation&rft.pages=261-263&rft.volume=Vol.+50" style="display:none">  </li> <li id="cite_note-Crandall-8"><a href="#cite_ref-Crandall_8-0">↑</a> <a href="/wiki/Richard_E._Crandall" class="mw-redirect" title="Richard E. Crandall">Richard E. Crandall</a>, <a href="/w/index.php?title=Ernst_W._Mayer&action=edit&redlink=1" class="new" title="Ernst W. Mayer (Seite nicht vorhanden)">Ernst W. Mayer</a>, <a href="/w/index.php?title=Jason_S._Papadopoulos&action=edit&redlink=1" class="new" title="Jason S. Papadopoulos (Seite nicht vorhanden)">Jason S. Papadopoulos</a>: <cite style="font-style:italic">The Twenty-Fourth Fermat Number is Composite</cite>. In: <cite style="font-style:italic"><a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a></cite>. Band 72, Nr. 243, 6. Dezember 2002, S. 1555–1572 (<a rel="nofollow" class="external text" href="http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01479-5/S0025-5718-02-01479-5.pdf">ams.org</a> [PDF; abgerufen am 14. August 2016]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=The+Twenty-Fourth+Fermat+Number+is+Composite&rft.au=Richard+E.+Crandall%2C+Ernst+W.+Mayer%2C+Jason+S.+Papadopoulos&rft.date=2002-12-06&rft.genre=journal&rft.issue=243&rft.jtitle=Mathematics+of+Computation&rft.pages=1555-1572&rft.volume=72" style="display:none">  </li> <li id="cite_note-MersenneForumF14-9"><a href="#cite_ref-MersenneForumF14_9-0">↑</a> <a rel="nofollow" class="external text" href="http://www.mersenneforum.org/showthread.php?t=13051">GIMPS’ second Fermat factor!</a> MersenneForum.org </li> <li id="cite_note-MersenneForumF22-10"><a href="#cite_ref-MersenneForumF22_10-0">↑</a> <a rel="nofollow" class="external text" href="http://www.mersenneforum.org/showthread.php?t=13209">F22 factored!</a> MersenneForum.org </li> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> <a rel="nofollow" class="external text" href="http://www.prothsearch.com/FMTcomp.html">When and how Fermat numbers Fm were proven composite (on the occasion of a remarkable discovery)</a> </li> <li id="cite_note-Rekordfaktor-12"><a href="#cite_ref-Rekordfaktor_12-0">↑</a> <a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=131289">7· 218233956 + 1 auf den Primepages</a>. </li> <li id="cite_note-MathWorld-13"><a href="#cite_ref-MathWorld_13-0">↑</a> Luigi Morelli: <a rel="nofollow" class="external text" href="http://www.fermatsearch.org/news.html">Distributed Search for Fermat Number Divisors – NEWS.</a> Abgerufen am 19. Dezember 2016.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Distributed+Search+for+Fermat+Number+Divisors+%E2%80%93+NEWS&rft.description=Distributed+Search+for+Fermat+Number+Divisors+%E2%80%93+NEWS&rft.identifier=http%3A%2F%2Fwww.fermatsearch.org%2Fnews.html&rft.creator=Luigi+Morelli">  </li> <li id="cite_note-Entdeckungsdaten-14"><a href="#cite_ref-Entdeckungsdaten_14-0">↑</a> Luigi Morelli: <a rel="nofollow" class="external text" href="http://www.fermatsearch.org/history.html">Distributed Search for Fermat Number Divisors – HISTORY.</a> Abgerufen am 25. Januar 2017.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Distributed+Search+for+Fermat+Number+Divisors+%E2%80%93+HISTORY&rft.description=Distributed+Search+for+Fermat+Number+Divisors+%E2%80%93+HISTORY&rft.identifier=http%3A%2F%2Fwww.fermatsearch.org%2Fhistory.html&rft.creator=Luigi+Morelli">  </li> <li id="cite_note-pseudoprim2-15"><a href="#cite_ref-pseudoprim2_15-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.12</cite>. Hrsg.: Canadian Mathematical Society. S. 31 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA31&lpg=PA31&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=onepage&q=No%20Fermat%20prime%20can%20be%20expressed%20as%20the%20difference%20of%20two%20pth%20powers%2C%20where%20p%20is%20an%20odd%20prime&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.12&rft.genre=book&rft.pages=31" style="display:none">  </li> <li id="cite_note-vorRemark3.7-16">↑ <a href="#cite_ref-vorRemark3.7_16-0">a</a> <a href="#cite_ref-vorRemark3.7_16-1">b</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, vor Remark 3.7</cite>. Hrsg.: Canadian Mathematical Society. S. 29 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA29&lpg=PA29&dq=No+Fermat+number+Fm+for+m+%3E+1+can+be+expressed+as+the+sum+of+two+primes.&source=bl&ots=OoVH_rdM5s&sig=TuyOXCdHeUJOhVjGRbvBIAORJts&hl=de&sa=X&ved=0ahUKEwjKrsvj5-TXAhVK2xoKHWB4DX8Q6AEINjAB#v=onepage&q&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+vor+Remark+3.7&rft.genre=book&rft.pages=29" style="display:none">  </li> <li id="cite_note-Proposition3.4-17"><a href="#cite_ref-Proposition3.4_17-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Proposition 3.4</cite>. Hrsg.: Canadian Mathematical Society. S. 28 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA30&lpg=PA30&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=snippet&q=Proposition%203.4&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Proposition+3.4&rft.genre=book&rft.pages=28" style="display:none">  </li> <li id="cite_note-Remark3.13-18">↑ <a href="#cite_ref-Remark3.13_18-0">a</a> <a href="#cite_ref-Remark3.13_18-1">b</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Remark 3.13</cite>. Hrsg.: Canadian Mathematical Society. S. 31 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA29&lpg=PA29&dq=No+Fermat+number+Fm+for+m+%3E+1+can+be+expressed+as+the+sum+of+two+primes.&source=bl&ots=OoVH_rdM5s&sig=TuyOXCdHeUJOhVjGRbvBIAORJts&hl=de&sa=X&ved=0ahUKEwjKrsvj5-TXAhVK2xoKHWB4DX8Q6AEINjAB#v=onepage&q=Remark%203.13&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Remark+3.13&rft.genre=book&rft.pages=31" style="display:none">  </li> <li id="cite_note-Primzahlsumme-19"><a href="#cite_ref-Primzahlsumme_19-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Proposition 3.8</cite>. Hrsg.: Canadian Mathematical Society. S. 29 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA29&lpg=PA29&dq=No+Fermat+number+Fm+for+m+%3E+1+can+be+expressed+as+the+sum+of+two+primes.&source=bl&ots=OoVH_rdM5s&sig=TuyOXCdHeUJOhVjGRbvBIAORJts&hl=de&sa=X&ved=0ahUKEwjKrsvj5-TXAhVK2xoKHWB4DX8Q6AEINjAB#v=onepage&q&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Proposition+3.8&rft.genre=book&rft.pages=29" style="display:none">  </li> <li id="cite_note-pseudoprim3-20"><a href="#cite_ref-pseudoprim3_20-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.14</cite>. Hrsg.: Canadian Mathematical Society. S. 31 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA31&lpg=PA31&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=onepage&q=No%20Fermat%20prime%20can%20be%20expressed%20as%20the%20difference%20of%20two%20pth%20powers%2C%20where%20p%20is%20an%20odd%20prime&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.14&rft.genre=book&rft.pages=31" style="display:none">  </li> <li id="cite_note-Remark3.7-21"><a href="#cite_ref-Remark3.7_21-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Remark 3.7</cite>. Hrsg.: Canadian Mathematical Society. S. 29 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA30&lpg=PA30&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=onepage&q=Remark%203.7&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Remark+3.7&rft.genre=book&rft.pages=29" style="display:none">  </li> <li id="cite_note-Theorem3.9-22">↑ <a href="#cite_ref-Theorem3.9_22-0">a</a> <a href="#cite_ref-Theorem3.9_22-1">b</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.9</cite>. Hrsg.: Canadian Mathematical Society. S. 29 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA29&lpg=PA29&dq=No+Fermat+number+Fm+for+m+%3E+1+can+be+expressed+as+the+sum+of+two+primes.&source=bl&ots=OoVH_rdM5s&sig=TuyOXCdHeUJOhVjGRbvBIAORJts&hl=de&sa=X&ved=0ahUKEwjKrsvj5-TXAhVK2xoKHWB4DX8Q6AEINjAB#v=onepage&q&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.9&rft.genre=book&rft.pages=29" style="display:none">  </li> <li id="cite_note-Theorem3.11-23"><a href="#cite_ref-Theorem3.11_23-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.11</cite>. Hrsg.: Canadian Mathematical Society. S. 30–31 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA30&lpg=PA30&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=onepage&q=No%20Fermat%20prime%20can%20be%20expressed%20as%20the%20difference%20of%20two%20pth%20powers%2C%20where%20p%20is%20an%20odd%20prime&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.11&rft.genre=book&rft.pages=30-31" style="display:none">  </li> <li id="cite_note-Proposition3.5-24"><a href="#cite_ref-Proposition3.5_24-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Proposition 3.5</cite>. Hrsg.: Canadian Mathematical Society. S. 28 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA29&lpg=PA29&dq=No+Fermat+number+Fm+for+m+%3E+1+can+be+expressed+as+the+sum+of+two+primes.&source=bl&ots=OoVH_rdM5s&sig=TuyOXCdHeUJOhVjGRbvBIAORJts&hl=de&sa=X&ved=0ahUKEwjKrsvj5-TXAhVK2xoKHWB4DX8Q6AEINjAB#v=onepage&q=Proposition%203.5&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Proposition+3.5&rft.genre=book&rft.pages=28" style="display:none">  </li> <li id="cite_note-Nielsen-25"><a href="#cite_ref-Nielsen_25-0">↑</a> <a href="/w/index.php?title=Jeppe_Stig_Nielsen&action=edit&redlink=1" class="new" title="Jeppe Stig Nielsen (Seite nicht vorhanden)">Jeppe Stig Nielsen</a>: <a rel="nofollow" class="external text" href="http://jeppesn.dk/nton.html">S(n) = n^n+1.</a> Abgerufen am 9. August 2016.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=S%28n%29+%3D+n%5En%2B1&rft.description=S%28n%29+%3D+n%5En%2B1&rft.identifier=http%3A%2F%2Fjeppesn.dk%2Fnton.html&rft.creator=%5B%5BJeppe+Stig+Nielsen%5D%5D">  </li> <li id="cite_note-Sierpinski-26">↑ <a href="#cite_ref-Sierpinski_26-0">a</a> <a href="#cite_ref-Sierpinski_26-1">b</a> <a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a>: <a rel="nofollow" class="external text" href="https://www.pdfdrive.com/waclaw-sierpinski-elementary-number-theory-e18826078.html">Elementary Theory of Numbers.</a> S. 375, abgerufen am 13. Juni 2019.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Elementary+Theory+of+Numbers&rft.description=Elementary+Theory+of+Numbers&rft.identifier=https%3A%2F%2Fwww.pdfdrive.com%2Fwaclaw-sierpinski-elementary-number-theory-e18826078.html&rft.creator=%5B%5BWac%C5%82aw+Sierpi%C5%84ski%5D%5D">  </li> <li id="cite_note-Theorem3.10-27"><a href="#cite_ref-Theorem3.10_27-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.10</cite>. Hrsg.: Canadian Mathematical Society. S. 30 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA30&lpg=PA30&dq=No+Fermat+prime+can+be+expressed+as+the+difference+of+two+pth+powers,+where+p+is+an+odd+prime&source=bl&ots=On-H_jaK6t&sig=VRZ9nRWMVfAIt_B8Udt_2QNcasI&hl=de&sa=X&ved=0ahUKEwja7uWhntbOAhXL2xoKHUAHA8UQ6AEINDAC#v=onepage&q=No%20Fermat%20prime%20can%20be%20expressed%20as%20the%20difference%20of%20two%20pth%20powers%2C%20where%20p%20is%20an%20odd%20prime&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.10&rft.genre=book&rft.pages=30" style="display:none">  </li> <li id="cite_note-Golomb-28"><a href="#cite_ref-Golomb_28-0">↑</a> <a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a>: <cite style="font-style:italic">On the sum of the reciprocals of the Fermat numbers and related irrationalities</cite>. In: <cite style="font-style:italic"><a href="/wiki/Canadian_Mathematical_Society" title="Canadian Mathematical Society">Canad. J. Math.</a></cite> Vol. 15, 1963, S. 475–478 (<style data-mw-deduplicate="TemplateStyles:r246413598">.mw-parser-output .webarchiv-memento{color:var(--color-base,#202122)!important}</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160321234733/https://cms.math.ca/openaccess/cjm/v15/cjm1963v15.0475-0478.pdf">cms.math.ca</a> (<a href="/wiki/Web-Archivierung#Begrifflichkeiten" title="Web-Archivierung">Memento</a> vom 21. März 2016 im <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>) [PDF; abgerufen am 9. August 2016]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=On+the+sum+of+the+reciprocals+of+the+Fermat+numbers+and+related+irrationalities&rft.au=Solomon+W.+Golomb&rft.btitle=Canad.+J.+Math.&rft.date=1963&rft.genre=book&rft.pages=475-478&rft.volume=Vol.+15" style="display:none">  </li> <li id="cite_note-Luca-29"><a href="#cite_ref-Luca_29-0">↑</a> <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>: <cite style="font-style:italic">The Anti-Social Fermat Number</cite>. In: <cite style="font-style:italic">The <a href="/wiki/American_Mathematical_Monthly" title="American Mathematical Monthly">American Mathematical Monthly</a></cite>. Vol. 07, Nr. 2, Februar 2000, S. 171–173, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2589441">2589441</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=The+Anti-Social+Fermat+Number&rft.au=Florian+Luca&rft.date=2000-02&rft.genre=journal&rft.issue=2&rft.jtitle=The+American+Mathematical+Monthly&rft.pages=171-173&rft.volume=Vol.+07" style="display:none">  </li> <li id="cite_note-Somer-30"><a href="#cite_ref-Somer_30-0">↑</a> <a href="/w/index.php?title=Michal_Kr%C3%AD%C5%BEek&action=edit&redlink=1" class="new" title="Michal Krížek (Seite nicht vorhanden)">Michal Krížek</a>, <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>, <a href="/w/index.php?title=Lawrence_Somer&action=edit&redlink=1" class="new" title="Lawrence Somer (Seite nicht vorhanden)">Lawrence Somer</a>: <cite style="font-style:italic">On the Convergence of Series of Reciprocals of Primes Related to the Fermat Numbers</cite>. In: <cite style="font-style:italic"><a href="/wiki/Journal_of_Number_Theory" title="Journal of Number Theory">Journal of Number Theory</a></cite>. Band 97, Nr. 1, November 2002, S. 95–112 (<a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S0022314X02927824">sciencedirect.com</a> [abgerufen am 9. August 2016]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=On+the+Convergence+of+Series+of+Reciprocals+of+Primes+Related+to+the+Fermat+Numbers&rft.au=Michal+Kr%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.date=2002-11&rft.genre=journal&rft.issue=1&rft.jtitle=Journal+of+Number+Theory&rft.pages=95-112&rft.volume=97" style="display:none">  </li> <li id="cite_note-Wojtowicz-31"><a href="#cite_ref-Wojtowicz_31-0">↑</a> <a href="/w/index.php?title=Aleksander_Grytczuk&action=edit&redlink=1" class="new" title="Aleksander Grytczuk (Seite nicht vorhanden)">Aleksander Grytczuk</a>, <a href="/w/index.php?title=Florian_Luca&action=edit&redlink=1" class="new" title="Florian Luca (Seite nicht vorhanden)">Florian Luca</a>, <a href="/w/index.php?title=Marek_W%C3%B3jtowicz&action=edit&redlink=1" class="new" title="Marek Wójtowicz (Seite nicht vorhanden)">Marek Wójtowicz</a>: <cite style="font-style:italic">Another note on the greatest prime factors of Fermat numbers</cite>. In: <cite style="font-style:italic"><a href="/wiki/Southeast_Asian_Bulletin_of_Mathematics" title="Southeast Asian Bulletin of Mathematics">Southeast Asian Bulletin of Mathematics</a></cite>. Band 25, Nr. 1, Juli 2001, S. 111–115 (<a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/250602035_Another_Note_on_the_Greatest_Prime_Factors_of_Fermat_Numbers">researchgate.net</a> [abgerufen am 9. August 2016]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.atitle=Another+note+on+the+greatest+prime+factors+of+Fermat+numbers&rft.au=Aleksander+Grytczuk%2C+Florian+Luca%2C+Marek+W%C3%B3jtowicz&rft.date=2001-07&rft.genre=journal&rft.issue=1&rft.jtitle=Southeast+Asian+Bulletin+of+Mathematics&rft.pages=111-115&rft.volume=25" style="display:none">  </li> <li id="cite_note-pseudoprim-32">↑ <a href="#cite_ref-pseudoprim_32-0">a</a> <a href="#cite_ref-pseudoprim_32-1">b</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 12.16</cite>. Hrsg.: <a href="/wiki/Canadian_Mathematical_Society" title="Canadian Mathematical Society">Canadian Mathematical Society</a>. S. 138 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA138&lpg=PA138&dq=Fermat-number+strong+Pseudoprimes&source=bl&ots=On-HSkeNZt&sig=5IAGQ412nnVhVYEiV85oU_Nu5yE&hl=de&sa=X&ved=0ahUKEwij1_X7rr_OAhXLBsAKHbsmDOkQ6AEIOzAE#v=onepage&q=Fermat-number%20strong%20Pseudoprimes&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+12.16&rft.genre=book&rft.pages=138" style="display:none">  </li> <li id="cite_note-Kennard-33"><a href="#cite_ref-Kennard_33-0">↑</a> <a href="/w/index.php?title=Fredrick_Kennard&action=edit&redlink=1" class="new" title="Fredrick Kennard (Seite nicht vorhanden)">Fredrick Kennard</a>: <cite style="font-style:italic">Unsolved Problems in Mathematics</cite>. S. 56 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=OaNsCQAAQBAJ&pg=PA56&lpg=PA56&dq=A+Fermat+prime+cannot+be+a+Wieferich+prime&source=bl&ots=P8ebc3lVVI&sig=IPt_Crj0z0ql0VuRMVg4fxtENgw&hl=de&sa=X&ved=0ahUKEwj8rM-NrqHPAhUIIMAKHXimDU8Q6AEIPDAE#v=onepage&q=A%20Fermat%20prime%20cannot%20be%20a%20Wieferich%20prime&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Fredrick+Kennard&rft.btitle=Unsolved+Problems+in+Mathematics&rft.genre=book&rft.pages=56" style="display:none">  </li> <li id="cite_note-Somer2-34"><a href="#cite_ref-Somer2_34-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 12.1</cite>. Hrsg.: Canadian Mathematical Society. S. 132 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&lpg=PA132&ots=On_KUoiNZp&dq=cipolla+fermat+1904&pg=PA132&redir_esc=y#v=onepage&q&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+12.1&rft.genre=book&rft.pages=132" style="display:none">  </li> <li id="cite_note-pseudoprim4-35"><a href="#cite_ref-pseudoprim4_35-0">↑</a> Michal Křížek, Florian Luca, Lawrence Somer: <cite style="font-style:italic">17 Lectures on Fermat Numbers: From Number Theory to Geometry, Theorem 3.17</cite>. Hrsg.: Canadian Mathematical Society. S. 32 (<a rel="nofollow" class="external text" href="https://books.google.at/books?id=hgfSBwAAQBAJ&pg=PA32&lpg=PA32&f=false">google.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fermat-Zahl&rft.au=Michal+K%C5%99%C3%AD%C5%BEek%2C+Florian+Luca%2C+Lawrence+Somer&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry%2C+Theorem+3.17&rft.genre=book&rft.pages=32" style="display:none">  </li> <li id="cite_note-36"><a href="#cite_ref-36">↑</a> Kent D. Boklan, <a href="/wiki/John_H._Conway" class="mw-redirect" title="John H. Conway">John H. Conway</a>: <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1605.01371">Expect at most one billionth of a new Fermat Prime!</a> <a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a> 39, 3–5 (2017), 9. Mai 2016, S. 1–7, abgerufen am 23. März 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Expect+at+most+one+billionth+of+a+new+Fermat+Prime%21&rft.description=Expect+at+most+one+billionth+of+a+new+Fermat+Prime%21&rft.identifier=https%3A%2F%2Farxiv.org%2Fabs%2F1605.01371&rft.creator=Kent+D.+Boklan%2C+%5B%5BJohn+H.+Conway%5D%5D&rft.publisher=%5B%5BThe+Mathematical+Intelligencer%5D%5D+%27%27%2739%27%27%27%2C+3%E2%80%935+%282017%29">  </li> <li id="cite_note-EmilArtin-37"><a href="#cite_ref-EmilArtin_37-0">↑</a> <a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a>: Galoissche Theorie. Verlag Harri Deutsch, Zürich 1973, <a href="/wiki/Spezial:ISBN-Suche/3871441678" class="internal mw-magiclink-isbn">ISBN 3-87144-167-8</a>, S. 85. </li> <li id="cite_note-prothsearch-38"><a href="#cite_ref-prothsearch_38-0">↑</a> <a rel="nofollow" class="external text" href="http://www.prothsearch.com/JFougGFNf.html">Faktoren von verallgemeinerten Fermat-Zahlen, die von Björn und Riesel gefunden wurden.</a> Abgerufen am 15. Dezember 2018.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Faktoren+von+verallgemeinerten+Fermat-Zahlen%2C+die+von+Bj%C3%B6rn+und+Riesel+gefunden+wurden&rft.description=Faktoren+von+verallgemeinerten+Fermat-Zahlen%2C+die+von+Bj%C3%B6rn+und+Riesel+gefunden+wurden&rft.identifier=http%3A%2F%2Fwww.prothsearch.com%2FJFougGFNf.html">  </li> <li id="cite_note-prothsearch2-39"><a href="#cite_ref-prothsearch2_39-0">↑</a> <a rel="nofollow" class="external text" href="http://www.prothsearch.com/GFNfacs.html">Faktoren von verallgemeinerten Fermat-Zahlen, die nach Björn und Riesel gefunden wurden.</a> Abgerufen am 15. Dezember 2018.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Faktoren+von+verallgemeinerten+Fermat-Zahlen%2C+die+nach+Bj%C3%B6rn+und+Riesel+gefunden+wurden&rft.description=Faktoren+von+verallgemeinerten+Fermat-Zahlen%2C+die+nach+Bj%C3%B6rn+und+Riesel+gefunden+wurden&rft.identifier=http%3A%2F%2Fwww.prothsearch.com%2FGFNfacs.html">  </li> <li id="cite_note-40"><a href="#cite_ref-40">↑</a> Jeppe Stig Salling Nielsen: <a rel="nofollow" class="external text" href="http://jeppesn.dk/generalized-fermat.html">Generalized Fermat Primes sorted by base.</a> Abgerufen am 6. Mai 2018.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Generalized+Fermat+Primes+sorted+by+base&rft.description=Generalized+Fermat+Primes+sorted+by+base&rft.identifier=http%3A%2F%2Fjeppesn.dk%2Fgeneralized-fermat.html&rft.creator=Jeppe+Stig+Salling+Nielsen">  </li> <li id="cite_note-F15Prime-41"><a href="#cite_ref-F15Prime_41-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN32768&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=32768.</a> PrimeGrid, abgerufen am 19. März 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D32768&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D32768&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN32768%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-F16Prime-42"><a href="#cite_ref-F16Prime_42-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN65536&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=65536.</a> PrimeGrid, abgerufen am 19. März 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D65536&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D65536&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN65536%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-F17Prime-43"><a href="#cite_ref-F17Prime_43-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN131072&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=131072.</a> PrimeGrid, abgerufen am 19. März 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D131072&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D131072&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN131072%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-F18Prime-44"><a href="#cite_ref-F18Prime_44-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN262144&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=262144.</a> PrimeGrid, abgerufen am 19. März 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D262144&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D262144&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN262144%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-F19Prime-45"><a href="#cite_ref-F19Prime_45-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN524288&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=524288.</a> PrimeGrid, abgerufen am 18. Juli 2023.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D524288&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D524288&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN524288%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-F20Prime-46"><a href="#cite_ref-F20Prime_46-0">↑</a> Rytis Slatkevičius: <a rel="nofollow" class="external text" href="http://www.primegrid.com/primes/primes.php?project=GFN1048576&factors=+&only=ALL&announcements=ALL&sortby=size&dc=no&search=">PrimeGrid: Generalized Fermat Prime Search n=1048576.</a> PrimeGrid, abgerufen am 19. März 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D1048576&rft.description=PrimeGrid%3A+Generalized+Fermat+Prime+Search+n%3D1048576&rft.identifier=http%3A%2F%2Fwww.primegrid.com%2Fprimes%2Fprimes.php%3Fproject%3DGFN1048576%26factors%3D%2B%26only%3DALL%26announcements%3DALL%26sortby%3Dsize%26dc%3Dno%26search%3D&rft.creator=Rytis+Slatkevi%C4%8Dius&rft.publisher=PrimeGrid">  </li> <li id="cite_note-primesutmedu1-47"><a href="#cite_ref-primesutmedu1_47-0">↑</a> <a rel="nofollow" class="external text" href="https://primes.utm.edu/top20/page.php?id=12">Die 20 größten verallgemeinerten Fermatschen Primzahlen.</a> Abgerufen am 24. Juli 2017 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Die+20+gr%C3%B6%C3%9Ften+verallgemeinerten+Fermatschen+Primzahlen&rft.description=Die+20+gr%C3%B6%C3%9Ften+verallgemeinerten+Fermatschen+Primzahlen&rft.identifier=https%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D12&rft.language=en">  </li> <li id="cite_note-primesutmedu2-48"><a href="#cite_ref-primesutmedu2_48-0">↑</a> <a rel="nofollow" class="external text" href="https://t5k.org/primes/lists/all.txt">Liste der größten bekannten Primzahlen.</a> Abgerufen am 26. Oktober 2024 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Liste+der+gr%C3%B6%C3%9Ften+bekannten+Primzahlen&rft.description=Liste+der+gr%C3%B6%C3%9Ften+bekannten+Primzahlen&rft.identifier=https%3A%2F%2Ft5k.org%2Fprimes%2Flists%2Fall.txt&rft.language=en">  </li> <li id="cite_note-listefermat-49"><a href="#cite_ref-listefermat_49-0">↑</a> <a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/search.php?Comment=Generalized+Fermat&OnList=yes&Number=100&Style=HTML">Liste der größten bekannten verallgemeinerten Fermatschen Primzahlen.</a> Abgerufen am 7. Januar 2023 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFermat-Zahl&rft.title=Liste+der+gr%C3%B6%C3%9Ften+bekannten+verallgemeinerten+Fermatschen+Primzahlen&rft.description=Liste+der+gr%C3%B6%C3%9Ften+bekannten+verallgemeinerten+Fermatschen+Primzahlen&rft.identifier=https%3A%2F%2Fprimes.utm.edu%2Fprimes%2Fsearch.php%3FComment%3DGeneralized%2BFermat%26OnList%3Dyes%26Number%3D100%26Style%3DHTML&rft.language=en">  </li> <li id="cite_note-primegrid11786358-50"><a href="#cite_ref-primegrid11786358_50-0">↑</a> <a rel="nofollow" class="external text" href="https://t5k.org/primes/page.php?id=138596">4 · 5 11786358 + 1 auf den PrimePages.</a> </li> <li id="cite_note-primegrid1963736-51"><a href="#cite_ref-primegrid1963736_51-0">↑</a> <a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=134423">19637361048576 + 1 auf den PrimePages.</a> </li> <li id="cite_note-primegrid1951734-52"><a href="#cite_ref-primegrid1951734_52-0">↑</a> <a rel="nofollow" class="external text" href="https://www.primegrid.com/download/GFN-1951734_1048576.pdf">19517341048576 + 1 auf primegrid.com</a> (PDF). </li> <li id="cite_note-primegrid1059094-53"><a href="#cite_ref-primegrid1059094_53-0">↑</a> <a rel="nofollow" class="external text" href="http://www.primegrid.com/download/GFN-1059094_1048576.pdf">10590941048576 + 1 auf primegrid.com</a> (PDF). </li> <li id="cite_note-primegrid919444-54"><a href="#cite_ref-primegrid919444_54-0">↑</a> <a rel="nofollow" class="external text" href="http://www.primegrid.com/download/GFN-919444_1048576.pdf">9194441048576 + 1 auf primegrid.com</a> (PDF). </li> <li id="cite_note-primegrid20498148-55"><a href="#cite_ref-primegrid20498148_55-0">↑</a> <a rel="nofollow" class="external text" href="https://t5k.org/primes/page.php?id=136165">81 · 2 20498148 + 1 auf den PrimePages.</a> </li> <li id="cite_note-primegrid4215589-56"><a 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</div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > Drucken/exportieren </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Spezial:DownloadAsPdf&page=Fermat-Zahl&action=show-download-screen">Als PDF herunterladen</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Fermat-Zahl&printable=yes" title="Druckansicht dieser Seite [p]" accesskey="p">Druckversion</a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > In anderen Projekten </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207264" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g">Wikidata-Datenobjekt</a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > In anderen Sprachen </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/Fermat_t%C3%A6l" title="Fermat tæl – Altenglisch" lang="ang" hreflang="ang" data-title="Fermat tæl" data-language-autonym="Ænglisc" data-language-local-name="Altenglisch" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%81%D9%8A%D8%B1%D9%85%D8%A7" title="عدد فيرما – Arabisch" lang="ar" hreflang="ar" data-title="عدد فيرما" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ferma_%C9%99d%C9%99dl%C9%99ri" title="Ferma ədədləri – Aserbaidschanisch" lang="az" hreflang="az" data-title="Ferma ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BD%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число на Ферма – Bulgarisch" lang="bg" hreflang="bg" data-title="Число на Ферма" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%AE%E0%A6%BE_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="ফার্মা সংখ্যা – Bengalisch" lang="bn" hreflang="bn" data-title="ফার্মা সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Fermat" title="Nombre de Fermat – Katalanisch" lang="ca" hreflang="ca" data-title="Nombre de Fermat" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D9%81%DB%8E%D8%B1%D9%85%D8%A7" title="ژمارەی فێرما – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="ژمارەی فێرما" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fermatovo_%C4%8D%C3%ADslo" title="Fermatovo číslo – Tschechisch" lang="cs" hreflang="cs" data-title="Fermatovo číslo" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fermatprimtal" title="Fermatprimtal – Dänisch" lang="da" hreflang="da" data-title="Fermatprimtal" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%A6%CE%B5%CF%81%CE%BC%CE%AC" title="Αριθμός Φερμά – Griechisch" lang="el" hreflang="el" data-title="Αριθμός Φερμά" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fermat_number" title="Fermat number – Englisch" lang="en" hreflang="en" data-title="Fermat number" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nombro_de_Fermat" title="Nombro de Fermat – Esperanto" lang="eo" hreflang="eo" data-title="Nombro de Fermat" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – Spanisch" lang="es" hreflang="es" data-title="Número de Fermat" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%81%D8%B1%D9%85%D8%A7" title="اعداد فرما – Persisch" lang="fa" hreflang="fa" data-title="اعداد فرما" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fermat%E2%80%99n_luku" title="Fermat’n luku – Finnisch" lang="fi" hreflang="fi" data-title="Fermat’n luku" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Fermat" title="Nombre de Fermat – Französisch" lang="fr" hreflang="fr" data-title="Nombre de Fermat" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – Galicisch" lang="gl" hreflang="gl" data-title="Número de Fermat" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94" title="מספר פרמה – Hebräisch" lang="he" hreflang="he" data-title="מספר פרמה" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fermat-sz%C3%A1mok" title="Fermat-számok – Ungarisch" lang="hu" hreflang="hu" data-title="Fermat-számok" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%A5%D6%80%D5%B4%D5%A1%D5%B5%D5%AB_%D5%A9%D5%AB%D5%BE" title="Ֆերմայի թիվ – Armenisch" lang="hy" hreflang="hy" data-title="Ֆերմայի թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Fermat" title="Numero di Fermat – Italienisch" lang="it" hreflang="it" data-title="Numero di Fermat" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9E%E3%83%BC%E6%95%B0" title="フェルマー数 – Japanisch" lang="ja" hreflang="ja" data-title="フェルマー数" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%98%EB%A5%B4%EB%A7%88_%EC%88%98" title="페르마 수 – Koreanisch" lang="ko" hreflang="ko" data-title="페르마 수" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ferma_skai%C4%8Dius" title="Ferma skaičius – Litauisch" lang="lt" hreflang="lt" data-title="Ferma skaičius" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fermatgetal" title="Fermatgetal – Niederländisch" lang="nl" hreflang="nl" data-title="Fermatgetal" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fermattal" title="Fermattal – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Fermattal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fermat-tallene" title="Fermat-tallene – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Fermat-tallene" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Fermata" title="Liczby Fermata – Polnisch" lang="pl" hreflang="pl" data-title="Liczby Fermata" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_%C3%ABd_Fermat" title="Nùmer ëd Fermat – Piemontesisch" lang="pms" hreflang="pms" data-title="Nùmer ëd Fermat" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – Portugiesisch" lang="pt" hreflang="pt" data-title="Número de Fermat" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число Ферма – Russisch" lang="ru" hreflang="ru" data-title="Число Ферма" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fermat_number" title="Fermat number – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Fermat number" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fermatovo_pra%C5%A1tevilo" title="Fermatovo praštevilo – Slowenisch" lang="sl" hreflang="sl" data-title="Fermatovo praštevilo" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fermattal" title="Fermattal – Schwedisch" lang="sv" hreflang="sv" data-title="Fermattal" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%83%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="ஃபெர்மா எண் – Tamil" lang="ta" hreflang="ta" data-title="ஃபெர்மா எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%81%E0%B8%9F%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%B2" title="จำนวนแฟร์มา – Thailändisch" lang="th" hreflang="th" data-title="จำนวนแฟร์มา" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fermat_say%C4%B1lar%C4%B1" title="Fermat sayıları – Türkisch" lang="tr" hreflang="tr" data-title="Fermat sayıları" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Числа Ферма – Ukrainisch" lang="uk" hreflang="uk" data-title="Числа Ферма" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_Fermat" title="Số Fermat – Vietnamesisch" lang="vi" 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Fermat-Zahl – Wikipedia