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width:9.274ex; height:2.509ex;" alt="{\displaystyle k\cdot 2^{n}+1}"></span>,</dd></dl> <p>где <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> является <a href="/wiki/%D0%9D%D0%B5%D1%87%D1%91%D1%82%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" class="mw-redirect" title="Нечётное число">нечётным</a> положительным <a href="/wiki/%D0%A6%D0%B5%D0%BB%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Целое число">целым числом</a> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> — положительное целое число, причём <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cebbf502808b931e8bba3fb8a799bc0f4ed5df" 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}}"></span> (без последнего условия числами Прота были бы все нечётные целые числа больше 1<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>). </p><p>Названы в честь французского математика <span data-interwiki-lang="en" data-interwiki-article="François Proth"><a href="/w/index.php?title=%D0%9F%D1%80%D0%BE%D1%82,_%D0%A4%D1%80%D0%B0%D0%BD%D1%81%D1%83%D0%B0&action=edit&redlink=1" class="new" title="Прот, Франсуа (страница отсутствует)">Франсуа Прота</a></span><sup class="noprint" style="font-style:normal; font-weight:normal;"><a href="https://en.wikipedia.org/wiki/Fran%C3%A7ois_Proth" class="extiw" title="en:François Proth"><span title="François Proth — версия статьи «Прот, Франсуа» на английском языке">[англ.]</span></a></sup> (1852—1879). </p><p>Первые числа Прота<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd>3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, …</dd></dl> <p>Наибольший интерес представляют <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%8B%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" class="mw-redirect" title="Простые числа">простые числа</a> Прота, первые таковые<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd>3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, …</dd></dl> <p>Простота чисел Прота может проверяться с помощью <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%80%D0%BE%D1%82%D0%B0" title="Теорема Прота">теоремы Прота</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>, которая утверждает, что число Прота <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> является простым, только если существует целое <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, для которого справедливо следующее сравнение: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {p-1}{2}}\equiv -1\ {\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <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^{\frac {p-1}{2}}\equiv -1\ {\pmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baaab8adf557f3876d647851f4461adf7c2f1462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.178ex; height:4.343ex;" alt="{\displaystyle a^{\frac {p-1}{2}}\equiv -1\ {\pmod {p}}}"></span>.</dd></dl> <p>На ноябрь 2016 года наибольшим известным простым числом Прота является <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10223\cdot 2^{31172165}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10223</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>31172165</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10223\cdot 2^{31172165}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffe8973c183d50da47c945004df9db8ea99d050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.465ex; height:2.843ex;" alt="{\displaystyle 10223\cdot 2^{31172165}+1}"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>, обнаруженное Петером Сабольчем (<i>Peter Szabolcs</i>) в <a href="/wiki/%D0%94%D0%BE%D0%B1%D1%80%D0%BE%D0%B2%D0%BE%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5_%D0%B2%D1%8B%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Добровольные вычисления">проекте добровольных вычислений</a> <a href="/wiki/Seventeen_or_Bust" title="Seventeen or Bust">Seventeen or Bust</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>, притом оно же является крупнейшим известным простым числом, не являющееся <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD%D0%BD%D0%B0" class="mw-redirect" title="Числа Мерсенна">числом Мерсенна</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>. </p><p><a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%9A%D0%B0%D0%BB%D0%BB%D0%B5%D0%BD%D0%B0" title="Числа Каллена">Числа Каллена</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\cdot 2^{n}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n\cdot 2^{n}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e9e6b97738ef48e8096a5942bce7ced58b3a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.267ex; height:2.843ex;" alt="{\displaystyle (n\cdot 2^{n}+1)}"></span> и <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" class="mw-redirect" title="Числа Ферма">числа Ферма</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <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> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{2^{n}}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93eb116c31324099b69d936d520e6ef3fcda921d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.994ex; height:3.176ex;" alt="{\displaystyle (2^{2^{n}}+1)}"></span> представляют собой частные случаи чисел Прота. </p><p>Каждый делитель <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" class="mw-redirect" title="Числа Ферма">числа Ферма</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>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>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd" 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>2}"></span> может быть представлен в виде <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> (<a href="/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80,_%D0%9B%D0%B5%D0%BE%D0%BD%D0%B0%D1%80%D0%B4" title="Эйлер, Леонард">Эйлер</a>, <a href="/wiki/%D0%9B%D1%8E%D0%BA%D0%B0,_%D0%A4%D1%80%D0%B0%D0%BD%D1%81%D1%83%D0%B0_%D0%AD%D0%B4%D1%83%D0%B0%D1%80%D0%B4_%D0%90%D0%BD%D0%B0%D1%82%D0%BE%D0%BB%D1%8C" title="Люка, Франсуа Эдуард Анатоль">Люка</a>, 1878). Однако, неравенство <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<2^{n+2}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<2^{n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311e32eecfa4049f4636334e046736b9e3a0b1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.791ex; height:2.676ex;" alt="{\displaystyle k<2^{n+2}}"></span> здесь может не выполняться. </p> <div class="mw-heading mw-heading2"><h2 id="См._также"><span id=".D0.A1.D0.BC._.D1.82.D0.B0.D0.BA.D0.B6.D0.B5"></span>См. также</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&veaction=edit&section=1" title="Редактировать раздел «См. также»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&action=edit&section=1" title="Редактировать код раздела «См. также»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A1%D0%B5%D1%80%D0%BF%D0%B8%D0%BD%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" class="mw-redirect" title="Число Серпинского">Число Серпинского</a></li> <li><a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> — проект добровольных вычислений по поиску больших простых Прота</li></ul> <div class="mw-heading mw-heading2"><h2 id="Примечания"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.87.D0.B0.D0.BD.D0.B8.D1.8F"></span>Примечания</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&veaction=edit&section=2" title="Редактировать раздел «Примечания»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&action=edit&section=2" title="Редактировать код раздела «Примечания»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><i>Weisstein, Eric W.</i> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ProthNumber.html">Proth Number</a> <small class="ref-info" style="cursor:help;" title="на английском языке">(англ.)</small> на сайте Wolfram <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">последовательность <a href="//oeis.org/A080075" class="extiw" title="oeis:A080075">A080075</a> в <a href="/wiki/%D0%9E%D0%BD%D0%BB%D0%B0%D0%B9%D0%BD-%D1%8D%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F_%D1%86%D0%B5%D0%BB%D0%BE%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8B%D1%85_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Онлайн-энциклопедия целочисленных последовательностей">OEIS</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">последовательность <a href="//oeis.org/A080076" class="extiw" title="oeis:A080076">A080076</a> в <a href="/wiki/%D0%9E%D0%BD%D0%BB%D0%B0%D0%B9%D0%BD-%D1%8D%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F_%D1%86%D0%B5%D0%BB%D0%BE%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8B%D1%85_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Онлайн-энциклопедия целочисленных последовательностей">OEIS</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><i>Weisstein, Eric W.</i> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ProthsTheorem.html">Proth's Theorem</a> <small class="ref-info" style="cursor:help;" title="на английском языке">(англ.)</small> на сайте Wolfram <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Chris Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=66">The Top Twenty: Proth</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171011072640/http://primes.utm.edu/top20/page.php?id=66">Архивная копия</a> от 11 октября 2017 на <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/w/index.php?title=Prime_Pages&action=edit&redlink=1" class="new" title="Prime Pages (страница отсутствует)">Prime Pages</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.primegrid.com/download/SOB-31172165.pdf">Press Release by Seventeen or Bust</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161112051125/https://www.primegrid.com/download/SOB-31172165.pdf">Архивная копия</a> от 12 ноября 2016 на <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 31 October 2016.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Chris Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=3">The Top Twenty: Largest Known Primes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120716181435/http://primes.utm.edu/top20/page.php?id=3">Архивная копия</a> от 16 июля 2012 на <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/w/index.php?title=Prime_Pages&action=edit&redlink=1" class="new" title="Prime Pages (страница отсутствует)">Prime Pages</a></span> </li> </ol></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div 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случайно выбранную страницу [x]" accesskey="x"><span>Случайная статья</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB:%D0%A2%D0%B5%D0%BA%D1%83%D1%89%D0%B8%D0%B5_%D1%81%D0%BE%D0%B1%D1%8B%D1%82%D0%B8%D1%8F" title="Статьи о текущих событиях в мире"><span>Текущие события</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ru.wikipedia.org&uselang=ru" title="Поддержите нас"><span>Пожертвовать</span></a></li> </ul> </div> </nav> <nav id="p-participation" class="mw-portlet mw-portlet-participation vector-menu-portal portal vector-menu" aria-labelledby="p-participation-label" > <h3 id="p-participation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Участие</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-bug_in_article" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%B8%D1%8F_%D0%BE%D0%B1_%D0%BE%D1%88%D0%B8%D0%B1%D0%BA%D0%B0%D1%85" title="Сообщить об ошибке в этой статье"><span>Сообщить об ошибке</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B0:%D0%92%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5"><span>Как править статьи</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BE%D0%B1%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="О проекте, о том, чем здесь можно заниматься, а также — где что находится"><span>Сообщество</span></a></li><li id="n-forum" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A4%D0%BE%D1%80%D1%83%D0%BC" title="Форум участников Википедии"><span>Форум</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D0%B5%D0%B6%D0%B8%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8" title="Список последних изменений [r]" accesskey="r"><span>Свежие правки</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9D%D0%BE%D0%B2%D1%8B%D0%B5_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B" title="Список недавно созданных страниц"><span>Новые страницы</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B0" title="Место расположения Справки"><span>Справка</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D1%81%D1%8B%D0%BB%D0%BA%D0%B8_%D1%81%D1%8E%D0%B4%D0%B0/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0" title="Список всех страниц, ссылающихся на данную [j]" accesskey="j"><span>Ссылки сюда</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D1%8F%D0%B7%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0" rel="nofollow" title="Последние изменения в страницах, на которые ссылается эта страница [k]" accesskey="k"><span>Связанные правки</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BF%D0%B5%D1%86%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B" title="Список служебных страниц [q]" accesskey="q"><span>Служебные страницы</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&oldid=135261755" title="Постоянная ссылка на эту версию страницы"><span>Постоянная ссылка</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&action=info" title="Подробнее об этой странице"><span>Сведения о странице</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A6%D0%B8%D1%82%D0%B0%D1%82%D0%B0&page=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&id=135261755&wpFormIdentifier=titleform" title="Информация о том, как цитировать эту страницу"><span>Цитировать страницу</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:UrlShortener&url=https%3A%2F%2Fru.wikipedia.org%2Fwiki%2F%25D0%25A7%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE_%25D0%259F%25D1%2580%25D0%25BE%25D1%2582%25D0%25B0"><span>Получить короткий URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:QrCode&url=https%3A%2F%2Fru.wikipedia.org%2Fwiki%2F%25D0%25A7%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE_%25D0%259F%25D1%2580%25D0%25BE%25D1%2582%25D0%25B0"><span>Скачать QR-код</span></a></li> </ul> </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 " > <span class="vector-menu-heading-label">Печать/экспорт</span> </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=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:DownloadAsPdf&page=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&action=show-download-screen" title="Скачать эту страницу как файл PDF"><span>Скачать как PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D1%80%D0%BE%D1%82%D0%B0&printable=yes" title="Версия этой страницы для печати [p]" accesskey="p"><span>Версия для печати</span></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 " > <span class="vector-menu-heading-label">В других проектах</span> </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/Q593418" title="Ссылка на связанный элемент репозитория данных [g]" accesskey="g"><span>Элемент Викиданных</span></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 " > <span class="vector-menu-heading-label">На других языках</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%88%D8%AB" title="عدد بروث — арабский" lang="ar" hreflang="ar" data-title="عدد بروث" data-language-autonym="العربية" data-language-local-name="арабский" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Proth" title="Nombre de Proth — каталанский" lang="ca" hreflang="ca" data-title="Nombre de Proth" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prothovo_%C4%8D%C3%ADslo" title="Prothovo číslo — чешский" lang="cs" hreflang="cs" data-title="Prothovo číslo" data-language-autonym="Čeština" data-language-local-name="чешский" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Prothsche_Primzahl" title="Prothsche Primzahl — немецкий" lang="de" hreflang="de" data-title="Prothsche Primzahl" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70893996 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Proth_number" title="Proth number — английский" lang="en" hreflang="en" data-title="Proth number" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Proth" title="Número de Proth — испанский" lang="es" hreflang="es" data-title="Número de Proth" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></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%BE%D8%B1%D9%88%D8%AA" title="اعداد پروت — персидский" lang="fa" hreflang="fa" data-title="اعداد پروت" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Prothin_alkuluku" title="Prothin alkuluku — финский" lang="fi" hreflang="fi" data-title="Prothin alkuluku" data-language-autonym="Suomi" data-language-local-name="финский" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Proth" title="Nombre de Proth — французский" lang="fr" hreflang="fr" data-title="Nombre de Proth" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Proth-sz%C3%A1mok" title="Proth-számok — венгерский" lang="hu" hreflang="hu" data-title="Proth-számok" data-language-autonym="Magyar" data-language-local-name="венгерский" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Proth" title="Numero di Proth — итальянский" lang="it" hreflang="it" data-title="Numero di Proth" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%AD%E3%82%B9%E6%95%B0" title="プロス数 — японский" lang="ja" hreflang="ja" data-title="プロス数" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%EB%A1%9C%ED%8A%B8_%EC%88%98" title="프로트 수 — корейский" lang="ko" hreflang="ko" data-title="프로트 수" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Prothgetal" title="Prothgetal — нидерландский" lang="nl" hreflang="nl" data-title="Prothgetal" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_de_Proth" title="Número de Proth — португальский" lang="pt" hreflang="pt" data-title="Número de Proth" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prothovo_%C5%A1tevilo" title="Prothovo število — словенский" lang="sl" hreflang="sl" data-title="Prothovo število" data-language-autonym="Slovenščina" data-language-local-name="словенский" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Prothtal" title="Prothtal — шведский" lang="sv" hreflang="sv" data-title="Prothtal" data-language-autonym="Svenska" data-language-local-name="шведский" class="interlanguage-link-target"><span>Svenska</span></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%9F%D1%80%D0%BE%D1%82%D0%B0" title="Числа Прота — украинский" lang="uk" hreflang="uk" data-title="Числа Прота" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%99%AE%E7%BD%97%E6%96%AF%E6%95%B0" title="普罗斯数 — китайский" lang="zh" hreflang="zh" data-title="普罗斯数" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%99%AE%E7%BE%85%E6%96%AF%E6%95%B8" title="普羅斯數 — кантонский" lang="yue" hreflang="yue" data-title="普羅斯數" data-language-autonym="粵語" data-language-local-name="кантонский" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q593418#sitelinks-wikipedia" title="Править ссылки на другие языки" class="wbc-editpage">Править ссылки</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 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