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class="vector-toc-numb">2</span> <span>Minimal primes in AP</span> </div> </a> <ul id="toc-Minimal_primes_in_AP-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Largest_known_primes_in_AP" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Largest_known_primes_in_AP"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Largest known primes in AP</span> </div> </a> <ul id="toc-Largest_known_primes_in_AP-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consecutive_primes_in_arithmetic_progression" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Consecutive_primes_in_arithmetic_progression"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Consecutive primes in arithmetic progression</span> </div> </a> <ul id="toc-Consecutive_primes_in_arithmetic_progression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimal_consecutive_primes_in_AP" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Minimal_consecutive_primes_in_AP"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Minimal consecutive primes in AP</span> </div> </a> <ul id="toc-Minimal_consecutive_primes_in_AP-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Largest_known_consecutive_primes_in_AP" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Largest_known_consecutive_primes_in_AP"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Largest known consecutive primes in AP</span> </div> </a> <ul id="toc-Largest_known_consecutive_primes_in_AP-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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searchaux" style="display:none">Set of prime numbers linked by a linear relationship</div> <p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <b>primes in arithmetic progression</b> are any <a href="/wiki/Sequence" title="Sequence">sequence</a> of at least three <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> that are consecutive terms in an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a>. An example is the sequence of primes (3, 7, 11), which is given by <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_{n}=3+4n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=3+4n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04333cbb39efb64fdc27a9a5de0b051ddfc6edcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.107ex; height:2.509ex;" alt="{\displaystyle a_{n}=3+4n}"></span> for <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 0\leq n\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq n\leq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f3c30caff3263640676bce53682877c2bdea42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.917ex; height:2.343ex;" alt="{\displaystyle 0\leq n\leq 2}"></span>. </p><p>According to the <a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a>, there exist <a href="/wiki/Arbitrarily_large" title="Arbitrarily large">arbitrarily long</a> arithmetic progressions in the sequence of primes. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form <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 an+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>n</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle an+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87f6fb82578ac8bc0a7757eaf98fc5125ab1f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:2.343ex;" alt="{\displaystyle an+b}"></span>, where <i>a</i> and <i>b</i> are <a href="/wiki/Coprime_integers" title="Coprime integers">coprime</a> which according to <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a> contains infinitely many primes, along with infinitely many composites. </p><p>For <a href="/wiki/Integer" title="Integer">integer</a> <i>k</i> ≥ 3, an <b>AP-<i>k</i></b> (also called <b>PAP-<i>k</i></b>) is any sequence of <i>k</i> primes in arithmetic progression. An AP-<i>k</i> can be written as <i>k</i> primes of the form <i>a</i>·<i>n</i> + <i>b</i>, for fixed integers <i>a</i> (called the common difference) and <i>b</i>, and <i>k</i> consecutive integer values of <i>n</i>. An AP-<i>k</i> is usually expressed with <i>n</i> = 0 to <i>k</i> − 1. This can always be achieved by defining <i>b</i> to be the first prime in the arithmetic progression. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any given arithmetic progression of primes has a finite length. In 2004, <a href="/wiki/Ben_J._Green" class="mw-redirect" title="Ben J. Green">Ben J. Green</a> and <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> settled an old <a href="/wiki/Conjecture" title="Conjecture">conjecture</a> by proving the <a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a>: The primes contain <a href="/wiki/Arbitrarily_large" title="Arbitrarily large">arbitrarily long</a> arithmetic progressions.<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> It follows immediately that there are infinitely many AP-<i>k</i> for any <i>k</i>. </p><p>If an AP-<i>k</i> does not begin with the prime <i>k</i>, then the common difference is a multiple of the <a href="/wiki/Primorial" title="Primorial">primorial</a> <i>k</i># = 2·3·5·...·<i>j</i>, where <i>j</i> is the largest prime ≤ <i>k</i>. </p> <dl><dd><i>Proof</i>: Let the AP-<i>k</i> be <i>a</i>·<i>n</i> + <i>b</i> for <i>k</i> consecutive values of <i>n</i>. If a prime <i>p</i> does not divide <i>a</i>, then <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> says that <i>p</i> will divide every <i>p'</i>th term of the arithmetic progression. (From H.J. Weber, Cor.10 in ``Exceptional Prime Number Twins, Triplets and Multiplets," arXiv:1102.3075[math.NT]. See also Theor.2.3 in ``Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447[math.NT], Global J.P.A.Math 8(2012), in press.) If the AP is prime for <i>k</i> consecutive values, then <i>a</i> must therefore be divisible by all primes <i>p</i> ≤ <i>k</i>.</dd></dl> <p>This also shows that an AP with common difference <i>a</i> cannot contain more consecutive prime terms than the value of the smallest prime that does not divide <i>a</i>. </p><p>If <i>k</i> is prime then an AP-<i>k</i> can begin with <i>k</i> and have a common difference which is only a multiple of (<i>k</i>−1)# instead of <i>k</i>#. (From H. J. Weber, ``Less Regular Exceptional and Repeating Prime Number Multiplets," arXiv:1105.4092[math.NT], Sect.3.) For example, the AP-3 with primes {3, 5, 7} and common difference 2# = 2, or the AP-5 with primes {5, 11, 17, 23, 29} and common difference 4# = 6. It is conjectured that such examples exist for all primes <i>k</i>. As of 2018<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Primes_in_arithmetic_progression&action=edit">[update]</a></sup>, the largest prime for which this is confirmed is <i>k</i> = 19, for this AP-19 found by Wojciech Iżykowski in 2013: </p> <dl><dd>19 + 4244193265542951705·17#·n, for <i>n</i> = 0 to 18.<sup id="cite_ref-APrecords_2-0" class="reference"><a href="#cite_note-APrecords-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>It follows from widely believed conjectures, such as <a href="/wiki/Dickson%27s_conjecture" title="Dickson's conjecture">Dickson's conjecture</a> and some variants of the <a href="/wiki/First_Hardy%E2%80%93Littlewood_conjecture" title="First Hardy–Littlewood conjecture">prime k-tuple conjecture</a>, that if <i>p</i> > 2 is the smallest prime not dividing <i>a</i>, then there are infinitely many AP-(<i>p</i>−1) with common difference <i>a</i>. For example, 5 is the smallest prime not dividing 6, so there is expected to be infinitely many AP-4 with common difference 6, which is called a <a href="/wiki/Sexy_prime" title="Sexy prime">sexy prime</a> quadruplet. When <i>a</i> = 2, <i>p</i> = 3, it is the <a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">twin prime conjecture</a>, with an "AP-2" of 2 primes (<i>b</i>, <i>b</i> + 2). </p> <div class="mw-heading mw-heading2"><h2 id="Minimal_primes_in_AP">Minimal primes in AP</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=2" title="Edit section: Minimal primes in AP"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We minimize the last term.<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> <table class="wikitable"> <caption>Minimal AP-<i>k</i> </caption> <tbody><tr> <th><i>k</i></th> <th>Primes for <i>n</i> = 0 to <i>k</i>−1 </th></tr> <tr> <th>3 </th> <td>3 + 2<i>n</i> </td></tr> <tr> <th>4 </th> <td>5 + 6<i>n</i> </td></tr> <tr> <th>5 </th> <td>5 + 6<i>n</i> </td></tr> <tr> <th>6 </th> <td>7 + 30<i>n</i> </td></tr> <tr> <th>7 </th> <td>7 + 150<i>n</i> </td></tr> <tr> <th>8 </th> <td>199 + 210<i>n</i> </td></tr> <tr> <th>9 </th> <td>199 + 210<i>n</i> </td></tr> <tr> <th>10 </th> <td>199 + 210<i>n</i> </td></tr> <tr> <th>11 </th> <td>110437 + 13860<i>n</i> </td></tr> <tr> <th>12 </th> <td>110437 + 13860<i>n</i> </td></tr> <tr> <th>13 </th> <td>4943 + 60060<i>n</i> </td></tr> <tr> <th>14 </th> <td>31385539 + 420420<i>n</i> </td></tr> <tr> <th>15 </th> <td>115453391 + 4144140<i>n</i> </td></tr> <tr> <th>16 </th> <td>53297929 + 9699690<i>n</i> </td></tr> <tr> <th>17 </th> <td>3430751869 + 87297210<i>n</i> </td></tr> <tr> <th>18 </th> <td>4808316343 + 717777060<i>n</i> </td></tr> <tr> <th>19 </th> <td>8297644387 + 4180566390<i>n</i> </td></tr> <tr> <th>20 </th> <td>214861583621 + 18846497670<i>n</i> </td></tr> <tr> <th>21 </th> <td>5749146449311 + 26004868890<i>n</i> </td></tr> <tr> <th>22 </th> <td>19261849254523 + 784801917900<i>n</i> </td></tr> <tr> <th>23 </th> <td>403185216600637 + 2124513401010<i>n</i> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Largest_known_primes_in_AP">Largest known primes in AP</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=3" title="Edit section: Largest known primes in AP"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For prime <i>q</i>, <i>q</i># denotes the <a href="/wiki/Primorial" title="Primorial">primorial</a> 2·3·5·7·...·<i>q</i>. </p><p>As of September 2019<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Primes_in_arithmetic_progression&action=edit">[update]</a></sup>, the longest known AP-<i>k</i> is an AP-27. Several examples are known for AP-26. The first to be discovered was found on April 12, 2010, by Benoît Perichon on a <a href="/wiki/PlayStation_3" title="PlayStation 3">PlayStation 3</a> with software by Jarosław Wróblewski and Geoff Reynolds, ported to the PlayStation 3 by Bryan Little, in a distributed <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> project:<sup id="cite_ref-APrecords_2-1" class="reference"><a href="#cite_note-APrecords-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>43142746595714191 + 23681770·23#·<i>n</i>, for <i>n</i> = 0 to 25. (23# = 223092870) (sequence <span class="nowrap external"><a href="//oeis.org/A204189" class="extiw" title="oeis:A204189">A204189</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>By the time the first AP-26 was found the search was divided into 131,436,182 segments by <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a><sup id="cite_ref-PrimeGridForum_4-0" class="reference"><a href="#cite_note-PrimeGridForum-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and processed by 32/64bit CPUs, <a href="/wiki/Nvidia" title="Nvidia">Nvidia</a> <a href="/wiki/CUDA" title="CUDA">CUDA</a> GPUs, and <a href="/wiki/Cell_microprocessor" class="mw-redirect" title="Cell microprocessor">Cell microprocessors</a> around the world. </p><p>Before that, the record was an AP-25 found by Raanan Chermoni and Jarosław Wróblewski on May 17, 2008:<sup id="cite_ref-APrecords_2-2" class="reference"><a href="#cite_note-APrecords-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>6171054912832631 + 366384·23#·<i>n</i>, for <i>n</i> = 0 to 24. (23# = 223092870)</dd></dl> <p>The AP-25 search was divided into segments taking about 3 minutes on <a href="/wiki/Athlon_64" title="Athlon 64">Athlon 64</a> and Wróblewski reported "I think Raanan went through less than 10,000,000 such segments"<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> (this would have taken about 57 cpu years on Athlon 64). </p><p>The earlier record was an AP-24 found by Jarosław Wróblewski alone on January 18, 2007: </p> <dl><dd>468395662504823 + 205619·23#·<i>n</i>, for <i>n</i> = 0 to 23.</dd></dl> <p>For this Wróblewski reported he used a total of 75 computers: 15 64-bit <a href="/wiki/Athlon" title="Athlon">Athlons</a>, 15 dual core 64-bit <a href="/wiki/Pentium_D" title="Pentium D">Pentium D</a> 805, 30 32-bit Athlons 2500, and 15 <a href="/wiki/Duron" title="Duron">Durons</a> 900.<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> </p><p>The following table shows the largest known AP-<i>k</i> with the year of discovery and the number of <a href="/wiki/Decimal" title="Decimal">decimal</a> digits in the ending prime. Note that the largest known AP-<i>k</i> may be the end of an AP-(<i>k</i>+1). Some record setters choose to first compute a large set of primes of form <i>c</i>·<i>p</i>#+1 with fixed <i>p</i>, and then search for AP's among the values of <i>c</i> that produced a prime. This is reflected in the expression for some records. The expression can easily be rewritten as <i>a</i>·<i>n</i> + <i>b</i>. </p> <table class="wikitable"> <caption>Largest known AP-<i>k</i> as of December 2023<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Primes_in_arithmetic_progression&action=edit">[update]</a></sup><sup id="cite_ref-APrecords_2-3" class="reference"><a href="#cite_note-APrecords-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th><i>k</i></th> <th>Primes for <i>n</i> = 0 to <i>k</i>−1</th> <th>Digits</th> <th>Year</th> <th>Discoverer </th></tr> <tr> <th>3 </th> <td>(503·2<sup>1092022</sup>−1) + (1103·2<sup>3558176</sup> − 503·2<sup>1092022</sup>)·<i>n</i></td> <td align="right">1071122</td> <td>2022</td> <td>Ryan Propper, Serge Batalov </td></tr> <tr> <th>4 </th> <td>(263093407 + 928724769·<i>n</i>)·2<sup>99901</sup>−1</td> <td align="right">30083</td> <td>2022</td> <td>Serge Batalov </td></tr> <tr> <th>5 </th> <td>(440012137 + 18195056·<i>n</i>)·30941#+1</td> <td align="right">13338</td> <td>2022</td> <td>Serge Batalov </td></tr> <tr> <th>6 </th> <td>(1445494494 + 141836149·<i>n</i>)·16301# + 1</td> <td align="right">7036</td> <td>2018</td> <td>Ken Davis </td></tr> <tr> <th>7 </th> <td>(2554152639 + 577051223·<i>n</i>)·7927# + 1</td> <td align="right">3407</td> <td>2022</td> <td>Serge Batalov </td></tr> <tr> <th>8 </th> <td>(48098104751 + 3026809034·<i>n</i>)·5303# + 1</td> <td align="right">2271</td> <td>2019</td> <td>Norman Luhn, Paul Underwood, Ken Davis </td></tr> <tr> <th>9 </th> <td>(65502205462 + 6317280828·<i>n</i>)·2371# + 1</td> <td align="right">1014</td> <td>2012</td> <td>Ken Davis, Paul Underwood </td></tr> <tr> <th>10 </th> <td>(20794561384 + 1638155407·<i>n</i>)·1050# + 1</td> <td align="right">450</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>11 </th> <td>(16533786790 + 1114209832·<i>n</i>)·666# + 1</td> <td align="right">289</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>12 </th> <td>(15079159689 + 502608831·<i>n</i>)·420# + 1</td> <td align="right">180</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>13 </th> <td>(50448064213 + 4237116495·<i>n</i>)·229# + 1</td> <td align="right">103</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>14 </th> <td>(55507616633 + 670355577·<i>n</i>)·229# + 1</td> <td align="right">103</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>15 </th> <td>(14512034548 + 87496195·n)·149# + 1</td> <td align="right">68</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>16 </th> <td>(9700128038 + 75782144·(<i>n</i>+1))·83# + 1</td> <td align="right">43</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>17 </th> <td>(9700128038 + 75782144·<i>n</i>)·83# + 1</td> <td align="right">43</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>18 </th> <td>(33277396902 + 139569962·(<i>n</i>+1))·53# + 1</td> <td align="right">31</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>19 </th> <td>(33277396902 + 139569962·<i>n</i>)·53# + 1</td> <td align="right">31</td> <td>2019</td> <td>Norman Luhn </td></tr> <tr> <th>20 </th> <td>23 + 134181089232118748020·19#·<i>n</i></td> <td align="right">29</td> <td>2017</td> <td>Wojciech Izykowski </td></tr> <tr> <th>21 </th> <td>5547796991585989797641 + 29#·<i>n</i></td> <td align="right">22</td> <td>2014</td> <td>Jarosław Wróblewski </td></tr> <tr> <th>22 </th> <td>22231637631603420833 + 8·41#·(<i>n</i> + 1)</td> <td align="right">20</td> <td>2014</td> <td>Jarosław Wróblewski </td></tr> <tr> <th>23 </th> <td>22231637631603420833 + 8·41#·<i>n</i></td> <td align="right">20</td> <td>2014</td> <td>Jarosław Wróblewski </td></tr> <tr> <th>24 </th> <td>230885165611851841 + 297206938·23#·<i>n</i></td> <td align="right">19</td> <td>2023</td> <td>Rob Gahan, PrimeGrid </td></tr> <tr> <th>25 </th> <td>290969863970949269 + 322359616·23#·<i>n</i></td> <td align="right">19</td> <td>2024</td> <td>Rob Gahan, PrimeGrid </td></tr> <tr> <th>26 </th> <td>233313669346314209 + 331326280·23#·<i>n</i></td> <td align="right">19</td> <td>2024</td> <td>Rob Gahan, PrimeGrid </td></tr> <tr> <th>27 </th> <td>605185576317848261 + 155368778·23#·<i>n</i></td> <td align="right">19</td> <td>2023</td> <td>Michael Kwok, PrimeGrid </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Consecutive_primes_in_arithmetic_progression">Consecutive primes in arithmetic progression</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=4" title="Edit section: Consecutive primes in arithmetic progression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Consecutive primes in arithmetic progression</b> refers to at least three <i>consecutive</i> primes which are consecutive terms in an arithmetic progression. Note that unlike an AP-<i>k</i>, all the other numbers between the terms of the progression must be composite. For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime. </p><p>For an integer <i>k</i> ≥ 3, a <b>CPAP-<i>k</i></b> is <i>k</i> consecutive primes in arithmetic progression. It is conjectured there are arbitrarily long CPAP's. This would imply infinitely many CPAP-<i>k</i> for all <i>k</i>. The middle prime in a CPAP-3 is called a <a href="/wiki/Balanced_prime" title="Balanced prime">balanced prime</a>. The largest known as of 2022<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Primes_in_arithmetic_progression&action=edit">[update]</a></sup> has 15004 digits. </p><p>The first known CPAP-10 was found in 1998 by Manfred Toplic in the <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project CP10 which was organized by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.<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> This CPAP-10 has the smallest possible common difference, 7# = 210. The only other known CPAP-10 as of 2018 was found by the same people in 2008. </p><p>If a CPAP-11 exists then it must have a common difference which is a multiple of 11# = 2310. The difference between the first and last of the 11 primes would therefore be a multiple of 23100. The requirement for at least 23090 composite numbers between the 11 primes makes it appear extremely hard to find a CPAP-11. Dubner and Zimmermann estimate it would be at least 10<sup>12</sup> times harder than a CPAP-10.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Minimal_consecutive_primes_in_AP">Minimal consecutive primes in AP</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=5" title="Edit section: Minimal consecutive primes in AP"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first occurrence of a CPAP-<i>k</i> is only known for <i>k</i> ≤ 6 (sequence <span class="nowrap external"><a href="//oeis.org/A006560" class="extiw" title="oeis:A006560">A006560</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <table class="wikitable"> <caption>Minimal CPAP-<i>k</i><sup id="cite_ref-minitable_9-0" class="reference"><a href="#cite_note-minitable-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th><i>k</i></th> <th>Primes for <i>n</i> = 0 to <i>k</i>−1 </th></tr> <tr> <th>3 </th> <td>3 + 2<i>n</i> </td></tr> <tr> <th>4 </th> <td>251 + 6<i>n</i> </td></tr> <tr> <th>5 </th> <td>9843019 + 30<i>n</i> </td></tr> <tr> <th>6 </th> <td>121174811 + 30<i>n</i> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Largest_known_consecutive_primes_in_AP">Largest known consecutive primes in AP</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=6" title="Edit section: Largest known consecutive primes in AP"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table shows the largest known case of <i>k</i> consecutive primes in arithmetic progression, for <i>k</i> = 3 to 10. </p> <table class="wikitable"> <caption>Largest known CPAP-<i>k</i> as of June 2024<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Primes_in_arithmetic_progression&action=edit">[update]</a></sup>,<sup id="cite_ref-CPAPrecords_10-0" class="reference"><a href="#cite_note-CPAPrecords-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chris_K._Caldwell_11-0" class="reference"><a href="#cite_note-Chris_K._Caldwell-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th><i>k</i></th> <th>Primes for <i>n</i> = 0 to <i>k</i>−1</th> <th>Digits</th> <th>Year</th> <th>Discoverer </th></tr> <tr> <th>3 </th> <td>17484430616589 · 2<sup>54201</sup> - 7 + 6<i>n</i></td> <td align="right">16330</td> <td>2024</td> <td>Serge Batalov </td></tr> <tr> <th>4 </th> <td>35734184537 · 11677#/3 - 9 + 6<i>n</i></td> <td align="right">5002</td> <td>2024</td> <td>Serge Batalov </td></tr> <tr> <th>5 </th> <td>2738129459017 · 4211# + 3399421517 + 30<i>n</i></td> <td align="right">1805</td> <td>2022</td> <td>Serge Batalov </td></tr> <tr> <th>6 </th> <td>533098369554 · 2357# + 3399421517 + 30<i>n</i></td> <td align="right">1012</td> <td>2021</td> <td>Serge Batalov </td></tr> <tr> <th>7 </th> <td>145706980166212 · 1069# + <i>x</i><sub>253</sub> + 420 + 210<i>n</i></td> <td align="right">466</td> <td>2021</td> <td>Serge Batalov </td></tr> <tr> <th>8 </th> <td>8081110034864 · 619# + <i>x</i><sub>253</sub> + 210 + 210<i>n</i></td> <td align="right">272</td> <td>2021</td> <td>Serge Batalov </td></tr> <tr> <th>9 </th> <td>7661619169627 · 379# + <i>x</i><sub>153</sub> + 210<i>n</i></td> <td align="right">167</td> <td>2021</td> <td>Serge Batalov </td></tr> <tr> <th>10 </th> <td>189382061960492204 · 257# + <i>x</i><sub>106</sub> + 210<i>n</i></td> <td align="right">121</td> <td>2021</td> <td>Serge Batalov </td></tr></tbody></table> <p><i>x</i><sub><i>d</i></sub> is a <i>d</i>-digit number used in one of the above records to ensure a small factor in unusually many of the required composites between the primes.<br /> <small> <i>x</i><sub>106</sub> = 115376 22283279672627497420 78637565852209646810 56709682233916942487 50925234318597647097 08315833909447378791<br /> <i>x</i><sub>153</sub> = 9656383640115 03965472274037609810 69585305769447451085 87635040605371157826 98320398681243637298 57205796522034199218 09817841129732061363 55565433981118807417 = <i>x</i><sub>253</sub> % 379#<br /> <i>x</i><sub>253</sub> = 1617599298905 320471304802538356587398499979 836255156671030473751281181199 911312259550734373874520536148 519300924327947507674746679858 816780182478724431966587843672 408773388445788142740274329621 811879827349575247851843514012 399313201211101277175684636727<br /> </small> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham chain</a></li> <li><a href="/wiki/Szemer%C3%A9di%27s_theorem" title="Szemerédi's theorem">Szemerédi's theorem</a></li> <li><a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a></li> <li><a href="/wiki/Problems_involving_arithmetic_progressions" title="Problems involving arithmetic progressions">Problems involving arithmetic progressions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Primes_in_arithmetic_progression&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PrimeArithmeticProgression.html">"Prime Arithmetic Progression"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Prime+Arithmetic+Progression&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPrimeArithmeticProgression.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrimes+in+arithmetic+progression" class="Z3988"></span></span></li> <li>Jarosław Wróblewski, <a rel="nofollow" class="external text" href="http://www.math.uni.wroc.pl/~jwr/AP26/AP26v3.pdf"><i>How to search for 26 primes in arithmetic progression?</i></a></li> <li><a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">P. Erdős</a> and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fermat_number" title="Fermat number">Fermat (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>n</i></sup></sup> + 1</span>)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>p</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple"><i>k</i>-tuples</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twin_prime" title="Twin prime">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a class="mw-selflink selflink">Arithmetic progression (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i> + <i>a·n</i>, <i>n</i> = 0, 1, 2, 3, ...</span>)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (<span class="texhtml texhtml-big" style="font-size:110%;">consecutive <i>p</i> − <i>n</i>, <i>p</i>, <i>p</i> + <i>n</i></span>)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i> ± 1, 2<i>n</i> ± 1, 4<i>n</i> ± 1, …</span>)</a></li> <li><a href="/wiki/Chen_prime" title="Chen prime">Chen</a></li> <li><a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">Sophie Germain/Safe (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> + 1</span>)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> ± 1, 4<i>p</i> ± 3, 8<i>p</i> ± 7, ...</span>)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By size</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Megaprime" title="Megaprime">Mega (1,000,000+ digits)</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known</a> <ul><li><a href="/wiki/List_of_largest_known_primes_and_probable_primes" title="List of largest known primes and probable primes">list</a></li></ul></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Gaussian_integer#Gaussian_primes" title="Gaussian integer">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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