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class="vector-toc-list"> <li id="toc-Definition_via_binomial_coefficients" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_binomial_coefficients"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definition via binomial coefficients</span> </div> </a> <ul id="toc-Definition_via_binomial_coefficients-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_via_Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_Bernoulli_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definition via Bernoulli numbers</span> </div> </a> <ul id="toc-Definition_via_Bernoulli_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_via_irregular_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_irregular_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Definition via irregular pairs</span> </div> </a> <ul id="toc-Definition_via_irregular_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_via_harmonic_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_via_harmonic_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Definition via harmonic numbers</span> </div> </a> <ul id="toc-Definition_via_harmonic_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Search_and_current_status" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Search_and_current_status"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Search and current status</span> </div> </a> <ul id="toc-Search_and_current_status-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expected_number_of_Wolstenholme_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expected_number_of_Wolstenholme_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Expected number of Wolstenholme primes</span> </div> </a> <ul id="toc-Expected_number_of_Wolstenholme_primes-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">4</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">5</span> <span>Notes</span> </div> </a> <ul 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Special type of prime number</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme number</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title fn org">Wolstenholme prime</caption><tbody><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="/wiki/Joseph_Wolstenholme" title="Joseph Wolstenholme">Joseph Wolstenholme</a></td></tr><tr><th scope="row" class="infobox-label">Publication year</th><td class="infobox-data">1995<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></td></tr><tr><th scope="row" class="infobox-label">Author of publication</th><td class="infobox-data">McIntosh, R. J.</td></tr><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label">Conjectured <abbr title="number">no.</abbr> of terms</th><td class="infobox-data">Infinite</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Subsequence" title="Subsequence">Subsequence</a> of</th><td class="infobox-data"><a href="/wiki/Irregular_prime" class="mw-redirect" title="Irregular prime">Irregular primes</a></td></tr><tr><th scope="row" class="infobox-label">First terms</th><td class="infobox-data"><a href="/wiki/16843_(number)" class="mw-redirect" title="16843 (number)">16843</a>, <a href="/w/index.php?title=2124679_(number)&action=edit&redlink=1" class="new" title="2124679 (number) (page does not exist)">2124679</a></td></tr><tr><th scope="row" class="infobox-label">Largest known term</th><td class="infobox-data"><a href="/w/index.php?title=2124679_(number)&action=edit&redlink=1" class="new" title="2124679 (number) (page does not exist)">2124679</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> index</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><a rel="nofollow" class="external text" href="//oeis.org/A088164">A088164</a></li><li>Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4)</li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, a <b>Wolstenholme prime</b> is a special type of <a href="/wiki/Prime_number" title="Prime number">prime number</a> satisfying a stronger version of <a href="/wiki/Wolstenholme%27s_theorem" title="Wolstenholme's theorem">Wolstenholme's theorem</a>. Wolstenholme's theorem is a <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a> satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician <a href="/wiki/Joseph_Wolstenholme" title="Joseph Wolstenholme">Joseph Wolstenholme</a>, who first described this theorem in the 19th century. </p><p>Interest in these primes first arose due to their connection with <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. </p><p>The only two known Wolstenholme primes are 16843 and 2124679 (sequence <span class="nowrap external"><a href="//oeis.org/A088164" class="extiw" title="oeis:A088164">A088164 </a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). There are no other Wolstenholme primes less than 10<sup>11</sup>.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Are there any Wolstenholme primes other than 16843 and 2124679?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <p>Wolstenholme prime can be defined in a number of equivalent ways. </p> <div class="mw-heading mw-heading3"><h3 id="Definition_via_binomial_coefficients">Definition via binomial coefficients</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=2" title="Edit section: Definition via binomial coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Wolstenholme prime is a prime number <i>p</i> > 7 that satisfies the <a href="/wiki/Congruence_relation" title="Congruence relation">congruence</a> </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 {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b13571bbe849c2002383a3c22fe444eb9e0e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.571ex; height:6.176ex;" alt="{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}"></span></dd></dl> <p>where the expression in <a href="/wiki/Left-hand_side" class="mw-redirect" title="Left-hand side">left-hand side</a> denotes a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>.<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> In comparison, <a href="/wiki/Wolstenholme%27s_theorem" title="Wolstenholme's theorem">Wolstenholme's theorem</a> states that for every prime <i>p</i> > 3 the following congruence holds: </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 {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c125e688ba726cd79730275353ba870f800e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.571ex; height:6.176ex;" alt="{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Definition_via_Bernoulli_numbers">Definition via Bernoulli numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=3" title="Edit section: Definition via Bernoulli numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Wolstenholme prime is a prime <i>p</i> that divides the numerator of the <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a> <i>B</i><sub><i>p</i>−3</sub>.<sup id="cite_ref-FOOTNOTEClarkeJones2004553_4-0" class="reference"><a href="#cite_note-FOOTNOTEClarkeJones2004553-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMcIntosh1995387_5-0" class="reference"><a href="#cite_note-FOOTNOTEMcIntosh1995387-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEZhao200825_6-0" class="reference"><a href="#cite_note-FOOTNOTEZhao200825-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The Wolstenholme primes therefore form a subset of the <a href="/wiki/Irregular_prime" class="mw-redirect" title="Irregular prime">irregular primes</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Definition_via_irregular_pairs">Definition via irregular pairs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=4" title="Edit section: Definition via irregular pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Regular_prime" title="Regular prime">Irregular prime</a></div> <p>A Wolstenholme prime is a prime <i>p</i> such that (<i>p</i>, <i>p</i>–3) is an <a href="/wiki/Regular_prime#Irregular_pairs" title="Regular prime">irregular pair</a>.<sup id="cite_ref-FOOTNOTEJohnson1975114_7-0" class="reference"><a href="#cite_note-FOOTNOTEJohnson1975114-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBuhlerCrandallErnvallMetsänkylä1993152_8-0" class="reference"><a href="#cite_note-FOOTNOTEBuhlerCrandallErnvallMetsänkylä1993152-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Definition_via_harmonic_numbers">Definition via harmonic numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=5" title="Edit section: Definition via harmonic numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Wolstenholme prime is a prime <i>p</i> such that<sup id="cite_ref-FOOTNOTEZhao200718_9-0" class="reference"><a href="#cite_note-FOOTNOTEZhao200718-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </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 H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effe22f2b4909b06da128e9062e721a6b44e12b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.293ex; height:3.343ex;" alt="{\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}"></span></dd></dl> <p>i.e. the numerator of the <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</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 H_{p-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{p-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1d81a0bf4e1a02edf43809d750370032296ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.091ex; height:2.843ex;" alt="{\displaystyle H_{p-1}}"></span> expressed in lowest terms is divisible by <i>p</i><sup>3</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Search_and_current_status">Search and current status</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=6" title="Edit section: Search and current status"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2022. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993.<sup id="cite_ref-FOOTNOTERibenboim200423_11-0" class="reference"><a href="#cite_note-FOOTNOTERibenboim200423-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Up to 1.2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000700000000000000♠"></span>7</span></sup>, no further Wolstenholme primes were found.<sup id="cite_ref-FOOTNOTEZhao200725_12-0" class="reference"><a href="#cite_note-FOOTNOTEZhao200725-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This was later extended to 2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> by McIntosh in 1995 <sup id="cite_ref-FOOTNOTEMcIntosh1995387_5-1" class="reference"><a href="#cite_note-FOOTNOTEMcIntosh1995387-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and Trevisan & Weber were able to reach 2.5<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup>.<sup id="cite_ref-FOOTNOTETrevisanWeber2001283–284_13-0" class="reference"><a href="#cite_note-FOOTNOTETrevisanWeber2001283–284-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The latest result as of 2022 is that there are only those two Wolstenholme primes up to 10<sup>11</sup>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Expected_number_of_Wolstenholme_primes">Expected number of Wolstenholme primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=7" title="Edit section: Expected number of Wolstenholme primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ <i>x</i> is about <i>ln ln x</i>, where <i>ln</i> denotes the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. For each prime <i>p</i> ≥ 5, the <b>Wolstenholme quotient</b> is defined as </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 W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7024773bbdf022f35271ea3e945708dac3f7b220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.426ex; height:7.176ex;" alt="{\displaystyle W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.}"></span></dd></dl> <p>Clearly, <i>p</i> is a Wolstenholme prime if and only if <i>W</i><sub><i>p</i></sub> ≡ 0 (mod <i>p</i>). <a href="/wiki/Empirical_relationship" title="Empirical relationship">Empirically</a> one may assume that the remainders of <i>W</i><sub><i>p</i></sub> modulo <i>p</i> are <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniformly distributed</a> in the set {0, 1, ..., <i>p</i>–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/<i>p</i>.<sup id="cite_ref-FOOTNOTEMcIntosh1995387_5-2" class="reference"><a href="#cite_note-FOOTNOTEMcIntosh1995387-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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=Wolstenholme_prime&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich prime</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun prime</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson prime</a></li> <li><a href="/wiki/Table_of_congruences" title="Table of congruences">Table of congruences</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=Wolstenholme_prime&action=edit&section=9" 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 .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Wolstenholme primes were first described by McIntosh in <a href="#CITEREFMcIntosh1995">McIntosh 1995</a>, p. 385</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Wolstenholme_prime"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WolstenholmePrime.html">"Wolstenholme prime"</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=Wolstenholme+prime&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FWolstenholmePrime.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCook" class="citation cs2">Cook, J. D., <a rel="nofollow" class="external text" href="http://www.johndcook.com/binomial_coefficients.html"><i>Binomial coefficients</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">21 December</span> 2010</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Binomial+coefficients&rft.aulast=Cook&rft.aufirst=J.+D.&rft_id=http%3A%2F%2Fwww.johndcook.com%2Fbinomial_coefficients.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEClarkeJones2004553-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEClarkeJones2004553_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFClarkeJones2004">Clarke & Jones 2004</a>, p. 553.</span> </li> <li id="cite_note-FOOTNOTEMcIntosh1995387-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEMcIntosh1995387_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMcIntosh1995387_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMcIntosh1995387_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFMcIntosh1995">McIntosh 1995</a>, p. 387.</span> </li> <li id="cite_note-FOOTNOTEZhao200825-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZhao200825_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZhao2008">Zhao 2008</a>, p. 25.</span> </li> <li id="cite_note-FOOTNOTEJohnson1975114-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJohnson1975114_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohnson1975">Johnson 1975</a>, p. 114.</span> </li> <li id="cite_note-FOOTNOTEBuhlerCrandallErnvallMetsänkylä1993152-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBuhlerCrandallErnvallMetsänkylä1993152_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBuhlerCrandallErnvallMetsänkylä1993">Buhler et al. 1993</a>, p. 152.</span> </li> <li id="cite_note-FOOTNOTEZhao200718-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZhao200718_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZhao2007">Zhao 2007</a>, p. 18.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Selfridge and Pollack published the first Wolstenholme prime in <a href="#CITEREFSelfridgePollack1964">Selfridge & Pollack 1964</a>, p. 97 (see <a href="#CITEREFMcIntoshRoettger2007">McIntosh & Roettger 2007</a>, p. 2092).</span> </li> <li id="cite_note-FOOTNOTERibenboim200423-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERibenboim200423_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, p. 23.</span> </li> <li id="cite_note-FOOTNOTEZhao200725-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZhao200725_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZhao2007">Zhao 2007</a>, p. 25.</span> </li> <li id="cite_note-FOOTNOTETrevisanWeber2001283–284-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETrevisanWeber2001283–284_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTrevisanWeber2001">Trevisan & Weber 2001</a>, p. 283–284.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBookerHathiMossinghoffTrudgian2022" class="citation journal cs1">Booker, Andrew R.; Hathi, Shehzad; Mossinghoff, Michael J.; Trudgian, Timothy S. (1 July 2022). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s11139-021-00438-3">"Wolstenholme and Vandiver primes"</a>. <i>The Ramanujan Journal</i>. <b>58</b> (3): 913–941. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11139-021-00438-3">10.1007/s11139-021-00438-3</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9303">1572-9303</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Ramanujan+Journal&rft.atitle=Wolstenholme+and+Vandiver+primes&rft.volume=58&rft.issue=3&rft.pages=913-941&rft.date=2022-07-01&rft_id=info%3Adoi%2F10.1007%2Fs11139-021-00438-3&rft.issn=1572-9303&rft.aulast=Booker&rft.aufirst=Andrew+R.&rft.au=Hathi%2C+Shehzad&rft.au=Mossinghoff%2C+Michael+J.&rft.au=Trudgian%2C+Timothy+S.&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs11139-021-00438-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuhlerCrandallErnvallMetsänkylä1993" class="citation cs2">Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T. (1993), <a rel="nofollow" class="external text" href="http://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf">"Irregular Primes and Cyclotomic Invariants to Four Million"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a></i>, <b>61</b> (203): 151–153, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993MaCom..61..151B">1993MaCom..61..151B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2152942">10.2307/2152942</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2152942">2152942</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Irregular+Primes+and+Cyclotomic+Invariants+to+Four+Million&rft.volume=61&rft.issue=203&rft.pages=151-153&rft.date=1993&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2152942%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2152942&rft_id=info%3Abibcode%2F1993MaCom..61..151B&rft.aulast=Buhler&rft.aufirst=J.&rft.au=Crandall%2C+R.&rft.au=Ernvall%2C+R.&rft.au=Mets%C3%A4nkyl%C3%A4%2C+T.&rft_id=http%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1993-61-203%2FS0025-5718-1993-1197511-5%2FS0025-5718-1993-1197511-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.today/20210922005744/https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf">Archived</a> 22 September 2021 at <a href="/wiki/Archive.today" title="Archive.today">archive.today</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClarkeJones2004" class="citation cs2">Clarke, F.; Jones, C. (2004), <a rel="nofollow" class="external text" href="http://blms.oxfordjournals.org/content/36/4/553.full.pdf">"A Congruence for Factorials"</a> <span class="cs1-format">(PDF)</span>, <i>Bulletin of the London Mathematical Society</i>, <b>36</b> (4): 553–558, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FS0024609304003194">10.1112/S0024609304003194</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120202453">120202453</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=A+Congruence+for+Factorials&rft.volume=36&rft.issue=4&rft.pages=553-558&rft.date=2004&rft_id=info%3Adoi%2F10.1112%2FS0024609304003194&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120202453%23id-name%3DS2CID&rft.aulast=Clarke&rft.aufirst=F.&rft.au=Jones%2C+C.&rft_id=http%3A%2F%2Fblms.oxfordjournals.org%2Fcontent%2F36%2F4%2F553.full.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://www.webcitation.org/5vRE6GbVK?url=http://blms.oxfordjournals.org/content/36/4/553.full.pdf">Archived</a> 2 January 2011 at <a href="/wiki/WebCite" title="WebCite">WebCite</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson1975" class="citation cs2">Johnson, W. 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J. (1995), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf">"On the converse of Wolstenholme's Theorem"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Acta_Arithmetica" title="Acta Arithmetica">Acta Arithmetica</a></i>, <b>71</b> (4): 381–389, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-71-4-381-389">10.4064/aa-71-4-381-389</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=On+the+converse+of+Wolstenholme%27s+Theorem&rft.volume=71&rft.issue=4&rft.pages=381-389&rft.date=1995&rft_id=info%3Adoi%2F10.4064%2Faa-71-4-381-389&rft.aulast=McIntosh&rft.aufirst=R.+J.&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Faa%2Faa71%2Faa7144.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcIntoshRoettger2007" class="citation cs2">McIntosh, R. 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(2007), <a rel="nofollow" class="external text" href="http://www.ams.org/mcom/2007-76-260/S0025-5718-07-01955-2/S0025-5718-07-01955-2.pdf">"A search for Fibonacci-Wieferich and Wolstenholme primes"</a> <span class="cs1-format">(PDF)</span>, <i>Mathematics of Computation</i>, <b>76</b> (260): 2087–2094, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007MaCom..76.2087M">2007MaCom..76.2087M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-07-01955-2">10.1090/S0025-5718-07-01955-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=A+search+for+Fibonacci-Wieferich+and+Wolstenholme+primes&rft.volume=76&rft.issue=260&rft.pages=2087-2094&rft.date=2007&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-07-01955-2&rft_id=info%3Abibcode%2F2007MaCom..76.2087M&rft.aulast=McIntosh&rft.aufirst=R.+J.&rft.au=Roettger%2C+E.+L.&rft_id=http%3A%2F%2Fwww.ams.org%2Fmcom%2F2007-76-260%2FS0025-5718-07-01955-2%2FS0025-5718-07-01955-2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://www.webcitation.org/5usE0UWhy?url=http://citeseerx.ksu.edu.sa/viewdoc/download?doi=10.1.1.105.9393&rep=rep1&type=pdf">Archived</a> 10 December 2010 at <a href="/wiki/WebCite" title="WebCite">WebCite</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim2004" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, P.</a> (2004), "Chapter 2. How to Recognize Whether a Natural Number is a Prime", <i>The Little Book of Bigger Primes</i>, New York: Springer-Verlag New York, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20169-6" title="Special:BookSources/978-0-387-20169-6"><bdi>978-0-387-20169-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2.+How+to+Recognize+Whether+a+Natural+Number+is+a+Prime&rft.btitle=The+Little+Book+of+Bigger+Primes&rft.place=New+York&rft.pub=Springer-Verlag+New+York%2C+Inc.&rft.date=2004&rft.isbn=978-0-387-20169-6&rft.aulast=Ribenboim&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSelfridgePollack1964" class="citation cs2">Selfridge, J. L.; Pollack, B. W. (1964), "Fermat's last theorem is true for any exponent up to 25,000", <i>Notices of the American Mathematical Society</i>, <b>11</b>: 97</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Fermat%27s+last+theorem+is+true+for+any+exponent+up+to+25%2C000&rft.volume=11&rft.pages=97&rft.date=1964&rft.aulast=Selfridge&rft.aufirst=J.+L.&rft.au=Pollack%2C+B.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrevisanWeber2001" class="citation cs2">Trevisan, V.; Weber, K. E. (2001), <a rel="nofollow" class="external text" href="http://www.lume.ufrgs.br/bitstream/handle/10183/448/000317407.pdf?sequence=1">"Testing the Converse of Wolstenholme's Theorem"</a> <span class="cs1-format">(PDF)</span>, <i>Matemática Contemporânea</i>, <b>21</b> (16): 275–286, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.21711%2F231766362001%2Frmc2116">10.21711/231766362001/rmc2116</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Matem%C3%A1tica+Contempor%C3%A2nea&rft.atitle=Testing+the+Converse+of+Wolstenholme%27s+Theorem&rft.volume=21&rft.issue=16&rft.pages=275-286&rft.date=2001&rft_id=info%3Adoi%2F10.21711%2F231766362001%2Frmc2116&rft.aulast=Trevisan&rft.aufirst=V.&rft.au=Weber%2C+K.+E.&rft_id=http%3A%2F%2Fwww.lume.ufrgs.br%2Fbitstream%2Fhandle%2F10183%2F448%2F000317407.pdf%3Fsequence%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111006064608/http://www.lume.ufrgs.br/bitstream/handle/10183/448/000317407.pdf?sequence=1">Archived</a> 6 October 2011 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhao2007" class="citation cs2">Zhao, J. (2007), <a rel="nofollow" class="external text" href="http://home.eckerd.edu/~zhaoj/research/ZhaoJNTBern.pdf">"Bernoulli numbers, Wolstenholme's theorem, and p<sup>5</sup> variations of Lucas' theorem"</a> <span class="cs1-format">(PDF)</span>, <i>Journal of Number Theory</i>, <b>123</b>: 18–26, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2006.05.005">10.1016/j.jnt.2006.05.005</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:937685">937685</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Bernoulli+numbers%2C+Wolstenholme%27s+theorem%2C+and+p%3Csup%3E5%3C%2Fsup%3E+variations+of+Lucas%27+theorem&rft.volume=123&rft.pages=18-26&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2006.05.005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A937685%23id-name%3DS2CID&rft.aulast=Zhao&rft.aufirst=J.&rft_id=http%3A%2F%2Fhome.eckerd.edu%2F~zhaoj%2Fresearch%2FZhaoJNTBern.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100630160329/http://home.eckerd.edu/~zhaoj/research/ZhaoJNTBern.pdf">Archived</a> 30 June 2010 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhao2008" class="citation cs2">Zhao, J. (2008), <a rel="nofollow" class="external text" href="http://home.eckerd.edu/~zhaoj/research/ZhaoIJNT.pdf">"Wolstenholme Type Theorem for Multiple Harmonic Sums"</a> <span class="cs1-format">(PDF)</span>, <i>International Journal of Number Theory</i>, <b>4</b> (1): 73–106, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs1793042108001146">10.1142/s1793042108001146</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Number+Theory&rft.atitle=Wolstenholme+Type+Theorem+for+Multiple+Harmonic+Sums&rft.volume=4&rft.issue=1&rft.pages=73-106&rft.date=2008&rft_id=info%3Adoi%2F10.1142%2Fs1793042108001146&rft.aulast=Zhao&rft.aufirst=J.&rft_id=http%3A%2F%2Fhome.eckerd.edu%2F~zhaoj%2Fresearch%2FZhaoIJNT.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBabbage1819" class="citation cs2">Babbage, C. (1819), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KrA-AAAAYAAJ&pg=PA46">"Demonstration of a theorem relating to prime numbers"</a>, <i>The Edinburgh Philosophical Journal</i>, <b>1</b>: 46–49</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Edinburgh+Philosophical+Journal&rft.atitle=Demonstration+of+a+theorem+relating+to+prime+numbers&rft.volume=1&rft.pages=46-49&rft.date=1819&rft.aulast=Babbage&rft.aufirst=C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKrA-AAAAYAAJ%26pg%3DPA46&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrattenthalerRivoal2009" class="citation cs2">Krattenthaler, C.; Rivoal, T. (2009), "On the integrality of the Taylor coefficients of mirror maps, II", <i>Communications in Number Theory and Physics</i>, <b>3</b> (3): 555–591, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.2578">0907.2578</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009arXiv0907.2578K">2009arXiv0907.2578K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FCNTP.2009.v3.n3.a5">10.4310/CNTP.2009.v3.n3.a5</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Number+Theory+and+Physics&rft.atitle=On+the+integrality+of+the+Taylor+coefficients+of+mirror+maps%2C+II&rft.volume=3&rft.issue=3&rft.pages=555-591&rft.date=2009&rft_id=info%3Aarxiv%2F0907.2578&rft_id=info%3Adoi%2F10.4310%2FCNTP.2009.v3.n3.a5&rft_id=info%3Abibcode%2F2009arXiv0907.2578K&rft.aulast=Krattenthaler&rft.aufirst=C.&rft.au=Rivoal%2C+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolstenholme1862" class="citation cs2">Wolstenholme, J. (1862), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vL0KAAAAIAAJ&pg=PA35">"On Certain Properties of Prime Numbers"</a>, <i>The Quarterly Journal of Pure and Applied Mathematics</i>, <b>5</b>: 35–39</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Quarterly+Journal+of+Pure+and+Applied+Mathematics&rft.atitle=On+Certain+Properties+of+Prime+Numbers&rft.volume=5&rft.pages=35-39&rft.date=1862&rft.aulast=Wolstenholme&rft.aufirst=J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvL0KAAAAIAAJ%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWolstenholme+prime" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wolstenholme_prime&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Caldwell, Chris K. <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/xpage/Wolstenholme.html">Wolstenholme prime</a> from The Prime Glossary</li> <li>McIntosh, R. J. <a rel="nofollow" class="external text" href="http://www.loria.fr/~zimmerma/records/Wieferich.status">Wolstenholme Search Status as of March 2004</a> e-mail to <a href="/wiki/Paul_Zimmermann_(mathematician)" title="Paul Zimmermann (mathematician)">Paul Zimmermann</a></li> <li>Bruck, R. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130208001700/http://imperator.usc.edu/~bruck/research/stirling">Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients</a></li> <li>Conrad, K. <a rel="nofollow" class="external text" href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/padicharmonicsum.pdf">The <i>p</i>-adic Growth of Harmonic Sums</a> interesting observation involving the two Wolstenholme primes</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 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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 class="mw-selflink selflink">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 href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">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?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Wolstenholme_prime&oldid=1255812666">https://en.wikipedia.org/w/index.php?title=Wolstenholme_prime&oldid=1255812666</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Classes_of_prime_numbers" title="Category:Classes 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