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<span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factorization_of_decimal_repunits" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factorization_of_decimal_repunits"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Factorization of decimal repunits</span> </div> </a> <ul id="toc-Factorization_of_decimal_repunits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Repunit_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Repunit_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Repunit primes</span> </div> </a> <button aria-controls="toc-Repunit_primes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Repunit primes subsection</span> </button> <ul id="toc-Repunit_primes-sublist" class="vector-toc-list"> <li id="toc-Decimal_repunit_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decimal_repunit_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Decimal repunit primes</span> </div> </a> <ul id="toc-Decimal_repunit_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebra_factorization_of_generalized_repunit_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra_factorization_of_generalized_repunit_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Algebra factorization of generalized repunit numbers</span> </div> </a> <ul id="toc-Algebra_factorization_of_generalized_repunit_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_generalized_repunit_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_generalized_repunit_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>The generalized repunit conjecture</span> </div> </a> <ul id="toc-The_generalized_repunit_conjecture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Demlo_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Demlo_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Demlo numbers</span> </div> </a> <ul id="toc-Demlo_numbers-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-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Footnotes</span> </div> </a> <button aria-controls="toc-Footnotes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Footnotes subsection</span> </button> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</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-2"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> 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Available in 22 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-22" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">22 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A7%D9%84%D9%88%D8%A7%D8%AD%D8%AF_%D8%A7%D9%84%D9%85%D9%83%D8%B1%D8%B1" title="عدد الواحد المكرر – Arabic" lang="ar" hreflang="ar" data-title="عدد الواحد المكرر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BF%D1%8E%D0%BD%D0%B8%D1%82" title="Репюнит – Bulgarian" lang="bg" hreflang="bg" data-title="Репюнит" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Jedni%C4%8Dkov%C3%A9_%C4%8D%C3%ADslo" title="Jedničkové číslo – Czech" lang="cs" hreflang="cs" data-title="Jedničkové číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Repunit" title="Repunit – German" lang="de" hreflang="de" data-title="Repunit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Repunit" title="Repunit – Spanish" lang="es" hreflang="es" data-title="Repunit" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Repunito" title="Repunito – Esperanto" lang="eo" hreflang="eo" data-title="Repunito" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_repituno" title="Zenbaki repituno – Basque" lang="eu" hreflang="eu" data-title="Zenbaki repituno" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%8C%DA%A9%DB%8C%D8%AA%DA%A9%D8%B1%DB%8C" title="یکیتکری – Persian" lang="fa" hreflang="fa" data-title="یکیتکری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9punit" title="Répunit – French" lang="fr" hreflang="fr" data-title="Répunit" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%9C%84_%EB%B0%98%EB%B3%B5_%EC%86%8C%EC%88%98" title="단위 반복 소수 – Korean" lang="ko" hreflang="ko" data-title="단위 반복 소수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Repunit" title="Repunit – Italian" lang="it" hreflang="it" data-title="Repunit" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%99%D7%93%D7%94_%D7%97%D7%95%D7%96%D7%A8%D7%AA" title="יחידה חוזרת – Hebrew" lang="he" hreflang="he" data-title="יחידה חוזרת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Repunit" title="Repunit – Hungarian" lang="hu" hreflang="hu" data-title="Repunit" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AC%E3%83%94%E3%83%A5%E3%83%8B%E3%83%83%E3%83%88" title="レピュニット – Japanese" lang="ja" hreflang="ja" data-title="レピュニット" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_jedynkowe" title="Liczby jedynkowe – Polish" lang="pl" hreflang="pl" data-title="Liczby jedynkowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Repunit" title="Repunit – Romanian" lang="ro" hreflang="ro" data-title="Repunit" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BF%D1%8C%D1%8E%D0%BD%D0%B8%D1%82%D1%8B" title="Репьюниты – Russian" lang="ru" hreflang="ru" data-title="Репьюниты" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Repunit" title="Repunit – Simple English" lang="en-simple" hreflang="en-simple" data-title="Repunit" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BF%27%D1%8E%D0%BD%D1%96%D1%82%D0%B8" title="Реп'юніти – Ukrainian" lang="uk" hreflang="uk" data-title="Реп'юніти" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Repunit" title="Repunit – Vietnamese" lang="vi" hreflang="vi" data-title="Repunit" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AA%E7%92%B0%E5%96%AE%E4%BD%8D" title="循環單位 – Cantonese" lang="yue" hreflang="yue" data-title="循環單位" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%AA%E7%92%B0%E5%96%AE%E4%BD%8D" title="循環單位 – Chinese" lang="zh" hreflang="zh" data-title="循環單位" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-pp-default" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected."><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </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">Numbers that contain only the digit 1</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">Repunit prime</caption><tbody><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">11</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">First terms</th><td class="infobox-data"><a href="/wiki/11_(number)" title="11 (number)">11</a>, 1111111111111111111, 11111111111111111111111</td></tr><tr><th scope="row" class="infobox-label">Largest known term</th><td class="infobox-data">(10<sup>8177207</sup>−1)/9</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/A004022">A004022</a></li><li>Primes of the form (10^n − 1)/9</li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>, a <b>repunit</b> is a <a href="/wiki/Number" title="Number">number</a> like 11, 111, or 1111 that contains only the digit <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a> — a more specific type of <a href="/wiki/Repdigit" title="Repdigit">repdigit</a>. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book <i>Recreations in the Theory of Numbers</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p>A <b>repunit prime</b> is a repunit that is also a <a href="/wiki/Prime_number" title="Prime number">prime number</a>. Primes that are repunits in <a href="/wiki/Binary_number" title="Binary number">base-2</a> are <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne primes</a>. As of October 2024, the <a href="/wiki/Largest_known_prime_number" title="Largest known prime number">largest known prime number</a> <span class="nowrap">2<sup>136,279,841</sup> − 1</span>, the largest <a href="/wiki/Probable_prime" title="Probable prime">probable prime</a> <i>R</i><sub>8177207</sub> and the largest <a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">elliptic curve primality</a>-proven prime <i>R</i><sub>86453</sub> are all repunits in various bases. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div> <p>The base-<i>b</i> repunits are defined as (this <i>b</i> can be either positive or negative) </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 R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>≡</mo> <mn>1</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82db0cf04082c02a86b6641bf837d9a5eda0fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:61.746ex; height:5.676ex;" alt="{\displaystyle R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.}"></span></dd></dl> <p>Thus, the number <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> consists of <i>n</i> copies of the digit 1 in base-<i>b</i> representation. The first two repunits base-<i>b</i> for <i>n</i> = 1 and <i>n</i> = 2 are </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 R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em" /> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb943da34163b9652ff4eccb2afece020eb89e19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:67.495ex; height:6.009ex;" alt="{\displaystyle R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.}"></span></dd></dl> <p>In particular, the <i><a href="/wiki/Decimal" title="Decimal">decimal</a> (base-</i>10<i>) repunits</i> that are often referred to as simply <i>repunits</i> are 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 R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>9</mn> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2ea3ec0141ce2f78f1e2cc2b1be398355a591d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.276ex; height:5.509ex;" alt="{\displaystyle R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.}"></span></dd></dl> <p>Thus, the number <i>R</i><sub><i>n</i></sub> = <i>R</i><sub><i>n</i></sub><sup>(10)</sup> consists of <i>n</i> copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with </p> <dl><dd><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, <a href="/wiki/11_(number)" title="11 (number)">11</a>, <a href="/wiki/111_(number)" title="111 (number)">111</a>, 1111, 11111, 111111, ... (sequence <span class="nowrap external"><a href="//oeis.org/A002275" class="extiw" title="oeis:A002275">A002275</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>Similarly, the repunits base-2 are 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 R_{n}^{(2)}={2^{n}-1 \over {2-1}}={2^{n}-1}\qquad {\mbox{for }}n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}^{(2)}={2^{n}-1 \over {2-1}}={2^{n}-1}\qquad {\mbox{for }}n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67005ed8674900b0c75cfb806b29b56d150d5f8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.212ex; height:5.509ex;" alt="{\displaystyle R_{n}^{(2)}={2^{n}-1 \over {2-1}}={2^{n}-1}\qquad {\mbox{for }}n\geq 1.}"></span></dd></dl> <p>Thus, the number <i>R</i><sub><i>n</i></sub><sup>(2)</sup> consists of <i>n</i> copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne numbers</a> <i>M</i><sub><i>n</i></sub> = 2<sup><i>n</i></sup> − 1, they start with </p> <dl><dd>1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000225" class="extiw" title="oeis:A000225">A000225</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div> <ul><li>Any repunit in any base having a composite number of digits is necessarily composite. For example, <dl><dd><i>R</i><sub>35</sub><sup>(<i>b</i>)</sup> = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,</dd></dl></li></ul> <dl><dd>since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base-<i>b</i> in which the repunit is expressed.</dd> <dd>Only repunits (in any base) having a prime number of digits can be prime. This is a <a href="/wiki/Necessity_and_sufficiency" title="Necessity and sufficiency">necessary but not sufficient condition</a>. For example, <dl><dd><i>R</i><sub>11</sub><sup>(2)</sup> = 2<sup>11</sup> − 1 = 2047 = 23 × 89.</dd></dl></dd></dl> <ul><li>If <i>p</i> is an odd prime, then every prime <i>q</i> that divides <i>R</i><sub><i>p</i></sub><sup>(<i>b</i>)</sup> must be either 1 plus a multiple of 2<i>p,</i> or a factor of <i>b</i> − 1. For example, a prime factor of <i>R</i><sub>29</sub> is 62003 = 1 + 2·29·1069. The reason is that the prime <i>p</i> is the smallest exponent greater than 1 such that <i>q</i> divides <i>b<sup>p</sup></i> − 1, because <i>p</i> is prime. Therefore, unless <i>q</i> divides <i>b</i> − 1, <i>p</i> divides the <a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael function</a> of <i>q</i>, which is even and equal to <i>q</i> − 1.</li> <li>Any positive multiple of the repunit <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> contains at least <i>n</i> nonzero digits in base-<i>b</i>.</li> <li>Any number <i>x</i> is a two-digit repunit in base x − 1.</li> <li>The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base-5, 11111 in base-2) and 8191 (111 in base-90, 1111111111111 in base-2). The <a href="/wiki/Goormaghtigh_conjecture" title="Goormaghtigh conjecture">Goormaghtigh conjecture</a> says there are only these two cases.</li> <li>Using the <a href="/wiki/Pigeon-hole_principle" class="mw-redirect" title="Pigeon-hole principle">pigeon-hole principle</a> it can be easily shown that for <a href="/wiki/Coprime_integers" title="Coprime integers">relatively prime</a> natural numbers <i>n</i> and <i>b</i>, there exists a repunit in base-<i>b</i> that is a multiple of <i>n</i>. To see this consider repunits <i>R</i><sub>1</sub><sup>(<i>b</i>)</sup>,...,<i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup>. Because there are <i>n</i> repunits but only <i>n</i>−1 non-zero residues modulo <i>n</i> there exist two repunits <i>R</i><sub><i>i</i></sub><sup>(<i>b</i>)</sup> and <i>R</i><sub><i>j</i></sub><sup>(<i>b</i>)</sup> with 1 ≤ <i>i</i> < <i>j</i> ≤ <i>n</i> such that <i>R</i><sub><i>i</i></sub><sup>(<i>b</i>)</sup> and <i>R</i><sub><i>j</i></sub><sup>(<i>b</i>)</sup> have the same residue modulo <i>n</i>. It follows that <i>R</i><sub><i>j</i></sub><sup>(<i>b</i>)</sup> − <i>R</i><sub><i>i</i></sub><sup>(<i>b</i>)</sup> has residue 0 modulo <i>n</i>, i.e. is divisible by <i>n</i>. Since <i>R</i><sub><i>j</i></sub><sup>(<i>b</i>)</sup> − <i>R</i><sub><i>i</i></sub><sup>(<i>b</i>)</sup> consists of <i>j</i> − <i>i</i> ones followed by <i>i</i> zeroes, <span class="nowrap"><i>R</i><sub><i>j</i></sub><sup>(<i>b</i>)</sup> − <i>R</i><sub><i>i</i></sub><sup>(<i>b</i>)</sup> = <i>R</i><sub><i>j</i>−<i>i</i></sub><sup>(<i>b</i>)</sup> × <i>b</i><sup><i>i</i></sup></span>. Now <i>n</i> divides the left-hand side of this equation, so it also divides the right-hand side, but since <i>n</i> and <i>b</i> are relatively prime, <i>n</i> must divide <i>R</i><sub><i>j</i>−<i>i</i></sub><sup>(<i>b</i>)</sup>.</li> <li>The <a href="/wiki/Feit%E2%80%93Thompson_conjecture" title="Feit–Thompson conjecture">Feit–Thompson conjecture</a> is that <i>R</i><sub><i>q</i></sub><sup>(<i>p</i>)</sup> never divides <i>R</i><sub><i>p</i></sub><sup>(<i>q</i>)</sup> for two distinct primes <i>p</i> and <i>q</i>.</li> <li>Using the <a href="/wiki/Euclidean_Algorithm" class="mw-redirect" title="Euclidean Algorithm">Euclidean Algorithm</a> for repunits definition: <i>R</i><sub>1</sub><sup>(<i>b</i>)</sup> = 1; <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> = <i>R</i><sub><i>n</i>−1</sub><sup>(<i>b</i>)</sup> × <i>b</i> + 1, any consecutive repunits <i>R</i><sub><i>n</i>−1</sub><sup>(<i>b</i>)</sup> and <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> are relatively prime in any base-<i>b</i> for any <i>n</i>.</li> <li>If <i>m</i> and <i>n</i> have a common divisor <i>d</i>, <i>R</i><sub><i>m</i></sub><sup>(<i>b</i>)</sup> and <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> have the common divisor <i>R</i><sub><i>d</i></sub><sup>(<i>b</i>)</sup> in any base-<i>b</i> for any <i>m</i> and <i>n</i>. That is, the repunits of a fixed base form a <a href="/wiki/Divisibility_sequence" title="Divisibility sequence">strong divisibility sequence</a>. As a consequence, If <i>m</i> and <i>n</i> are relatively prime, <i>R</i><sub><i>m</i></sub><sup>(<i>b</i>)</sup> and <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> are relatively prime. The Euclidean Algorithm is based on <i>gcd</i>(<i>m</i>, <i>n</i>) = <i>gcd</i>(<i>m</i> − <i>n</i>, <i>n</i>) for <i>m</i> > <i>n</i>. Similarly, using <i>R</i><sub><i>m</i></sub><sup>(<i>b</i>)</sup> − <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> × <i>b</i><sup><i>m</i>−<i>n</i></sup> = <i>R</i><sub><i>m</i>−<i>n</i></sub><sup>(<i>b</i>)</sup>, it can be easily shown that <i>gcd</i>(<i>R</i><sub><i>m</i></sub><sup>(<i>b</i>)</sup>, <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup>) = <i>gcd</i>(<i>R</i><sub><i>m</i>−<i>n</i></sub><sup>(<i>b</i>)</sup>, <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup>) for <i>m</i> > <i>n</i>. Therefore, if <i>gcd</i>(<i>m</i>, <i>n</i>) = <i>d</i>, then <i>gcd</i>(<i>R</i><sub><i>m</i></sub><sup>(<i>b</i>)</sup>, <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup>) = <i>R<sub>d</sub></i><sup>(<i>b</i>)</sup>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Factorization_of_decimal_repunits">Factorization of decimal repunits</h2></div> <p>(Prime factors colored <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:red">red</span> means "new factors", i. e. the prime factor divides <i>R</i><sub><i>n</i></sub> but does not divide <i>R</i><sub><i>k</i></sub> for all <i>k</i> < <i>n</i>) (sequence <span class="nowrap external"><a href="//oeis.org/A102380" class="extiw" title="oeis:A102380">A102380</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <table> <tbody><tr> <td> <table> <tbody><tr> <td><i>R</i><sub>1</sub> =</td> <td>1 </td></tr> <tr> <td><i>R</i><sub>2</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">11</span> </td></tr> <tr> <td><i>R</i><sub>3</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">3</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">37</span> </td></tr> <tr> <td><i>R</i><sub>4</sub> =</td> <td>11 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">101</span> </td></tr> <tr> <td><i>R</i><sub>5</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">41</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">271</span> </td></tr> <tr> <td><i>R</i><sub>6</sub> =</td> <td>3 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">7</span> · 11 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">13</span> · 37 </td></tr> <tr> <td><i>R</i><sub>7</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">239</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">4649</span> </td></tr> <tr> <td><i>R</i><sub>8</sub> =</td> <td>11 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">73</span> · 101 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">137</span> </td></tr> <tr> <td><i>R</i><sub>9</sub> =</td> <td>3<sup>2</sup> · 37 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">333667</span> </td></tr> <tr> <td><i>R</i><sub>10</sub> =</td> <td>11 · 41 · 271 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">9091</span> </td></tr></tbody></table> </td> <td> <table> <tbody><tr> <td><i>R</i><sub>11</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">21649</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">513239</span> </td></tr> <tr> <td><i>R</i><sub>12</sub> =</td> <td>3 · 7 · 11 · 13 · 37 · 101 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">9901</span> </td></tr> <tr> <td><i>R</i><sub>13</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">53</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">79</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">265371653</span> </td></tr> <tr> <td><i>R</i><sub>14</sub> =</td> <td>11 · 239 · 4649 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">909091</span> </td></tr> <tr> <td><i>R</i><sub>15</sub> =</td> <td>3 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">31</span> · 37 · 41 · 271 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">2906161</span> </td></tr> <tr> <td><i>R</i><sub>16</sub> =</td> <td>11 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">17</span> · 73 · 101 · 137 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">5882353</span> </td></tr> <tr> <td><i>R</i><sub>17</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">2071723</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">5363222357</span> </td></tr> <tr> <td><i>R</i><sub>18</sub> =</td> <td>3<sup>2</sup> · 7 · 11 · 13 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">19</span> · 37 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">52579</span> · 333667 </td></tr> <tr> <td><i>R</i><sub>19</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">1111111111111111111</span> </td></tr> <tr> <td><i>R</i><sub>20</sub> =</td> <td>11 · 41 · 101 · 271 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">3541</span> · 9091 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">27961</span> </td></tr></tbody></table> </td> <td> <table> <tbody><tr> <td><i>R</i><sub>21</sub> =</td> <td>3 · 37 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">43</span> · 239 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">1933</span> · 4649 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">10838689</span> </td></tr> <tr> <td><i>R</i><sub>22</sub> =</td> <td>11<sup>2</sup> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">23</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">4093</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">8779</span> · 21649 · 513239 </td></tr> <tr> <td><i>R</i><sub>23</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">11111111111111111111111</span> </td></tr> <tr> <td><i>R</i><sub>24</sub> =</td> <td>3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">99990001</span> </td></tr> <tr> <td><i>R</i><sub>25</sub> =</td> <td>41 · 271 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">21401</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">25601</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">182521213001</span> </td></tr> <tr> <td><i>R</i><sub>26</sub> =</td> <td>11 · 53 · 79 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">859</span> · 265371653 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">1058313049</span> </td></tr> <tr> <td><i>R</i><sub>27</sub> =</td> <td>3<sup>3</sup> · 37 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">757</span> · 333667 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">440334654777631</span> </td></tr> <tr> <td><i>R</i><sub>28</sub> =</td> <td>11 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">29</span> · 101 · 239 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">281</span> · 4649 · 909091 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">121499449</span> </td></tr> <tr> <td><i>R</i><sub>29</sub> =</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">3191</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">16763</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">43037</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">62003</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">77843839397</span> </td></tr> <tr> <td><i>R</i><sub>30</sub> =</td> <td>3 · 7 · 11 · 13 · 31 · 37 · 41 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">211</span> · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">241</span> · 271 · <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">2161</span> · 9091 · 2906161 </td></tr></tbody></table> </td></tr></tbody></table> <p>Smallest prime factor of <i>R</i><sub><i>n</i></sub> for <i>n</i> > 1 are </p> <dl><dd>11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... (sequence <span class="nowrap external"><a href="//oeis.org/A067063" class="extiw" title="oeis:A067063">A067063</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Repunit_primes">Repunit primes</h2></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">For a more comprehensive list, see <a href="/wiki/List_of_repunit_primes" title="List of repunit primes">List of repunit primes</a>.</div> <p>The definition of repunits was motivated by recreational mathematicians looking for <a href="/wiki/Integer_factorization" title="Integer factorization">prime factors</a> of such numbers. </p><p>It is easy to show that if <i>n</i> is divisible by <i>a</i>, then <i>R</i><sub><i>n</i></sub><sup>(<i>b</i>)</sup> is divisible by <i>R</i><sub><i>a</i></sub><sup>(<i>b</i>)</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 R_{n}^{(b)}={\frac {1}{b-1}}\prod _{d|n}\Phi _{d}(b),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mrow> </munder> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}^{(b)}={\frac {1}{b-1}}\prod _{d|n}\Phi _{d}(b),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d732d51bff698c8e4567105e18dc4a66f42398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.883ex; height:6.843ex;" alt="{\displaystyle R_{n}^{(b)}={\frac {1}{b-1}}\prod _{d|n}\Phi _{d}(b),}"></span></dd></dl> <p>where <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 \Phi _{d}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{d}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a30cb223d12e7cb0cdac9a0cffd5f59e94451c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.909ex; height:2.843ex;" alt="{\displaystyle \Phi _{d}(x)}"></span> is the <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 d^{\mathrm {th} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{\mathrm {th} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6d04da5e55ddb7b340fbbffedc38d6cf3e45f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.004ex; height:2.676ex;" alt="{\displaystyle d^{\mathrm {th} }}"></span> <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomial</a> and <i>d</i> ranges over the divisors of <i>n</i>. For <i>p</i> prime, </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 \Phi _{p}(x)=\sum _{i=0}^{p-1}x^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{p}(x)=\sum _{i=0}^{p-1}x^{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2cc66498ed5ce37a10742ae00a6bbbad63d723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.493ex; height:7.343ex;" alt="{\displaystyle \Phi _{p}(x)=\sum _{i=0}^{p-1}x^{i},}"></span></dd></dl> <p>which has the expected form of a repunit when <i>x</i> is substituted with <i>b</i>. </p><p>For example, 9 is divisible by 3, and thus <i>R</i><sub>9</sub> is divisible by <i>R</i><sub>3</sub>—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials <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 \Phi _{3}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{3}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367a0b0ebb1a6c1fb11e98f085ccf9955759a0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \Phi _{3}(x)}"></span> and <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 \Phi _{9}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{9}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e149ab26c190cdd81bfc6a2c20e2db87c1696cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \Phi _{9}(x)}"></span> are <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 x^{2}+x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78235883dfed13f5c0c7b6fb5aa82c002a1ac649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.557ex; height:2.843ex;" alt="{\displaystyle x^{2}+x+1}"></span> and <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 x^{6}+x^{3}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{6}+x^{3}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4672a19cbff72e32d609a58b51a887a95a58f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.611ex; height:2.843ex;" alt="{\displaystyle x^{6}+x^{3}+1}"></span>, respectively. Thus, for <i>R</i><sub><i>n</i></sub> to be prime, <i>n</i> must necessarily be prime, but it is not sufficient for <i>n</i> to be prime. For example, <i>R</i><sub>3</sub> = 111 = 3 · 37 is not prime. Except for this case of <i>R</i><sub>3</sub>, <i>p</i> can only divide <i>R</i><sub><i>n</i></sub> for prime <i>n</i> if <i>p</i> = 2<i>kn</i> + 1 for some <i>k</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Decimal_repunit_primes">Decimal repunit primes</h3></div> <p><i>R</i><sub><i>n</i></sub> is prime for <i>n</i> = 2, 19, 23, 317, 1031, 49081, 86453 ... (sequence <a href="//oeis.org/A004023" class="extiw" title="oeis:A004023">A004023</a> in <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a>). On April 3, 2007 <a href="/wiki/Harvey_Dubner" title="Harvey Dubner">Harvey Dubner</a> (who also found <i>R</i><sub>49081</sub>) announced that <i>R</i><sub>109297</sub> is a probable prime.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> On July 15, 2007, Maksym Voznyy announced <i>R</i><sub>270343</sub> to be probably prime.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Serge Batalov and Ryan Propper found <i>R</i><sub>5794777</sub> and <i>R</i><sub>8177207</sub> to be probable primes on April 20 and May 8, 2021, respectively.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> As of their discovery each was the largest known probable prime. On March 22, 2022 probable prime <i>R</i><sub>49081</sub> was eventually proven to be a prime.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> On May 15, 2023 probable prime <i>R</i><sub>86453</sub> was eventually proven to be a prime.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>It has been conjectured that there are infinitely many repunit primes<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and they seem to occur roughly as often as the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> would predict: the exponent of the <i>N</i>th repunit prime is generally around a fixed multiple of the exponent of the (<i>N</i>−1)th. </p><p>The prime repunits are a trivial subset of the <a href="/wiki/Permutable_prime" title="Permutable prime">permutable primes</a>, i.e., primes that remain prime after any <a href="/wiki/Permutation" title="Permutation">permutation</a> of their digits. </p><p>Particular properties are </p> <ul><li>The remainder of <i>R</i><sub><i>n</i></sub> modulo 3 is equal to the remainder of <i>n</i> modulo 3. Using 10<sup><i>a</i></sup> ≡ 1 (mod 3) for any <i>a</i> ≥ 0,<br /><i>n</i> ≡ 0 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ 0 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ 0 (mod <i>R</i><sub>3</sub>),<br /><i>n</i> ≡ 1 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ 1 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ <i>R</i><sub>1</sub> ≡ 1 (mod <i>R</i><sub>3</sub>),<br /><i>n</i> ≡ 2 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ 2 (mod 3) ⇔ <i>R</i><sub><i>n</i></sub> ≡ <i>R</i><sub>2</sub> ≡ 11 (mod <i>R</i><sub>3</sub>).<br />Therefore, 3 | <i>n</i> ⇔ 3 | <i>R</i><sub><i>n</i></sub> ⇔ <i>R</i><sub>3</sub> | <i>R</i><sub><i>n</i></sub>.</li> <li>The remainder of <i>R</i><sub><i>n</i></sub> modulo 9 is equal to the remainder of <i>n</i> modulo 9. Using 10<sup><i>a</i></sup> ≡ 1 (mod 9) for any <i>a</i> ≥ 0,<br /><i>n</i> ≡ <i>r</i> (mod 9) ⇔ <i>R</i><sub><i>n</i></sub> ≡ <i>r</i> (mod 9) ⇔ <i>R</i><sub><i>n</i></sub> ≡ <i>R</i><sub><i>r</i></sub> (mod <i>R</i><sub>9</sub>),<br />for 0 ≤ <i>r</i> < 9.<br />Therefore, 9 | <i>n</i> ⇔ 9 | <i>R</i><sub><i>n</i></sub> ⇔ <i>R</i><sub>9</sub> | <i>R</i><sub><i>n</i></sub>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebra_factorization_of_generalized_repunit_numbers">Algebra factorization of generalized repunit numbers</h3></div> <p>If <i>b</i> is a <a href="/wiki/Perfect_power" title="Perfect power">perfect power</a> (can be written as <i>m</i><sup><i>n</i></sup>, with <i>m</i>, <i>n</i> integers, <i>n</i> > 1) differs from 1, then there is at most one repunit in base-<i>b</i>. If <i>n</i> is a <a href="/wiki/Prime_power" title="Prime power">prime power</a> (can be written as <i>p</i><sup><i>r</i></sup>, with <i>p</i> prime, <i>r</i> integer, <i>p</i>, <i>r</i> >0), then all repunit in base-<i>b</i> are not prime aside from <i>R<sub>p</sub></i> and <i>R<sub>2</sub></i>. <i>R<sub>p</sub></i> can be either prime or composite, the former examples, <i>b</i> = −216, −128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the latter examples, <i>b</i> = −243, −125, −64, −32, −27, −8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and <i>R<sub>2</sub></i> can be prime (when <i>p</i> differs from 2) only if <i>b</i> is negative, a power of −2, for example, <i>b</i> = −8, −32, −128, −8192, etc., in fact, the <i>R<sub>2</sub></i> can also be composite, for example, <i>b</i> = −512, −2048, −32768, etc. If <i>n</i> is not a prime power, then no base-<i>b</i> repunit prime exists, for example, <i>b</i> = 64, 729 (with <i>n</i> = 6), <i>b</i> = 1024 (with <i>n</i> = 10), and <i>b</i> = −1 or 0 (with <i>n</i> any natural number). Another special situation is <i>b</i> = −4<i>k</i><sup>4</sup>, with <i>k</i> positive integer, which has the <a href="/wiki/Aurifeuillean_factorization" title="Aurifeuillean factorization">aurifeuillean factorization</a>, for example, <i>b</i> = −4 (with <i>k</i> = 1, then <i>R<sub>2</sub></i> and <i>R<sub>3</sub></i> are primes), and <i>b</i> = −64, −324, −1024, −2500, −5184, ... (with <i>k</i> = 2, 3, 4, 5, 6, ...), then no base-<i>b</i> repunit prime exists. It is also conjectured that when <i>b</i> is neither a perfect power nor −4<i>k</i><sup>4</sup> with <i>k</i> positive integer, then there are infinity many base-<i>b</i> repunit primes. </p> <div class="mw-heading mw-heading3"><h3 id="The_generalized_repunit_conjecture">The generalized repunit conjecture</h3></div> <p>A conjecture related to the generalized repunit primes:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> (the conjecture predicts where is the next <a href="/wiki/Generalized_Mersenne_prime" class="mw-redirect" title="Generalized Mersenne prime">generalized Mersenne prime</a>, if the conjecture is true, then there are infinitely many repunit primes for all bases <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>) </p><p>For any integer <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, which satisfies the conditions: </p> <ol><li><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 |b|>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b|>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87838d1a2091562cfe8bdead8d53039d460518a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.552ex; height:2.843ex;" alt="{\displaystyle |b|>1}"></span>.</li> <li><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is not a <a href="/wiki/Perfect_power" title="Perfect power">perfect power</a>. (since when <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is a perfect <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>th power, it can be shown that there is at most one <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> value such that <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 {\frac {b^{n}-1}{b-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{n}-1}{b-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662c3a415dc9367419e534ff0301dd1fa61bd9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.055ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{n}-1}{b-1}}}"></span> is prime, and this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> value is <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> itself or a <a href="/wiki/Nth_root" title="Nth root">root</a> of <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>)</li> <li><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is not in 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 -4k^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4k^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c946dc22e8c2310749f5c56f694d09bf905d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle -4k^{4}}"></span>. (if so, then the number has <a href="/wiki/Aurifeuillean_factorization" title="Aurifeuillean factorization">aurifeuillean factorization</a>)</li></ol> <p>has generalized repunit primes of the form </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 R_{p}(b)={\frac {b^{p}-1}{b-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{p}(b)={\frac {b^{p}-1}{b-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a525481625cbf4b241dcb1e3f9d54627705a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.624ex; height:5.676ex;" alt="{\displaystyle R_{p}(b)={\frac {b^{p}-1}{b-1}}}"></span></dd></dl> <p>for prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, the prime numbers will be distributed near the best fit line </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 Y=G\cdot \log _{|b|}\left(\log _{|b|}\left(R_{(b)}(n)\right)\right)+C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>G</mi> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=G\cdot \log _{|b|}\left(\log _{|b|}\left(R_{(b)}(n)\right)\right)+C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/555a7e8bee6f5ea198f909c24e4076afbc145d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.724ex; height:3.343ex;" alt="{\displaystyle Y=G\cdot \log _{|b|}\left(\log _{|b|}\left(R_{(b)}(n)\right)\right)+C,}"></span></dd></dl> <p>where limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>, <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 G={\frac {1}{e^{\gamma }}}=0.561459483566...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0.561459483566...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G={\frac {1}{e^{\gamma }}}=0.561459483566...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19145c3679286f5975b7774808afbc16056dd803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.767ex; height:5.176ex;" alt="{\displaystyle G={\frac {1}{e^{\gamma }}}=0.561459483566...}"></span> </p><p>and there are about </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 \left(\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}{\big (}\log _{e}(N){\big )}+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(|b|)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>m</mi> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <mo>−</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}{\big (}\log _{e}(N){\big )}+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(|b|)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8edf25ebdbc7ccff0da49f62d97aa84a6ca3cee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.313ex; height:6.509ex;" alt="{\displaystyle \left(\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}{\big (}\log _{e}(N){\big )}+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(|b|)}}}"></span></dd></dl> <p>base-<i>b</i> repunit primes less than <i>N</i>. </p> <ul><li><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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> is the <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">base of natural logarithm</a>.</li> <li><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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>.</li> <li><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 \log _{|b|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{|b|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9997950651d9b501282bf0b7bf6739b29166ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.824ex; height:3.009ex;" alt="{\displaystyle \log _{|b|}}"></span> is the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> in <a href="/wiki/Base_(exponentiation)" title="Base (exponentiation)">base</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 |b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881f49e94388a46a05d329251551ce20baf4f05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.291ex; height:2.843ex;" alt="{\displaystyle |b|}"></span></li> <li><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 R_{(b)}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{(b)}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ea4310ca268fa70daf3c28c3513a88e2ae511d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.185ex; height:3.176ex;" alt="{\displaystyle R_{(b)}(n)}"></span> is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>th generalized repunit prime in base<i>b</i> (with prime <i>p</i>)</li> <li><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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is a data fit constant which varies with <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>.</li> <li><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 \delta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c5c1a63ccd876384e6cf0a31296e3aee31ac84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta =1}"></span> if <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 b>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}"></span>, <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 \delta =1.6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mn>1.6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =1.6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04ec7a7207c04b4c30947134d40aa426e4b884f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.119ex; height:2.343ex;" alt="{\displaystyle \delta =1.6}"></span> if <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 b<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1319de18afa795d6489f49303614c84472f6d1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b<0}"></span>.</li> <li><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the largest natural number such that <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 -b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133483a1fba17ce25b728fa3448ea784c5ba3630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.806ex; height:2.343ex;" alt="{\displaystyle -b}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa87a0e12a49fd39ebc49301a85a531a8b9e8d94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.938ex; height:2.676ex;" alt="{\displaystyle 2^{m-1}}"></span>th power.</li></ul> <p>We also have the following 3 properties: </p> <ol><li>The number of prime numbers 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 {\frac {b^{n}-1}{b-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{n}-1}{b-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662c3a415dc9367419e534ff0301dd1fa61bd9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.055ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{n}-1}{b-1}}}"></span> (with prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) less than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is about <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 e^{\gamma }\cdot \log _{|b|}{\big (}\log _{|b|}(n){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\gamma }\cdot \log _{|b|}{\big (}\log _{|b|}(n){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fbe95e11e0ee4af34a427d12b7144109df9fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.644ex; height:3.343ex;" alt="{\displaystyle e^{\gamma }\cdot \log _{|b|}{\big (}\log _{|b|}(n){\big )}}"></span>.</li> <li>The expected number of prime numbers 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 {\frac {b^{n}-1}{b-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{n}-1}{b-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662c3a415dc9367419e534ff0301dd1fa61bd9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.055ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{n}-1}{b-1}}}"></span> with prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <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 |b|\cdot n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b|\cdot n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d119fe7dda8b206dbc6a2c258ad9ba974369e172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.365ex; height:2.843ex;" alt="{\displaystyle |b|\cdot n}"></span> is about <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 e^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15141c220223dbd2dcad746e6907b20ff6b1328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.208ex; height:2.343ex;" alt="{\displaystyle e^{\gamma }}"></span>.</li> <li>The probability that number 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 {\frac {b^{n}-1}{b-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{n}-1}{b-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662c3a415dc9367419e534ff0301dd1fa61bd9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.055ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{n}-1}{b-1}}}"></span> is prime (for prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) is about <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 {\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mrow> <mi>p</mi> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05caba40ef325f965b12c4918ea335cb6b048cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.755ex; height:6.009ex;" alt="{\displaystyle {\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}}"></span>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <p>Although they were not then known by that name, repunits in base-10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimals</a>.<sup id="cite_ref-Dickson1999_12-0" class="reference"><a href="#cite_note-Dickson1999-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>It was found very early on that for any prime <i>p</i> greater than 5, the <a href="/wiki/Repeating_decimal" title="Repeating decimal">period</a> of the decimal expansion of 1/<i>p</i> is equal to the length of the smallest repunit number that is divisible by <i>p</i>. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the <a href="/wiki/Integer_factorization" title="Integer factorization">factorization</a> by such mathematicians as Reuschle of all repunits up to <i>R<sub>16</sub></i> and many larger ones. By 1880, even <i>R<sub>17</sub></i> to <i>R<sub>36</sub></i> had been factored<sup id="cite_ref-Dickson1999_12-1" class="reference"><a href="#cite_note-Dickson1999-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and it is curious that, though <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a> showed no prime below three million had period <a href="/wiki/19_(number)" title="19 (number)">nineteen</a>, there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician Oscar Hoppe proved <i>R<sub>19</sub></i> to be prime in 1916<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and Lehmer and Kraitchik independently found <i>R<sub>23</sub></i> to be prime in 1929. </p><p>Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected. <i>R<sub>317</sub></i> was found to be a <a href="/wiki/Probable_prime" title="Probable prime">probable prime</a> circa 1966 and was proved prime eleven years later, when <i>R<sub>1031</sub></i> was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes. </p><p>Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size. </p><p>The <a href="/wiki/Cunningham_project" class="mw-redirect" title="Cunningham project">Cunningham project</a> endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12. </p> <div class="mw-heading mw-heading2"><h2 id="Demlo_numbers">Demlo numbers</h2></div> <p><a href="/wiki/D._R._Kaprekar" title="D. R. Kaprekar">D. R. Kaprekar</a> has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> They are named after <a href="/w/index.php?title=Demlo&action=edit&redlink=1" class="new" title="Demlo (page does not exist)">Demlo</a> railway station (now called <a href="/wiki/Dombivli_railway_station" title="Dombivli railway station">Dombivili</a>) 30 miles from Bombay on the then <a href="/wiki/G.I.P._Railway" class="mw-redirect" title="G.I.P. Railway">G.I.P. Railway</a>, where Kaprekar started investigating them. He calls <i>Wonderful Demlo numbers</i> those of the form 1, 121, 12321, 1234321, ..., 12345678987654321. The fact that these are the squares of the repunits has led some authors to call Demlo numbers the infinite sequence of these,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> 1, 121, 12321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequence <span class="nowrap external"><a href="//oeis.org/A002477" class="extiw" title="oeis:A002477">A002477</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), although one can check these are not Demlo numbers for <i>p</i> = 10, 19, 28, ... </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/All_one_polynomial" title="All one polynomial">All one polynomial</a> — Another generalization</li> <li><a href="/wiki/Goormaghtigh_conjecture" title="Goormaghtigh conjecture">Goormaghtigh conjecture</a></li> <li><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff prime</a> — can be thought of as repunit primes with <a href="/wiki/Negative_base" title="Negative base">negative base</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 b=-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85e516027c85ac29982ef9557d5e1c7a30a9941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle b=-2}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></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"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Albert H. Beiler coined the term "repunit number" as follows:<blockquote><p>A number which consists of a repeated of a single digit is sometimes called a monodigit number, and for convenience the author has used the term "repunit number" (repeated unit) to represent monodigit numbers consisting solely of the digit 1.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></p></blockquote></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="References">References</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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"><a href="#CITEREFBeiler2013">Beiler 2013</a>, pp. 83</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">For more information, see <a rel="nofollow" class="external text" href="http://stdkmd.net/nrr/repunit/">Factorization of repunit numbers</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Harvey Dubner, <i><a rel="nofollow" class="external text" href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178">New Repunit R(109297)</a></i></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Maksym Voznyy, <i><a rel="nofollow" class="external text" href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=NMBRTHRY&P=5903">New PRP Repunit R(270343)</a></i></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><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="CITEREFSloane_"A004023"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A004023">"Sequence A004023 (Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime.)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA004023%26%23x20%3B%28Indices+of+prime+repunits%3A+numbers+n+such+that+11...111+%28with+n+1%27s%29+%3D+%2810%5En+-+1%29%2F9+is+prime.%29&rft_id=https%3A%2F%2Foeis.org%2FA004023&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=133761">"PrimePage Primes: R(49081)"</a>. <i>PrimePage Primes</i>. 2022-03-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-03-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PrimePage+Primes&rft.atitle=PrimePage+Primes%3A+R%2849081%29&rft.date=2022-03-21&rft_id=https%3A%2F%2Fprimes.utm.edu%2Fprimes%2Fpage.php%3Fid%3D133761&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=136044">"PrimePage Primes: R(86453)"</a>. <i>PrimePage Primes</i>. 2023-05-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-05-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PrimePage+Primes&rft.atitle=PrimePage+Primes%3A+R%2886453%29&rft.date=2023-05-16&rft_id=https%3A%2F%2Fprimes.utm.edu%2Fprimes%2Fpage.php%3Fid%3D136044&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChris_Caldwell" class="citation web cs1">Chris Caldwell. <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/page.php?sort=Repunit">"repunit"</a>. <i>The Prime Glossary</i>. <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Prime+Glossary&rft.atitle=repunit&rft.au=Chris+Caldwell&rft_id=http%3A%2F%2Fprimes.utm.edu%2Fglossary%2Fpage.php%3Fsort%3DRepunit&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://primes.utm.edu/mersenne/heuristic.html">Deriving the Wagstaff Mersenne Conjecture</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No+Match%3BMatch%3BMatches&z=4">Generalized Repunit Conjecture</a></span> </li> <li id="cite_note-Dickson1999-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dickson1999_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dickson1999_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDicksonCresse1999">Dickson & Cresse 1999</a>, pp. 164–167</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFFrancis1988">Francis 1988</a>, pp. 240–246</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">Kaprekar <a href="#CITEREFKaprekar1938a">1938a</a>, <a href="#CITEREFKaprekar1938b">1938b</a>, <a href="#CITEREFGunjikarKaprekar1939">Gunjikar & Kaprekar 1939</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Demlo_Number"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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Cresse (page does not exist)">Cresse, G.H.</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XnwsAQAAIAAJ"><i>History of the Theory of Numbers</i></a>, Volume I: Divisibility and primality (2nd Reprinted ed.), Providence, RI: AMS Chelsea Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1934-0" title="Special:BookSources/978-0-8218-1934-0"><bdi>978-0-8218-1934-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+the+Theory+of+Numbers&rft.place=Providence%2C+RI&rft.series=Volume+I%3A+Divisibility+and+primality&rft.edition=2nd+Reprinted&rft.pub=AMS+Chelsea+Publishing&rft.date=1999&rft.isbn=978-0-8218-1934-0&rft.aulast=Dickson&rft.aufirst=Leonard+Eugene&rft.au=Cresse%2C+G.H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXnwsAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrancis1988" class="citation cs2"><a href="/w/index.php?title=Richard_L._Francis&action=edit&redlink=1" class="new" title="Richard L. 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R.</a> (1938a), <a rel="nofollow" class="external text" href="http://www.indianmathsociety.org.in/">"On Wonderful Demlo numbers"</a>, <i>The Mathematics Student</i>, <b>6</b>: 68</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Student&rft.atitle=On+Wonderful+Demlo+numbers&rft.volume=6&rft.pages=68&rft.date=1938&rft.aulast=Kaprekar&rft.aufirst=D.+R.&rft_id=http%3A%2F%2Fwww.indianmathsociety.org.in%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaprekar1938b" class="citation cs2"><a href="/wiki/D._R._Kaprekar" title="D. R. Kaprekar">Kaprekar, D. R.</a> (1938b), "Demlo numbers", <i>J. Phys. Sci. Univ. Bombay</i>, <b>VII</b> (3)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+Sci.+Univ.+Bombay&rft.atitle=Demlo+numbers&rft.volume=VII&rft.issue=3&rft.date=1938&rft.aulast=Kaprekar&rft.aufirst=D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaprekar1948" class="citation cs2"><a href="/wiki/D._R._Kaprekar" title="D. R. Kaprekar">Kaprekar, D. R.</a> (1948), <i>Demlo numbers</i>, Devlali, India: Khareswada</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Demlo+numbers&rft.place=Devlali%2C+India&rft.pub=Khareswada&rft.date=1948&rft.aulast=Kaprekar&rft.aufirst=D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996-02-02), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2VTSBwAAQBAJ"><i>The New Book of Prime Number Records</i></a>, Computers and Medicine (3rd ed.), New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94457-9" title="Special:BookSources/978-0-387-94457-9"><bdi>978-0-387-94457-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.series=Computers+and+Medicine&rft.edition=3rd&rft.pub=Springer&rft.date=1996-02-02&rft.isbn=978-0-387-94457-9&rft.aulast=Ribenboim&rft.aufirst=Paulo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2VTSBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYates1982" class="citation cs2"><a href="/wiki/Samuel_Yates" title="Samuel Yates">Yates, Samuel</a> (1982), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3_vuAAAAMAAJ"><i>Repunits and repetends</i></a>, FL: Delray Beach, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9608652-0-8" title="Special:BookSources/978-0-9608652-0-8"><bdi>978-0-9608652-0-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Repunits+and+repetends&rft.place=FL&rft.pub=Delray+Beach&rft.date=1982&rft.isbn=978-0-9608652-0-8&rft.aulast=Yates&rft.aufirst=Samuel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3_vuAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Repunit"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><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/Repunit.html">"Repunit"</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=Repunit&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRepunit.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepunit" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.cerias.purdue.edu/homes/ssw/cun/third/pmain901">The main tables</a> of the <a rel="nofollow" class="external text" href="http://www.cerias.purdue.edu/homes/ssw/cun/">Cunningham project</a>.</li> <li><a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/page.php?sort=Repunit">Repunit</a> at <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">The Prime Pages</a> by Chris Caldwell.</li> <li><a rel="nofollow" class="external text" href="http://www.worldofnumbers.com/repunits.htm">Repunits and their prime factors</a> at <a rel="nofollow" class="external text" href="http://www.worldofnumbers.com">World!Of Numbers</a>.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html">Prime generalized repunits</a> of at least 1000 decimal digits by Andy Steward</li> <li><a rel="nofollow" class="external text" href="http://www.elektrosoft.it/matematica/repunit/repunit.htm">Repunit Primes Project</a> Giovanni Di Maria's repunit primes page.</li> <li><a rel="nofollow" class="external text" href="https://docs.google.com/document/d/e/2PACX-1vRha5-vd9Covl4k6I02lOEdWdhsXnBCeFT3FHyhMYTO1i7jtZdTV3wQw2yXgTNja5k_-XINgqL9VNmo/pub">Smallest odd prime p such that (b^p-1)/(b-1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024</a></li> <li><a rel="nofollow" class="external text" href="http://stdkmd.net/nrr/repunit/">Factorization of repunit numbers</a></li> <li><a rel="nofollow" class="external text" href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Generalized repunit primes in base -50 to 50</a></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 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3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</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/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</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/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</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/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</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 prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</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/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</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/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</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/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</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/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</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/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</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/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a class="mw-selflink selflink">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</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/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Prime_number_classes" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><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 class="mw-selflink selflink">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>    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