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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-About_Mersenne_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#About_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>About Mersenne primes</span> </div> </a> <ul id="toc-About_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Perfect_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Perfect_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Perfect numbers</span> </div> </a> <ul id="toc-Perfect_numbers-sublist" class="vector-toc-list"> </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">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Searching_for_Mersenne_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Searching_for_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Searching for Mersenne primes</span> </div> </a> <ul id="toc-Searching_for_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theorems_about_Mersenne_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theorems_about_Mersenne_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Theorems about Mersenne numbers</span> </div> </a> <ul id="toc-Theorems_about_Mersenne_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-List_of_known_Mersenne_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#List_of_known_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>List of known Mersenne primes</span> </div> </a> <ul id="toc-List_of_known_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factorization_of_composite_Mersenne_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factorization_of_composite_Mersenne_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Factorization of composite Mersenne numbers</span> </div> </a> <ul id="toc-Factorization_of_composite_Mersenne_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mersenne_numbers_in_nature_and_elsewhere" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mersenne_numbers_in_nature_and_elsewhere"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Mersenne numbers in nature and elsewhere</span> </div> </a> <ul id="toc-Mersenne_numbers_in_nature_and_elsewhere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mersenne–Fermat_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mersenne–Fermat_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Mersenne–Fermat primes</span> </div> </a> <ul id="toc-Mersenne–Fermat_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-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 Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> <li id="toc-Gaussian_Mersenne_primes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gaussian_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1.1</span> <span>Gaussian Mersenne primes</span> </div> </a> <ul id="toc-Gaussian_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eisenstein_Mersenne_primes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Eisenstein_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1.2</span> <span>Eisenstein Mersenne primes</span> </div> </a> <ul id="toc-Eisenstein_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Divide_an_integer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divide_an_integer"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Divide an integer</span> </div> </a> <ul id="toc-Divide_an_integer-sublist" class="vector-toc-list"> <li id="toc-Repunit_primes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Repunit_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2.1</span> <span>Repunit primes</span> </div> </a> <ul id="toc-Repunit_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_generalized_Mersenne_primes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other_generalized_Mersenne_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2.2</span> <span>Other generalized Mersenne primes</span> </div> </a> <ul id="toc-Other_generalized_Mersenne_primes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </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">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-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">14</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-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 External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-MathWorld_links" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#MathWorld_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.1</span> <span>MathWorld links</span> </div> </a> <ul id="toc-MathWorld_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mersenne prime</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 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-56" 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">56 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_%D9%85%D8%B1%D8%B3%D9%8A%D9%86_%D8%A7%D9%84%D8%A3%D9%88%D9%84%D9%8A" title="عدد مرسين الأولي – Arabic" lang="ar" hreflang="ar" data-title="عدد مرسين الأولي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%84%D5%A5%D6%80%D5%BD%D5%A7%D5%B6%D5%AB_%D5%A9%D5%AB%D6%82" title="Մերսէնի թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Մերսէնի թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Mersenn_%C9%99d%C9%99di" title="Mersenn ədədi – Azerbaijani" lang="az" hreflang="az" data-title="Mersenn ədədi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%B8%E0%A7%87%E0%A6%A8_%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95" title="মার্সেন মৌলিক – Bangla" lang="bn" hreflang="bn" data-title="মার্সেন মৌলিক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BD%D0%B0_%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD" title="Число на Мерсен – Bulgarian" lang="bg" hreflang="bg" data-title="Число на Мерсен" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Mersenneovi_prosti_brojevi" title="Mersenneovi prosti brojevi – Bosnian" lang="bs" hreflang="bs" data-title="Mersenneovi prosti brojevi" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_primer_de_Mersenne" title="Nombre primer de Mersenne – Catalan" lang="ca" hreflang="ca" data-title="Nombre primer de Mersenne" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Mersennovo_prvo%C4%8D%C3%ADslo" title="Mersennovo prvočíslo – Czech" lang="cs" hreflang="cs" data-title="Mersennovo prvočíslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Mersenne-primtal" title="Mersenne-primtal – Danish" lang="da" hreflang="da" data-title="Mersenne-primtal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Mersenne-Primzahl" title="Mersenne-Primzahl – German" lang="de" hreflang="de" data-title="Mersenne-Primzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Mersenne%27i_arvud" title="Mersenne'i arvud – Estonian" lang="et" hreflang="et" data-title="Mersenne'i arvud" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CF%8E%CF%84%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%9C%CE%B5%CF%81%CF%83%CE%AD%CE%BD" title="Πρώτος αριθμός Μερσέν – Greek" lang="el" hreflang="el" data-title="Πρώτος αριθμός Μερσέν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_pr%C3%ACm_ed_Mersenne" title="Nùmer prìm ed Mersenne – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer prìm ed Mersenne" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_primo_de_Mersenne" title="Número primo de Mersenne – Spanish" lang="es" hreflang="es" data-title="Número primo de Mersenne" 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/Primo_de_Mersenne" title="Primo de Mersenne – Esperanto" lang="eo" hreflang="eo" data-title="Primo de Mersenne" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A7%D9%88%D9%84_%D9%85%D8%B1%D8%B3%D9%86" title="عدد اول مرسن – Persian" lang="fa" hreflang="fa" data-title="عدد اول مرسن" 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/Nombre_de_Mersenne_premier" title="Nombre de Mersenne premier – French" lang="fr" hreflang="fr" data-title="Nombre de Mersenne premier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_primo_de_Mersenne" title="Número primo de Mersenne – Galician" lang="gl" hreflang="gl" data-title="Número primo de Mersenne" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A9%94%EB%A5%B4%EC%84%BC_%EC%86%8C%EC%88%98" title="메르센 소수 – Korean" lang="ko" hreflang="ko" data-title="메르센 소수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A5%D6%80%D5%BD%D5%A5%D5%B6%D5%AB_%D5%A9%D5%AB%D5%BE" title="Մերսենի թիվ – Armenian" lang="hy" hreflang="hy" data-title="Մերսենի թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Mersenneovi_brojevi" title="Mersenneovi brojevi – Croatian" lang="hr" hreflang="hr" data-title="Mersenneovi brojevi" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_prima_Mersenne" title="Bilangan prima Mersenne – Indonesian" lang="id" hreflang="id" data-title="Bilangan prima Mersenne" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Mersenne-frumt%C3%B6lur" title="Mersenne-frumtölur – Icelandic" lang="is" hreflang="is" data-title="Mersenne-frumtölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_primo_di_Mersenne" title="Numero primo di Mersenne – Italian" lang="it" hreflang="it" data-title="Numero primo di Mersenne" 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%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A8%D7%A1%D7%9F" title="מספר מרסן – Hebrew" lang="he" hreflang="he" data-title="מספר מרסן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Nonm_premye_Mersenne" title="Nonm premye Mersenne – Haitian Creole" lang="ht" hreflang="ht" data-title="Nonm premye Mersenne" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mersenne-Primzuel" title="Mersenne-Primzuel – Luxembourgish" lang="lb" hreflang="lb" data-title="Mersenne-Primzuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Merseno_skai%C4%8Diai" title="Merseno skaičiai – Lithuanian" lang="lt" hreflang="lt" data-title="Merseno skaičiai" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_primm_de_Mersenne" title="Numer primm de Mersenne – Lombard" lang="lmo" hreflang="lmo" data-title="Numer primm de Mersenne" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mersenne-pr%C3%ADmek" title="Mersenne-prímek – Hungarian" lang="hu" hreflang="hu" data-title="Mersenne-prímek" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD%D0%BE%D0%B2%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%B8_%D0%B1%D1%80%D0%BE%D0%B5%D0%B2%D0%B8" title="Мерсенови прости броеви – Macedonian" lang="mk" hreflang="mk" data-title="Мерсенови прости броеви" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B5%86%E0%B4%B4%E0%B5%8D%E0%B4%B8%E0%B5%86%E0%B5%BB_%E0%B4%85%E0%B4%AD%E0%B4%BE%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="മെഴ്സെൻ അഭാജ്യസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="മെഴ്സെൻ അഭാജ്യസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Mersennepriemgetal" title="Mersennepriemgetal – Dutch" lang="nl" hreflang="nl" data-title="Mersennepriemgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mersenne-primtall" title="Mersenne-primtall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Mersenne-primtall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Mersenne%E2%80%99a" title="Liczby Mersenne’a – Polish" lang="pl" hreflang="pl" data-title="Liczby Mersenne’a" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Primo_de_Mersenne" title="Primo de Mersenne – Portuguese" lang="pt" hreflang="pt" data-title="Primo de Mersenne" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Prim_Mersenne" title="Prim Mersenne – Romanian" lang="ro" hreflang="ro" data-title="Prim Mersenne" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE_%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD%D0%BD%D0%B0" title="Простое число Мерсенна – Russian" lang="ru" hreflang="ru" data-title="Простое число Мерсенна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_primu_di_Mersenne" title="Nùmmuru primu di Mersenne – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru primu di Mersenne" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mersenne_prime" title="Mersenne prime – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mersenne prime" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mersennovo_prvo%C4%8D%C3%ADslo" title="Mersennovo prvočíslo – Slovak" lang="sk" hreflang="sk" data-title="Mersennovo prvočíslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mersennovo_%C5%A1tevilo" title="Mersennovo število – Slovenian" lang="sl" hreflang="sl" data-title="Mersennovo število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%B3%DB%95%D8%B1%DB%95%D8%AA%D8%A7%DB%8C%DB%8C%DB%8C_%D9%85%DB%8E%D8%B1%D8%B3%DB%8E%D9%86" title="ژمارەی سەرەتاییی مێرسێن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی سەرەتاییی مێرسێن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD%D0%BE%D0%B2%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%B8_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B8" title="Мерсенови прости бројеви – Serbian" lang="sr" hreflang="sr" data-title="Мерсенови прости бројеви" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Mersenneovi_prosti_brojevi" title="Mersenneovi prosti brojevi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Mersenneovi prosti brojevi" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Mersennen_alkuluku" title="Mersennen alkuluku – Finnish" lang="fi" hreflang="fi" data-title="Mersennen alkuluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Mersenneprimtal" title="Mersenneprimtal – Swedish" lang="sv" hreflang="sv" data-title="Mersenneprimtal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AF%86%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AE%95%E0%AE%BE%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%A9%E0%AE%BF" title="மெர்சென் பகாத்தனி – Tamil" lang="ta" hreflang="ta" data-title="மெர்சென் பகாத்தனி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0%E0%B9%81%E0%B8%A1%E0%B8%A3%E0%B9%8C%E0%B9%81%E0%B8%8B%E0%B8%99" title="จำนวนเฉพาะแมร์แซน – Thai" lang="th" hreflang="th" data-title="จำนวนเฉพาะแมร์แซน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mersenne_say%C4%B1s%C4%B1" title="Mersenne sayısı – Turkish" lang="tr" hreflang="tr" data-title="Mersenne sayısı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B1%D8%B3%D9%86_%D9%85%D9%81%D8%B1%D8%AF_%D8%B9%D8%AF%D8%AF" title="مرسن مفرد عدد – Urdu" lang="ur" hreflang="ur" data-title="مرسن مفرد عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91_Mersenne" title="Số nguyên tố Mersenne – Vietnamese" lang="vi" hreflang="vi" data-title="Số nguyên tố Mersenne" 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-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%A2%85%E6%A3%AE%E8%B3%AA%E6%95%B8" title="梅森質數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="梅森質數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%A2%85%E6%A3%AE%E7%B4%A0%E6%95%B0" title="梅森素数 – Wu" lang="wuu" hreflang="wuu" data-title="梅森素数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A2%85%E6%A3%AE%E8%B3%AA%E6%95%B8" title="梅森質數 – Cantonese" lang="yue" hreflang="yue" data-title="梅森質數" 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/%E6%A2%85%E6%A3%AE%E7%B4%A0%E6%95%B0" title="梅森素数 – Chinese" lang="zh" hreflang="zh" data-title="梅森素数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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<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">Prime number of the form 2^n – 1</div> <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">Mersenne prime</caption><tbody><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a></td></tr><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">52</td></tr><tr><th scope="row" class="infobox-label">Conjectured <abbr title="number">no.</abbr> of terms</th><td class="infobox-data">Infinite</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Subsequence" title="Subsequence">Subsequence</a> of</th><td class="infobox-data">Mersenne numbers</td></tr><tr><th scope="row" class="infobox-label">First terms</th><td class="infobox-data"><a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a>, <a href="/wiki/7_(number)" class="mw-redirect" title="7 (number)">7</a>, <a href="/wiki/31_(number)" title="31 (number)">31</a>, <a href="/wiki/127_(number)" title="127 (number)">127</a>, 8191</td></tr><tr><th scope="row" class="infobox-label">Largest known term</th><td class="infobox-data"><span class="nowrap">2<sup>136,279,841</sup> − 1</span> (October 12, 2024)</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/A000668">A000668</a></li><li>Mersenne primes (primes of the form 2^n - 1).</li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Mersenne prime</b> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> that is one less than a <a href="/wiki/Power_of_two" title="Power of two">power of two</a>. That is, it is a prime number <a href="/wiki/Of_the_form" title="Of the form">of the form</a> <span class="texhtml"><i>M<sub>n</sub></i> = 2<sup><i>n</i></sup> − 1</span> for some <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml"><i>n</i></span>. They are named after <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a>, a French <a href="/wiki/Minims_(religious_order)" class="mw-redirect" title="Minims (religious order)">Minim friar</a>, who studied them in the early 17th century. If <span class="texhtml"><i>n</i></span> is a <a href="/wiki/Composite_number" title="Composite number">composite number</a> then so is <span class="texhtml">2<sup><i>n</i></sup> − 1</span>. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form <span class="texhtml"><i>M<sub>p</sub></i> = 2<sup><i>p</i></sup> − 1</span> for some prime <span class="texhtml"><i>p</i></span>. </p><p>The <a href="/wiki/Exponentiation" title="Exponentiation">exponents</a> <span class="texhtml"><i>n</i></span> which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000043" class="extiw" title="oeis:A000043">A000043</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) and the resulting Mersenne primes are <a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a>, <a href="/wiki/7_(number)" class="mw-redirect" title="7 (number)">7</a>, <a href="/wiki/31_(number)" title="31 (number)">31</a>, <a href="/wiki/127_(number)" title="127 (number)">127</a>, 8191, 131071, 524287, <a href="/wiki/2,147,483,647" title="2,147,483,647">2147483647</a>, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000668" class="extiw" title="oeis:A000668">A000668</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>Numbers of the form <span class="texhtml"><i>M<sub>n</sub></i> = 2<sup><i>n</i></sup> − 1</span> without the primality requirement may be called <b>Mersenne numbers</b>. Sometimes, however, Mersenne numbers are defined to have the additional requirement that <span class="texhtml"><i>n</i></span> should be prime. The smallest composite Mersenne number with prime exponent <i>n</i> is <span class="nowrap">2<sup>11</sup> − 1 = 2047 = 23 × 89</span>. </p><p>Mersenne primes were studied in antiquity because of their close <a href="#Perfect_numbers">connection to perfect numbers</a>: the <a href="/wiki/Euclid%E2%80%93Euler_theorem" title="Euclid–Euler theorem">Euclid–Euler theorem</a> asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality. </p><p>As of 2024<sup class="plainlinks noexcerpt noprint asof-tag ref" style="display:none;"><a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=4">[ref]</a></sup>, 52 Mersenne primes are known. 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>, is a Mersenne prime.<sup id="cite_ref-GIMPS-2024_1-0" class="reference"><a href="#cite_note-GIMPS-2024-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-GIMPS-2018_2-0" class="reference"><a href="#cite_note-GIMPS-2018-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Since 1997, all newly found Mersenne primes have been discovered by the <a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a>, a <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="About_Mersenne_primes">About Mersenne primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=1" title="Edit section: About Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Are there infinitely many Mersenne primes?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <p>Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. </p><p>The <a href="/wiki/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture" class="mw-redirect" title="Lenstra–Pomerance–Wagstaff conjecture">Lenstra–Pomerance–Wagstaff conjecture</a> claims that there are infinitely many Mersenne primes and predicts their <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">order of growth</a> and frequency: For every number n, there should on average be 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 _{2}(10)}"> <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"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\gamma }\cdot \log _{2}(10)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d871abf34c7297dae090ca3ba17ad23dda1d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.048ex; height:2.843ex;" alt="{\displaystyle e^{\gamma }\cdot \log _{2}(10)}"></span> ≈ 5.92 primes <i>p</i> with n decimal digits (i.e. 10<sup>n-1</sup> < p < 10<sup>n</sup>) for which <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8004b0c032f7742153c5994fb87d923fdb4493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.313ex; height:2.843ex;" alt="{\displaystyle M_{p}}"></span> is prime. Here, γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. </p><p>It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed <a href="/wiki/Conjecture" title="Conjecture">conjectures</a> about prime numbers, for example, the infinitude of <a href="/wiki/Sophie_Germain_prime" class="mw-redirect" title="Sophie Germain prime">Sophie Germain primes</a> <a href="/wiki/Congruence_relation" title="Congruence relation">congruent</a> to 3 (<a href="/wiki/Modular_arithmetic" title="Modular arithmetic">mod 4</a>). For these primes <span class="texhtml mvar" style="font-style:italic;">p</span>, <span class="texhtml">2<i>p</i> + 1</span> (which is also prime) will divide <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span>, for example, <span class="texhtml">23 | <i>M</i><sub>11</sub></span>, <span class="texhtml">47 | <i>M</i><sub>23</sub></span>, <span class="texhtml">167 | <i>M</i><sub>83</sub></span>, <span class="texhtml">263 | <i>M</i><sub>131</sub></span>, <span class="texhtml">359 | <i>M</i><sub>179</sub></span>, <span class="texhtml">383 | <i>M</i><sub>191</sub></span>, <span class="texhtml">479 | <i>M</i><sub>239</sub></span>, and <span class="texhtml">503 | <i>M</i><sub>251</sub></span> (sequence <span class="nowrap external"><a href="//oeis.org/A002515" class="extiw" title="oeis:A002515">A002515</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). Since for these primes <span class="texhtml mvar" style="font-style:italic;">p</span>, <span class="texhtml">2<i>p</i> + 1</span> is congruent to 7 mod 8, so 2 is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> mod <span class="texhtml">2<i>p</i> + 1</span>, and the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of 2 mod <span class="texhtml">2<i>p</i> + 1</span> must divide <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="{\textstyle {\frac {(2p+1)-1}{2}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {(2p+1)-1}{2}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82a61b0293c2eb102740af92aa01cdd4726dd48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.233ex; height:4.176ex;" alt="{\textstyle {\frac {(2p+1)-1}{2}}=p}"></span>. Since <span class="texhtml mvar" style="font-style:italic;">p</span> is a prime, it must be <span class="texhtml mvar" style="font-style:italic;">p</span> or 1. However, it cannot be 1 since <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 _{1}(2)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{1}(2)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029cdc4ffeed8281e6d67c625adb871e8fa096e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.965ex; height:2.843ex;" alt="{\displaystyle \Phi _{1}(2)=1}"></span> and 1 has no <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a>, so it must be <span class="texhtml mvar" style="font-style:italic;">p</span>. Hence, <span class="texhtml">2<i>p</i> + 1</span> divides <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}(2)=2^{p}-1}"> <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> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{p}(2)=2^{p}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866151727414c825aa4143e9c7068f3af4dc5d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.032ex; height:3.009ex;" alt="{\displaystyle \Phi _{p}(2)=2^{p}-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 2^{p}-1=M_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}-1=M_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0ce75dcda473aed1c0e52937beb3e2fcdc7cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.636ex; height:3.009ex;" alt="{\displaystyle 2^{p}-1=M_{p}}"></span> cannot be prime. The first four Mersenne primes are <span class="texhtml"><i>M</i><sub>2</sub> = 3</span>, <span class="texhtml"><i>M</i><sub>3</sub> = 7</span>, <span class="texhtml"><i>M</i><sub>5</sub> = 31</span> and <span class="texhtml"><i>M</i><sub>7</sub> = 127</span> and because the first Mersenne prime starts at <span class="texhtml"><i>M</i><sub>2</sub></span>, all Mersenne primes are congruent to 3 (mod 4). Other than <span class="texhtml"><i>M</i><sub>0</sub> = 0</span> and <span class="texhtml"><i>M</i><sub>1</sub> = 1</span>, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the <a href="/wiki/Prime_factorization" class="mw-redirect" title="Prime factorization">prime factorization</a> of a Mersenne number ( <span class="texhtml">≥ <i>M</i><sub>2</sub></span> ) there must be at least one prime factor congruent to 3 (mod 4). </p><p>A basic <a href="/wiki/Theorem" title="Theorem">theorem</a> about Mersenne numbers states that if <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> is prime, then the exponent <span class="texhtml"><i>p</i></span> must also be prime. This follows from the identity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>b</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>b</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c3bfe3d18527583a9b29e3360f1f7c4fc5d162" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:56.605ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}}"></span> This rules out primality for Mersenne numbers with a composite exponent, such as <span class="texhtml"><i>M</i><sub>4</sub> = 2<sup>4</sup> − 1 = 15 = 3 × 5 = (2<sup>2</sup> − 1) × (1 + 2<sup>2</sup>)</span>. </p><p>Though the above examples might suggest that <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> is prime for all primes <span class="texhtml mvar" style="font-style:italic;">p</span>, this is not the case, and the smallest counterexample is the Mersenne number </p> <dl><dd><span class="texhtml"><i>M</i><sub>11</sub> = 2<sup>11</sup> − 1 = 2047 = 23 × 89</span>.</dd></dl> <p>The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size.<sup id="cite_ref-Wagstaff_4-0" class="reference"><a href="#cite_note-Wagstaff-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Nonetheless, prime values of <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> appear to grow increasingly sparse as <span class="texhtml mvar" style="font-style:italic;">p</span> increases. For example, eight of the first 11 primes <span class="texhtml mvar" style="font-style:italic;">p</span> give rise to a Mersenne prime <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> (the correct terms on Mersenne's original list), while <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> is prime for only 43 of the first two million prime numbers (up to 32,452,843). </p><p>Since Mersenne numbers grow very rapidly, the search for Mersenne primes is a difficult task, even though there is a simple efficient test to determine whether a given Mersenne number is prime: the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer primality test</a> (LLT), which makes it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a <a href="/wiki/Cult_following" title="Cult following">cult following</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2023)">citation needed</span></a></i>]</sup> Consequently, a large amount of computer power has been expended searching for new Mersenne primes, much of which is now done using <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a>. </p><p>Arithmetic modulo a Mersenne number is particularly efficient on a <a href="/wiki/Binary_number" title="Binary number">binary computer</a>, making them popular choices when a prime modulus is desired, such as the <a href="/wiki/Park%E2%80%93Miller_random_number_generator" class="mw-redirect" title="Park–Miller random number generator">Park–Miller random number generator</a>. To find a <a href="/wiki/Primitive_polynomial_(field_theory)" title="Primitive polynomial (field theory)">primitive polynomial</a> of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find primitive polynomials of very high order. Such <a href="/wiki/Primitive_polynomial_(field_theory)#Primitive_trinomials" title="Primitive polynomial (field theory)">primitive trinomials</a> are used in <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a> with very large periods such as the <a href="/wiki/Mersenne_twister" class="mw-redirect" title="Mersenne twister">Mersenne twister</a>, generalized shift register and <a href="/wiki/Lagged_Fibonacci_generator" title="Lagged Fibonacci generator">Lagged Fibonacci generators</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Perfect_numbers">Perfect numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=2" title="Edit section: Perfect numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Euclid%E2%80%93Euler_theorem" title="Euclid–Euler theorem">Euclid–Euler theorem</a></div> <p>Mersenne primes <span class="texhtml"><i>M</i><sub><i>p</i></sub></span> are closely connected to <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a>. In the 4th century BC, <a href="/wiki/Euclid" title="Euclid">Euclid</a> proved that if <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is prime, then <span class="texhtml">2<sup><i>p</i> − 1</sup>(2<sup><i>p</i></sup> − 1</span>) is a perfect number. In the 18th century, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> proved that, conversely, all even perfect numbers have this form.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This is known as the <a href="/wiki/Euclid%E2%80%93Euler_theorem" title="Euclid–Euler theorem">Euclid–Euler theorem</a>. It is unknown whether there are any <a href="/wiki/Odd_perfect_number" class="mw-redirect" title="Odd perfect number">odd perfect numbers</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=3" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable infobox" style="width:1px;padding-left:1em;"> <tbody><tr> <th style="color:#dd0066;background:#99ffff;text-align:right;">2 </th> <th style="color:#dd0066;background:#99ffff;text-align:right;">3 </th> <th style="color:#dd0066;background:#99ffff;text-align:right;">5 </th> <th style="color:#dd0066;background:#99ffff;text-align:right;">7 </th> <td style="text-align: right;">11 </td> <th style="color:#dd0066;background:#99ffff;text-align:right;">13 </th> <th style="color:#dd0066;background:#99ffff;text-align:right;">17 </th> <th style="color:#dd0066;background:#99ffff;text-align:right;">19 </th></tr> <tr> <td style="text-align: right;">23 </td> <td style="text-align: right;">29 </td> <th style="color:#dd0066;background:#99ffff;text-align:right;">31 </th> <td style="text-align: right;">37 </td> <td style="text-align: right;">41 </td> <td style="text-align: right;">43 </td> <td style="text-align: right;">47 </td> <td style="text-align: right;">53 </td></tr> <tr> <td style="text-align: right;">59 </td> <th style="background:#99ffff;text-align:right;">61 </th> <th style="color:#dd0066;text-align:right;">67 </th> <td style="text-align: right;">71 </td> <td style="text-align: right;">73 </td> <td style="text-align: right;">79 </td> <td style="text-align: right;">83 </td> <th style="background:#99ffff;text-align:right;">89 </th></tr> <tr> <td style="text-align: right;">97 </td> <td style="text-align: right;">101 </td> <td style="text-align: right;">103 </td> <th style="background:#99ffff;text-align:right;">107 </th> <td style="text-align: right;">109 </td> <td style="text-align: right;">113 </td> <th style="color:#dd0066;background:#99ffff;text-align:right;">127 </th> <td style="text-align: right;">131 </td></tr> <tr> <td style="text-align: right;">137 </td> <td style="text-align: right;">139 </td> <td style="text-align: right;">149 </td> <td style="text-align: right;">151 </td> <td style="text-align: right;">157 </td> <td style="text-align: right;">163 </td> <td style="text-align: right;">167 </td> <td style="text-align: right;">173 </td></tr> <tr> <td style="text-align: right;">179 </td> <td style="text-align: right;">181 </td> <td style="text-align: right;">191 </td> <td style="text-align: right;">193 </td> <td style="text-align: right;">197 </td> <td style="text-align: right;">199 </td> <td style="text-align: right;">211 </td> <td style="text-align: right;">223 </td></tr> <tr> <td style="text-align: right;">227 </td> <td style="text-align: right;">229 </td> <td style="text-align: right;">233 </td> <td style="text-align: right;">239 </td> <td style="text-align: right;">241 </td> <td style="text-align: right;">251 </td> <th style="color:#dd0066;">257 </th> <td style="text-align: right;">263 </td></tr> <tr> <td style="text-align: right;">269 </td> <td style="text-align: right;">271 </td> <td style="text-align: right;">277 </td> <td style="text-align: right;">281 </td> <td style="text-align: right;">283 </td> <td style="text-align: right;">293 </td> <td style="text-align: right;">307 </td> <td style="text-align: right;">311 </td></tr> <tr> <td colspan="8">The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold </td></tr></tbody></table> <p>Mersenne primes take their name from the 17th-century <a href="/wiki/French_people" title="French people">French</a> scholar <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a>, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne in 1644 were as follows: </p> <dl><dd><dl><dd>2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.</dd></dl></dd></dl> <p>His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included <span class="texhtml"><i>M</i><sub>67</sub></span> and <span class="texhtml"><i>M</i><sub>257</sub></span> (which are composite) and omitted <span class="texhtml"><i>M</i><sub>61</sub></span>, <span class="texhtml"><i>M</i><sub>89</sub></span>, and <span class="texhtml"><i>M</i><sub>107</sub></span> (which are prime). Mersenne gave little indication of how he came up with his list.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a> proved in 1876 that <span class="texhtml"><i>M</i><sub>127</sub></span> is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Aimé Ferrier found a larger 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 (2^{148}+1)/17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>148</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{148}+1)/17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3021f0b1f0abf192f23db74d56744253b01a258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.16ex; height:3.176ex;" alt="{\displaystyle (2^{148}+1)/17}"></span>, using a desk calculating machine.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: page 22">: page 22 </span></sup> <span class="texhtml"><i>M</i><sub>61</sub></span> was determined to be prime in 1883 by <a href="/wiki/Ivan_Mikheevich_Pervushin" class="mw-redirect" title="Ivan Mikheevich Pervushin">Ivan Mikheevich Pervushin</a>, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2025)">citation needed</span></a></i>]</sup>. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876 by demonstrating that <span class="texhtml"><i>M</i><sub>67</sub></span> was composite without finding a factor. No factor was found until a famous talk by <a href="/wiki/Frank_Nelson_Cole" title="Frank Nelson Cole">Frank Nelson Cole</a> in 1903.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one, resulting in the number <span class="nowrap">147,573,952,589,676,412,927</span>. On the other side of the board, he multiplied <span class="nowrap">193,707,721 × 761,838,257,287</span> and got the same number, then returned to his seat (to applause) without speaking.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> He later said that the result had taken him "three years of Sundays" to find.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list. </p> <div class="mw-heading mw-heading2"><h2 id="Searching_for_Mersenne_primes">Searching for Mersenne primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=4" title="Edit section: Searching for Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fast algorithms for finding Mersenne primes are available, and as of October 2024<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&action=edit">[update]</a></sup>, the seven <a href="/wiki/Largest_known_prime_number" title="Largest known prime number">largest known prime numbers</a> are Mersenne primes. </p><p>The first four Mersenne primes <span class="texhtml"><i>M</i><sub>2</sub> = 3</span>, <span class="texhtml"><i>M</i><sub>3</sub> = 7</span>, <span class="texhtml"><i>M</i><sub>5</sub> = 31</span> and <span class="texhtml"><i>M</i><sub>7</sub> = 127</span> were known in antiquity. The fifth, <span class="texhtml"><i>M</i><sub>13</sub> = 8191</span>, was discovered anonymously before 1461; the next two (<span class="texhtml"><i>M</i><sub>17</sub></span> and <span class="texhtml"><i>M</i><sub>19</sub></span>) were found by <a href="/wiki/Pietro_Cataldi" title="Pietro Cataldi">Pietro Cataldi</a> in 1588. After nearly two centuries, <span class="texhtml"><i>M</i><sub>31</sub></span> was verified to be prime by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1772. The next (in historical, not numerical order) was <span class="texhtml"><i>M</i><sub>127</sub></span>, found by <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a> in 1876, then <span class="texhtml"><i>M</i><sub>61</sub></span> by <a href="/wiki/Ivan_Mikheevich_Pervushin" class="mw-redirect" title="Ivan Mikheevich Pervushin">Ivan Mikheevich Pervushin</a> in 1883. Two more (<span class="texhtml"><i>M</i><sub>89</sub></span> and <span class="texhtml"><i>M</i><sub>107</sub></span>) were found early in the 20th century, by <a href="/wiki/R._E._Powers" class="mw-redirect" title="R. E. Powers">R. E. Powers</a> in 1911 and 1914, respectively. </p><p>The most efficient method presently known for testing the primality of Mersenne numbers is the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer primality test</a>. Specifically, it can be shown that for prime <span class="texhtml"><i>p</i> > 2</span>, <span class="texhtml"><i>M<sub>p</sub></i> = 2<sup><i>p</i></sup> − 1</span> is prime <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml"><i>M<sub>p</sub></i></span> divides <span class="texhtml"><i>S</i><sub><i>p</i> − 2</sub></span>, where <span class="texhtml"><i>S</i><sub>0</sub> = 4</span> and <span class="texhtml"><i>S<sub>k</sub></i> = (<i>S</i><sub><i>k</i> − 1</sub>)<sup>2</sup> − 2</span> for <span class="texhtml"><i>k</i> > 0</span>. </p><p>During the era of manual calculation, all previously untested exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times as large as the previous record of 127. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Digits_in_largest_prime_found_as_a_function_of_time.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Digits_in_largest_prime_found_as_a_function_of_time.svg/400px-Digits_in_largest_prime_found_as_a_function_of_time.svg.png" decoding="async" width="400" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Digits_in_largest_prime_found_as_a_function_of_time.svg/600px-Digits_in_largest_prime_found_as_a_function_of_time.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Digits_in_largest_prime_found_as_a_function_of_time.svg/800px-Digits_in_largest_prime_found_as_a_function_of_time.svg.png 2x" data-file-width="614" data-file-height="461" /></a><figcaption>Graph of number of digits in largest known Mersenne prime by year – electronic era. The vertical scale is logarithmic in the number of digits, thus being 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 \log(\log(y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(\log(y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70cf9b9a4d88913870ab809ed644c156c7c3844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.718ex; height:2.843ex;" alt="{\displaystyle \log(\log(y))}"></span> function in the value of the prime.</figcaption></figure> <p>The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> searched for them on the <a href="/wiki/Manchester_Mark_1" title="Manchester Mark 1">Manchester Mark 1</a> in 1949,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> but the first successful identification of a Mersenne prime, <span class="texhtml"><i>M</i><sub>521</sub></span>, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S. <a href="/wiki/National_Bureau_of_Standards" class="mw-redirect" title="National Bureau of Standards">National Bureau of Standards</a> <a href="/wiki/SWAC_(computer)" title="SWAC (computer)">Western Automatic Computer (SWAC)</a> at the Institute for Numerical Analysis at the <a href="/wiki/University_of_California,_Los_Angeles" title="University of California, Los Angeles">University of California, Los Angeles</a> (UCLA), under the direction of <a href="/wiki/Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">D. H. Lehmer</a>, with a computer search program written and run by Prof. <a href="/wiki/Raphael_M._Robinson" title="Raphael M. Robinson">R. M. Robinson</a>. It was the first Mersenne prime to be identified in thirty-eight years; the next one, <span class="texhtml"><i>M</i><sub>607</sub></span>, was found by the computer a little less than two hours later. Three more — <span class="texhtml"><i>M</i><sub>1279</sub></span>, <span class="texhtml"><i>M</i><sub>2203</sub></span>, and <span class="texhtml"><i>M</i><sub>2281</sub></span> — were found by the same program in the next several months. <span class="texhtml"><i>M</i><sub>4,423</sub></span> was the first prime discovered with more than 1000 digits, <span class="texhtml"><i>M</i><sub>44,497</sub></span> was the first with more than 10,000, and <span class="texhtml"><i>M</i><sub>6,972,593</sub></span> was the first with more than a million. In general, the number of digits in the decimal representation of <span class="texhtml"><i>M<sub>n</sub></i></span> equals <span class="texhtml">⌊<i>n</i> × log<sub>10</sub>2⌋ + 1</span>, where <span class="texhtml">⌊<i>x</i>⌋</span> denotes the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> (or equivalently <span class="texhtml">⌊log<sub>10</sub><i>M<sub>n</sub></i>⌋ + 1</span>). </p><p>In September 2008, mathematicians at UCLA participating in the <a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a> (GIMPS) won part of a $100,000 prize from the <a href="/wiki/Electronic_Frontier_Foundation" title="Electronic Frontier Foundation">Electronic Frontier Foundation</a> for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a <a href="/wiki/Dell_OptiPlex" title="Dell OptiPlex">Dell OptiPlex</a> 745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is <span class="nowrap">2<sup>42,643,801</sup> − 1</span>. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered. </p><p>On January 25, 2013, <a href="/wiki/Curtis_Cooper_(mathematician)" title="Curtis Cooper (mathematician)">Curtis Cooper</a>, a mathematician at the <a href="/wiki/University_of_Central_Missouri" title="University of Central Missouri">University of Central Missouri</a>, discovered a 48th Mersenne prime, <span class="nowrap">2<sup>57,885,161</sup> − 1</span> (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, <span class="nowrap">2<sup>74,207,281</sup> − 1</span> (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network.<sup id="cite_ref-GIMPS-2016_15-0" class="reference"><a href="#cite_note-GIMPS-2016-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-NYT-20160121_17-0" class="reference"><a href="#cite_note-NYT-20160121-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years. </p><p>On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below <span class="texhtml"><i>M</i><sub>37,156,667</sub></span>, thus officially confirming its position as the 45th Mersenne prime.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in <a href="/wiki/Germantown,_Tennessee" title="Germantown, Tennessee">Germantown, Tennessee</a>, had found a 50th Mersenne prime, <span class="nowrap">2<sup>77,232,917</sup> − 1</span> (a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The discovery was made by a computer in the offices of a church in the same town.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered a new prime number, <span class="nowrap">2<sup>82,589,933</sup> − 1</span>, having 24,862,048 digits. A computer volunteered by Patrick Laroche from <a href="/wiki/Ocala,_Florida" title="Ocala, Florida">Ocala, Florida</a> made the find on December 7, 2018.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the <a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a> (PRP) test, based on development from Robert Gerbicz in 2017, and a simple way to verify tests developed by Krzysztof Pietrzak in 2018. Due to the low error rate and ease of proof, this nearly halved the computing time to rule out potential primes over the Lucas-Lehmer test (as two users would no longer have to perform the same test to confirm the other's result), although exponents passing the PRP test still require one to confirm their primality.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>On October 12, 2024, a user named Luke Durant from San Jose, California, found the current largest known Mersenne prime, <span class="nowrap">2<sup>136,279,841</sup> − 1</span>, having 41,024,320 digits. This marks the first Mersenne prime with an exponent surpassing 8 digits. This was announced on October 21, 2024.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theorems_about_Mersenne_numbers">Theorems about Mersenne numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=5" title="Edit section: Theorems about Mersenne numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mersenne numbers are 0, 1, 3, 7, 15, 31, 63, ... (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>). </p> <ol><li>If <span class="texhtml"><i>a</i></span> and <span class="texhtml mvar" style="font-style:italic;">p</span> are natural numbers such that <span class="texhtml"><i>a<sup>p</sup></i> − 1</span> is prime, then <span class="texhtml"><i>a</i> = 2</span> or <span class="texhtml"><i>p</i> = 1</span>. <ul><li><b>Proof</b>: <span class="texhtml"><i>a</i> ≡ 1 (<a href="/wiki/Modular_arithmetic" title="Modular arithmetic">mod</a> <i>a</i> − 1)</span>. Then <span class="texhtml"><i>a<sup>p</sup></i> ≡ 1 (mod <i>a</i> − 1)</span>, so <span class="texhtml"><i>a<sup>p</sup></i> − 1 ≡ 0 (mod <i>a</i> − 1)</span>. Thus <span class="texhtml"><i>a</i> − 1 | <i>a<sup>p</sup></i> − 1</span>. However, <span class="texhtml"><i>a<sup>p</sup></i> − 1</span> is prime, so <span class="texhtml"><i>a</i> − 1 = <i>a<sup>p</sup></i> − 1</span> or <span class="texhtml"><i>a</i> − 1 = ±1</span>. In the former case, <span class="texhtml"><i>a</i> = <i>a<sup>p</sup></i></span>, hence <span class="texhtml"><i>a</i> = 0, 1</span> (which is a contradiction, as neither −1 nor 0 is prime) or <span class="texhtml"><i>p</i> = 1.</span> In the latter case, <span class="texhtml"><i>a</i> = 2</span> or <span class="texhtml"><i>a</i> = 0</span>. If <span class="texhtml"><i>a</i> = 0</span>, however, <span class="texhtml">0<sup><i>p</i></sup> − 1 = 0 − 1 = −1</span> which is not prime. Therefore, <span class="texhtml"><i>a</i> = 2</span>.</li></ul></li> <li>If <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is prime, then <span class="texhtml mvar" style="font-style:italic;">p</span> is prime. <ul><li><b>Proof</b>: Suppose that <span class="texhtml mvar" style="font-style:italic;">p</span> is composite, hence can be written <span class="texhtml"><i>p</i> = <i>ab</i></span> with <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml"><i>b</i> > 1</span>. Then <span class="texhtml">2<sup><i>p</i></sup> − 1</span> <span class="texhtml">= 2<sup><i>ab</i></sup> − 1</span> <span class="texhtml">= (2<sup><i>a</i></sup>)<sup><i>b</i></sup> − 1</span> <span class="texhtml">= (2<sup><i>a</i></sup> − 1)<big>(</big>(2<sup><i>a</i></sup>)<sup><i>b</i>−1</sup> + (2<sup><i>a</i></sup>)<sup><i>b</i>−2</sup> + ... + 2<sup><i>a</i></sup> + 1<big>)</big></span> so <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is composite. By contraposition, if <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is prime then <i>p</i> is prime.</li></ul></li> <li>If <span class="texhtml mvar" style="font-style:italic;">p</span> is an odd prime, then every prime <span class="texhtml mvar" style="font-style:italic;">q</span> that divides <span class="texhtml">2<sup><i>p</i></sup> − 1</span> must be 1 plus a multiple of <span class="texhtml">2<i>p</i></span>. This holds even when <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is prime. <ul><li>For example, <span class="nowrap">2<sup>5</sup> − 1 = 31</span> is prime, and <span class="nowrap">31 = 1 + 3 × (2 × 5)</span>. A composite example is <span class="nowrap">2<sup>11</sup> − 1 = 23 × 89</span>, where <span class="nowrap">23 = 1 + (2 × 11)</span> and <span class="nowrap">89 = 1 + 4 × (2 × 11)</span>.</li> <li><b>Proof</b>: By <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>, <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>q</i>−1</sup> − 1</span>. Since <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>p</i></sup> − 1</span>, for all positive integers <span class="texhtml"><i>c</i></span>, <span class="texhtml mvar" style="font-style:italic;">q</span> is also a factor of <span class="texhtml">2<sup><i>pc</i></sup> − 1</span>. Since <span class="texhtml mvar" style="font-style:italic;">p</span> is prime and <span class="texhtml mvar" style="font-style:italic;">q</span> is not a factor of <span class="nowrap">2<sup>1</sup> − 1</span>, <span class="texhtml mvar" style="font-style:italic;">p</span> is also the smallest positive integer <span class="texhtml mvar" style="font-style:italic;">x</span> such that <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>x</i></sup> − 1</span>. As a result, for all positive integers <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>x</i></sup> − 1</span> if and only if <span class="texhtml mvar" style="font-style:italic;">p</span> is a factor of <span class="texhtml mvar" style="font-style:italic;">x</span>. Therefore, since <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>q</i>−1</sup> − 1</span>, <span class="texhtml mvar" style="font-style:italic;">p</span> is a factor of <span class="texhtml"><i>q</i> − 1</span> so <span class="texhtml"><i>q</i> ≡ 1 (mod <i>p</i>)</span>. Furthermore, since <span class="texhtml mvar" style="font-style:italic;">q</span> is a factor of <span class="texhtml">2<sup><i>p</i></sup> − 1</span>, which is odd, <span class="texhtml mvar" style="font-style:italic;">q</span> is odd. Therefore, <span class="texhtml"><i>q</i> ≡ 1 (mod 2<i>p</i>)</span>.</li> <li>This fact leads to a proof of <a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a>, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime <span class="texhtml mvar" style="font-style:italic;">p</span>, all primes dividing <span class="texhtml">2<sup><i>p</i></sup> − 1</span> are larger than <span class="texhtml mvar" style="font-style:italic;">p</span>; thus there are always larger primes than any particular prime.</li> <li>It follows from this fact that for every prime <span class="texhtml"><i>p</i> > 2</span>, there is at least one prime of the form <span class="texhtml">2<i>kp</i>+1</span> less than or equal to <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span>, for some integer <span class="texhtml mvar" style="font-style:italic;">k</span>.</li></ul></li> <li>If <span class="texhtml mvar" style="font-style:italic;">p</span> is an odd prime, then every prime <span class="texhtml mvar" style="font-style:italic;">q</span> that divides <span class="texhtml">2<sup><i>p</i></sup> − 1</span> is congruent to <span class="nowrap">±1 (mod 8)</span>. <ul><li><b>Proof</b>: <span class="texhtml">2<sup><i>p</i>+1</sup> ≡ 2 (mod <i>q</i>)</span>, so <span class="texhtml">2<sup><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(p+1)</sup></span> is a square root of <span class="texhtml">2 mod <i>q</i></span>. By <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>, every prime modulus in which the number 2 has a square root is congruent to <span class="nowrap">±1 (mod 8)</span>.</li></ul></li> <li>A Mersenne prime cannot be a <a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich prime</a>. <ul><li><b>Proof</b>: We show if <span class="texhtml"><i>p</i> = 2<sup><i>m</i></sup> − 1</span> is a Mersenne prime, then the congruence <span class="texhtml">2<sup><i>p</i>−1</sup> ≡ 1 (mod <i>p</i><sup>2</sup>)</span> does not hold. By Fermat's little theorem, <span class="texhtml"><i>m</i> | <i>p</i> − 1</span>. Therefore, one can write <span class="texhtml"><i>p</i> − 1 = <i>mλ</i></span>. If the given congruence is satisfied, then <span class="texhtml"><i>p</i><sup>2</sup> | 2<sup><i>mλ</i></sup> − 1</span>, therefore <span class="texhtml">0 ≡ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2<sup><i>mλ</i></sup> − 1</span><span class="sr-only">/</span><span class="den">2<sup><i>m</i></sup> − 1</span></span>⁠</span></span> <span class="texhtml">= 1 + 2<sup><i>m</i></sup> + 2<sup>2<i>m</i></sup> + ... + 2<sup>(<i>λ</i> − 1)<i>m</i></sup></span> <span class="texhtml">≡ <i>λ</i> mod (2<sup><i>m</i></sup> − 1)</span>. Hence <span class="texhtml">p | <i>λ</i></span>, and therefore <span class="texhtml">−1 = 0 (mod p)</span> which is impossible.</li></ul></li> <li>If <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are natural numbers then <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> if and only if <span class="texhtml">2<sup><i>m</i></sup> − 1</span> and <span class="texhtml">2<sup><i>n</i></sup> − 1</span> are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> That is, the set of <a href="/wiki/Pernicious_number" title="Pernicious number">pernicious</a> Mersenne numbers is pairwise coprime.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml">2<i>p</i> + 1</span> are both prime (meaning that <span class="texhtml mvar" style="font-style:italic;">p</span> is a <a href="/wiki/Sophie_Germain_prime" class="mw-redirect" title="Sophie Germain prime">Sophie Germain prime</a>), and <span class="texhtml mvar" style="font-style:italic;">p</span> is <a href="/wiki/Congruence_relation" title="Congruence relation">congruent</a> to <span class="nowrap">3 (mod 4)</span>, then <span class="texhtml">2<i>p</i> + 1</span> divides <span class="texhtml">2<sup><i>p</i></sup> − 1</span>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <ul><li><b>Example</b>: 11 and 23 are both prime, and <span class="nowrap">11 = 2 × 4 + 3</span>, so 23 divides <span class="nowrap">2<sup>11</sup> − 1</span>.</li> <li><b>Proof</b>: Let <span class="texhtml mvar" style="font-style:italic;">q</span> be <span class="texhtml">2<i>p</i> + 1</span>. By Fermat's little theorem, <span class="texhtml">2<sup>2<i>p</i></sup> ≡ 1 (mod <i>q</i>)</span>, so either <span class="texhtml">2<sup><i>p</i></sup> ≡ 1 (mod <i>q</i>)</span> or <span class="texhtml">2<sup><i>p</i></sup> ≡ −1 (mod <i>q</i>)</span>. Supposing latter true, then <span class="texhtml">2<sup><i>p</i>+1</sup> = (2<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(<i>p</i> + 1)</sup>)<sup>2</sup> ≡ −2 (mod <i>q</i>)</span>, so −2 would be a quadratic residue mod <span class="texhtml mvar" style="font-style:italic;">q</span>. However, since <span class="texhtml mvar" style="font-style:italic;">p</span> is congruent to <span class="nowrap">3 (mod 4)</span>, <span class="texhtml mvar" style="font-style:italic;">q</span> is congruent to <span class="nowrap">7 (mod 8)</span> and therefore 2 is a quadratic residue mod <span class="texhtml mvar" style="font-style:italic;">q</span>. Also since <span class="texhtml mvar" style="font-style:italic;">q</span> is congruent to <span class="nowrap">3 (mod 4)</span>, −1 is a quadratic nonresidue mod <span class="texhtml mvar" style="font-style:italic;">q</span>, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and <span class="texhtml">2<i>p</i> + 1</span> divides <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span>.</li></ul></li> <li>All composite divisors of prime-exponent Mersenne numbers are <a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">strong pseudoprimes</a> to the base 2.</li> <li>With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with <a href="/wiki/Catalan%27s_conjecture" title="Catalan's conjecture">Mihăilescu's theorem</a>, the equation <span class="texhtml">2<sup><i>m</i></sup> − 1 = <i>n<sup>k</sup></i></span> has no solutions where <span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>, and <span class="texhtml mvar" style="font-style:italic;">k</span> are integers with <span class="texhtml"><i>m</i> > 1</span> and <span class="texhtml"><i>k</i> > 1</span>.</li> <li>The Mersenne number sequence is a member of the family of <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas sequences</a>. It is <span class="texhtml">U<sub><i>n</i></sub></span>(3, 2). That is, Mersenne number <span class="texhtml"><i>m</i><sub><i>n</i></sub> = 3<i>m</i><sub><i>n</i>-1</sub> - 2<i>m</i><sub><i>n</i>-2</sub></span> with <span class="texhtml"><i>m</i><sub>0</sub> = 0</span> and <span class="texhtml"><i>m</i><sub>1</sub> = 1</span>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="List_of_known_Mersenne_primes">List of known Mersenne primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=6" title="Edit section: List of known Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_Mersenne_primes_and_perfect_numbers" title="List of Mersenne primes and perfect numbers">List of Mersenne primes and perfect numbers</a></div> <p>As of 2024<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&action=edit">[update]</a></sup>, the 52 known Mersenne primes are 2<sup><i>p</i></sup> − 1 for the following <i>p</i>: </p> <dl><dd>2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933, 136279841. (sequence <span class="nowrap external"><a href="//oeis.org/A000043" class="extiw" title="oeis:A000043">A000043</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="Factorization_of_composite_Mersenne_numbers">Factorization of composite Mersenne numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=7" title="Edit section: Factorization of composite Mersenne numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all Mersenne numbers are Mersenne primes. Mersenne numbers are very good test cases for the <a href="/wiki/Special_number_field_sieve" title="Special number field sieve">special number field sieve</a> algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&action=edit">[update]</a></sup>, <span class="texhtml">2<sup>1,193</sup> − 1</span> is the record-holder,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See <a href="/wiki/Integer_factorization_records" title="Integer factorization records">integer factorization records</a> for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then running a <a href="/wiki/Primality_test" title="Primality test">primality test</a> on the cofactor. As of September 2022<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&action=edit">[update]</a></sup>, the largest completely factored number (with <a href="/wiki/Probable_prime" title="Probable prime">probable prime</a> factors allowed) is <span class="texhtml">2<sup>12,720,787</sup> − 1 = 1,119,429,257 × 175,573,124,547,437,977 × 8,480,999,878,421,106,991 × <span class="texhtml mvar" style="font-style:italic;">q</span></span>, where <span class="texhtml mvar" style="font-style:italic;">q</span> is a 3,829,294-digit probable prime. It was discovered by a GIMPS participant with nickname "Funky Waddle".<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> As of September 2022<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&action=edit">[update]</a></sup>, the Mersenne number <i>M</i><sub>1277</sub> is the smallest composite Mersenne number with no known factors; it has no prime factors below 2<sup>68</sup>,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> and is very unlikely to have any factors below 10<sup>65</sup> (~2<sup>216</sup>).<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>The table below shows factorizations for the first 20 composite Mersenne numbers (sequence <span class="nowrap external"><a href="//oeis.org/A244453" class="extiw" title="oeis:A244453">A244453</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <table class="wikitable" style="text-align:right;"> <tbody><tr> <th><span class="texhtml"><i>p</i></span> </th> <th><span class="texhtml"><i>M<sub>p</sub></i></span> </th> <th>Factorization of <span class="texhtml"><i>M<sub>p</sub></i></span> </th></tr> <tr> <td>11 </td> <td>2047 </td> <td>23 × 89 </td></tr> <tr> <td>23 </td> <td>8388607 </td> <td>47 × 178,481 </td></tr> <tr> <td>29 </td> <td>536870911 </td> <td>233 × 1,103 × 2,089 </td></tr> <tr> <td>37 </td> <td>137438953471 </td> <td>223 × 616,318,177 </td></tr> <tr> <td>41 </td> <td>2199023255551 </td> <td>13,367 × 164,511,353 </td></tr> <tr> <td>43 </td> <td>8796093022207 </td> <td>431 × 9,719 × 2,099,863 </td></tr> <tr> <td>47 </td> <td>140737488355327 </td> <td>2,351 × 4,513 × 13,264,529 </td></tr> <tr> <td>53 </td> <td>9007199254740991 </td> <td>6,361 × 69,431 × 20,394,401 </td></tr> <tr> <td>59 </td> <td>576460752303423487 </td> <td>179,951 × 3,203,431,780,337 (13 digits) </td></tr> <tr> <td>67 </td> <td>147573952589676412927 </td> <td>193,707,721 × 761,838,257,287 (12 digits) </td></tr> <tr> <td>71 </td> <td>2361183241434822606847 </td> <td>228,479 × 48,544,121 × 212,885,833 </td></tr> <tr> <td>73 </td> <td>9444732965739290427391 </td> <td>439 × 2,298,041 × 9,361,973,132,609 (13 digits) </td></tr> <tr> <td>79 </td> <td>604462909807314587353087 </td> <td>2,687 × 202,029,703 × 1,113,491,139,767 (13 digits) </td></tr> <tr> <td>83 </td> <td>9671406556917033397649407 </td> <td>167 × 57,912,614,113,275,649,087,721 (23 digits) </td></tr> <tr> <td>97 </td> <td>158456325028528675187087900671 </td> <td>11,447 × 13,842,607,235,828,485,645,766,393 (26 digits) </td></tr> <tr> <td>101 </td> <td>2535301200456458802993406410751 </td> <td>7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits) </td></tr> <tr> <td>103 </td> <td>10141204801825835211973625643007 </td> <td>2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits) </td></tr> <tr> <td>109 </td> <td>649037107316853453566312041152511 </td> <td>745,988,807 × 870,035,986,098,720,987,332,873 (24 digits) </td></tr> <tr> <td>113 </td> <td>10384593717069655257060992658440191 </td> <td>3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits) </td></tr> <tr> <td>131 </td> <td>2722258935367507707706996859454145691647 </td> <td>263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits) </td></tr></tbody></table> <p>The number of factors for the first 500 Mersenne numbers can be found at (sequence <span class="nowrap external"><a href="//oeis.org/A046800" class="extiw" title="oeis:A046800">A046800</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Mersenne_numbers_in_nature_and_elsewhere">Mersenne numbers in nature and elsewhere</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=8" title="Edit section: Mersenne numbers in nature and elsewhere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the mathematical problem <a href="/wiki/Tower_of_Hanoi" title="Tower of Hanoi">Tower of Hanoi</a>, solving a puzzle with an <span class="texhtml"><i>n</i></span>-disc tower requires <span class="texhtml"><i>M<sub>n</sub></i></span> steps, assuming no mistakes are made.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The number of rice grains on the whole chessboard in the <a href="/wiki/Wheat_and_chessboard_problem" title="Wheat and chessboard problem">wheat and chessboard problem</a> is <span class="texhtml"><i>M<sub>64</sub></i></span>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Asteroid" title="Asteroid">asteroid</a> with <a href="/wiki/Minor_planet" title="Minor planet">minor planet</a> number 8191 is named <a href="/wiki/8191_Mersenne" class="mw-redirect" title="8191 Mersenne">8191 Mersenne</a> after Marin Mersenne, because 8191 is a Mersenne prime.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, an integer <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> that is <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">primitive</a> and has its even leg a power of 2 ( <span class="texhtml">≥ 4</span> ) generates a unique right triangle such that its <a href="/wiki/Inradius" class="mw-redirect" title="Inradius">inradius</a> is always a Mersenne number. For example, if the even leg is <span class="texhtml">2<sup><i>n</i> + 1</sup></span> then because it is primitive it constrains the odd leg to be <span class="texhtml">4<sup><i>n</i></sup> − 1</span>, the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> to be <span class="texhtml">4<sup><i>n</i></sup> + 1</span> and its inradius to be <span class="texhtml">2<sup><i>n</i></sup> − 1</span>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mersenne–Fermat_primes"><span id="Mersenne.E2.80.93Fermat_primes"></span>Mersenne–Fermat primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=9" title="Edit section: Mersenne–Fermat primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Mersenne–Fermat number</b> is defined as <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2<i><sup>p<sup>r</sup></sup></i> − 1</span><span class="sr-only">/</span><span class="den">2<sup><i>p</i><sup><i>r</i> − 1</sup></sup> − 1</span></span>⁠</span></span> with <span class="texhtml"><i>p</i></span> prime, <span class="texhtml"><i>r</i></span> natural number, and can be written as <b><span class="texhtml">MF(<i>p</i>, <i>r</i>)</span></b>. When <span class="texhtml"><i>r</i> = 1</span>, it is a Mersenne number. When <span class="texhtml"><i>p</i> = 2</span>, it is a <a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a>. The only known Mersenne–Fermat primes with <span class="texhtml"><i>r</i> > 1</span> are </p> <dl><dd><span class="texhtml">MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2),</span> and <span class="texhtml">MF(59, 2)</span>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></dd></dl> <p>In fact, <span class="texhtml">MF(<i>p</i>, <i>r</i>) = <i>Φ<sub>p<sup>r</sup></sub></i>(2)</span>, where <span class="texhtml"><i>Φ</i></span> is the <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomial</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=10" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Generalized_Mersenne_prime" class="mw-redirect" title="Generalized Mersenne prime">Generalized Mersenne prime</a></div> <p>The simplest generalized Mersenne primes are prime numbers of the form <span class="texhtml"><i>f</i>(2<sup><i>n</i></sup>)</span>, where <span class="texhtml"><i>f</i>(<i>x</i>)</span> is a low-degree <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with small integer <a href="/wiki/Coefficients" class="mw-redirect" title="Coefficients">coefficients</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> An example is <span class="texhtml">2<sup>64</sup> − 2<sup>32</sup> + 1</span>, in this case, <span class="texhtml"><i>n</i> = 32</span>, and <span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> − <i>x</i> + 1</span>; another example is <span class="texhtml">2<sup>192</sup> − 2<sup>64</sup> − 1</span>, in this case, <span class="texhtml"><i>n</i> = 64</span>, and <span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup> − <i>x</i> − 1</span>. </p><p>It is also natural to try to generalize primes of the form <span class="texhtml">2<sup><i>n</i></sup> − 1</span> to primes of the form <span class="texhtml"><i>b</i><sup><i>n</i></sup> − 1</span> (for <span class="texhtml"><i>b</i> ≠ 2</span> and <span class="texhtml"><i>n</i> > 1</span>). However (see also <a href="#Theorems_about_Mersenne_numbers">theorems above</a>), <span class="texhtml"><i>b</i><sup><i>n</i></sup> − 1</span> is always divisible by <span class="texhtml"><i>b</i> − 1</span>, so unless the latter is a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>, the former is not a prime. This can be remedied by allowing <i>b</i> to be an algebraic integer instead of an integer: </p> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=11" title="Edit section: Complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of integers (on <a href="/wiki/Real_number" title="Real number">real numbers</a>), if <span class="texhtml"><i>b</i> − 1</span> is a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>, then <span class="texhtml"><i>b</i></span> is either 2 or 0. But <span class="texhtml">2<sup><i>n</i></sup> − 1</span> are the usual Mersenne primes, and the formula <span class="texhtml">0<sup><i>n</i></sup> − 1</span> does not lead to anything interesting (since it is always −1 for all <span class="texhtml"><i>n</i> > 0</span>). Thus, we can regard a ring of "integers" on <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> instead of <a href="/wiki/Real_number" title="Real number">real numbers</a>, like <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> and <a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Gaussian_Mersenne_primes">Gaussian Mersenne primes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=12" title="Edit section: Gaussian Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we regard the ring of <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>, we get the case <span class="texhtml"><i>b</i> = 1 + <i>i</i></span> and <span class="texhtml"><i>b</i> = 1 − <i>i</i></span>, and can ask (<a href="/wiki/Without_loss_of_generality" title="Without loss of generality">WLOG</a>) for which <span class="texhtml"><i>n</i></span> the number <span class="texhtml">(1 + <i>i</i>)<sup><i>n</i></sup> − 1</span> is a <a href="/wiki/Gaussian_prime" class="mw-redirect" title="Gaussian prime">Gaussian prime</a> which will then be called a <b>Gaussian Mersenne prime</b>.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p><span class="texhtml">(1 + <i>i</i>)<sup><i>n</i></sup> − 1</span> is a Gaussian prime for the following <span class="texhtml"><i>n</i></span>: </p> <dl><dd>2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (sequence <span class="nowrap external"><a href="//oeis.org/A057429" class="extiw" title="oeis:A057429">A057429</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>Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers. </p><p>As for all Gaussian primes, the <a href="/wiki/Gaussian_integer#Norm_of_a_Gaussian_integer" title="Gaussian integer">norms</a> (that is, squares of absolute values) of these numbers are rational primes: </p> <dl><dd>5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence <span class="nowrap external"><a href="//oeis.org/A182300" class="extiw" title="oeis:A182300">A182300</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-heading4"><h4 id="Eisenstein_Mersenne_primes">Eisenstein Mersenne primes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=13" title="Edit section: Eisenstein Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may encounter cases where such a Mersenne prime is also an <i>Eisenstein prime</i>, being of the form <span class="texhtml"><i>b</i> = 1 + <i>ω</i></span> and <span class="texhtml"><i>b</i> = 1 − <i>ω</i></span>. In these cases, such numbers are called <b>Eisenstein Mersenne primes</b>. </p><p><span class="texhtml">(1 + <i>ω</i>)<sup><i>n</i></sup> − 1</span> is an Eisenstein prime for the following <span class="texhtml"><i>n</i></span>: </p> <dl><dd>2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence <span class="nowrap external"><a href="//oeis.org/A066408" class="extiw" title="oeis:A066408">A066408</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>The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes: </p> <dl><dd>7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence <span class="nowrap external"><a href="//oeis.org/A066413" class="extiw" title="oeis:A066413">A066413</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-heading3"><h3 id="Divide_an_integer">Divide an integer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=14" title="Edit section: Divide an integer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Repunit_primes">Repunit primes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=15" title="Edit section: Repunit primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Repunit" title="Repunit">Repunit</a></div> <p>The other way to deal with the fact that <span class="texhtml"><i>b</i><sup><i>n</i></sup> − 1</span> is always divisible by <span class="texhtml"><i>b</i> − 1</span>, it is to simply take out this factor and ask which values of <span class="texhtml"><i>n</i></span> make </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 {\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></dd></dl> <p>be prime. (The integer <span class="texhtml"><i>b</i></span> can be either positive or negative.) If, for example, we take <span class="texhtml"><i>b</i> = 10</span>, we get <span class="texhtml"><i>n</i></span> values of: </p> <dl><dd>2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence <span class="nowrap external"><a href="//oeis.org/A004023" class="extiw" title="oeis:A004023">A004023</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>),<br />corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence <span class="nowrap external"><a href="//oeis.org/A004022" class="extiw" title="oeis:A004022">A004022</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>These primes are called repunit primes. Another example is when we take <span class="texhtml"><i>b</i> = −12</span>, we get <span class="texhtml"><i>n</i></span> values of: </p> <dl><dd>2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence <span class="nowrap external"><a href="//oeis.org/A057178" class="extiw" title="oeis:A057178">A057178</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>),<br />corresponding to primes −11, 19141, 57154490053, ....</dd></dl> <p>It is a conjecture that for every integer <span class="texhtml"><i>b</i></span> which is not a <a href="/wiki/Perfect_power" title="Perfect power">perfect power</a>, there are infinitely many values of <span class="texhtml"><i>n</i></span> such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i><sup><i>n</i></sup> − 1</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span></span> is prime. (When <span class="texhtml"><i>b</i></span> is a perfect power, it can be shown that there is at most one <span class="texhtml"><i>n</i></span> value such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i><sup><i>n</i></sup> − 1</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span></span> is prime) </p><p>Least <span class="texhtml"><i>n</i></span> such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i><sup><i>n</i></sup> − 1</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span></span> is prime are (starting with <span class="texhtml"><i>b</i> = 2</span>, <span class="texhtml">0</span> if no such <span class="texhtml"><i>n</i></span> exists) </p> <dl><dd>2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... (sequence <span class="nowrap external"><a href="//oeis.org/A084740" class="extiw" title="oeis:A084740">A084740</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>For negative bases <span class="texhtml"><i>b</i></span>, they are (starting with <span class="texhtml"><i>b</i> = −2</span>, <span class="texhtml">0</span> if no such <span class="texhtml"><i>n</i></span> exists) </p> <dl><dd>3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence <span class="nowrap external"><a href="//oeis.org/A084742" class="extiw" title="oeis:A084742">A084742</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) (notice this OEIS sequence does not allow <span class="texhtml"><i>n</i> = 2</span>)</dd></dl> <p>Least base <span class="texhtml"><i>b</i></span> such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>b</i><sup>prime(<i>n</i>)</sup> − 1</span><span class="sr-only">/</span><span class="den"><i>b</i> − 1</span></span>⁠</span></span> is prime are </p> <dl><dd>2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence <span class="nowrap external"><a href="//oeis.org/A066180" class="extiw" title="oeis:A066180">A066180</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>For negative bases <span class="texhtml"><i>b</i></span>, they are </p> <dl><dd>3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence <span class="nowrap external"><a href="//oeis.org/A103795" class="extiw" title="oeis:A103795">A103795</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-heading4"><h4 id="Other_generalized_Mersenne_primes">Other generalized Mersenne primes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=16" title="Edit section: Other generalized Mersenne primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another generalized Mersenne number is </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 {\frac {a^{n}-b^{n}}{a-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a^{n}-b^{n}}{a-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c213410281390beadae9e558eedf9ffbb13dc37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.341ex; height:5.676ex;" alt="{\displaystyle {\frac {a^{n}-b^{n}}{a-b}}}"></span></dd></dl> <p>with <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span> any <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> integers, <span class="texhtml"><i>a</i> > 1</span> and <span class="texhtml">−<i>a</i> < <i>b</i> < <i>a</i></span>. (Since <span class="texhtml"><i>a</i><sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span> is always divisible by <span class="texhtml"><i>a</i> − <i>b</i></span>, the division is necessary for there to be any chance of finding prime numbers.)<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> We can ask which <span class="texhtml mvar" style="font-style:italic;">n</span> makes this number prime. It can be shown that such <span class="texhtml mvar" style="font-style:italic;">n</span> must be primes themselves or equal to 4, and <span class="texhtml mvar" style="font-style:italic;">n</span> can be 4 if and only if <span class="texhtml"><i>a</i> + <i>b</i> = 1</span> and <span class="texhtml"><i>a</i><sup>2</sup> + <i>b</i><sup>2</sup></span> is prime.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> It is a conjecture that for any pair <span class="texhtml">(<i>a</i>, <i>b</i>)</span> such that <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are not both perfect <span class="texhtml mvar" style="font-style:italic;">r</span>th powers for any <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml">−4<i>ab</i></span> is not a perfect <a href="/wiki/Fourth_power" title="Fourth power">fourth power</a>, there are infinitely many values of <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i><sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span><span class="sr-only">/</span><span class="den"><i>a</i> − <i>b</i></span></span>⁠</span></span> is prime.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> However, this has not been proved for any single value of <span class="texhtml">(<i>a</i>, <i>b</i>)</span>. </p> <table class="wikitable"> <caption>For more information, see <sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">a</span></span> </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">b</span></span> </th> <th>numbers <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">n</span></span> such that <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i><sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span><span class="sr-only">/</span><span class="den"><i>a</i> − <i>b</i></span></span>⁠</span></span></span> is prime<br />(some large terms are only <a href="/wiki/Probable_prime" title="Probable prime">probable primes</a>, these <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">n</span></span> are checked up to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold">100000</span> for <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>b</i></span>| ≤ 5</span></span> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>b</i></span>| = <i>a</i> − 1</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold">20000</span> for <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml">5 < |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>b</i></span>| < <i>a</i> − 1</span></span>) </th> <th><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence </th></tr> <tr> <td style="text-align:right;">2 </td> <td style="text-align:right;">1 </td> <td>2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ..., 136279841, ... </td> <td><a href="//oeis.org/A000043" class="extiw" title="oeis:A000043">A000043</a> </td></tr> <tr> <td style="text-align:right;">2 </td> <td style="text-align:right;">−1 </td> <td>3, 4<sup>*</sup>, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... </td> <td><a href="//oeis.org/A000978" class="extiw" title="oeis:A000978">A000978</a> </td></tr> <tr> <td style="text-align:right;">3 </td> <td style="text-align:right;">2 </td> <td>2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ... </td> <td><a href="//oeis.org/A057468" class="extiw" title="oeis:A057468">A057468</a> </td></tr> <tr> <td style="text-align:right;">3 </td> <td style="text-align:right;">1 </td> <td>3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... </td> <td><a href="//oeis.org/A028491" class="extiw" title="oeis:A028491">A028491</a> </td></tr> <tr> <td style="text-align:right;">3 </td> <td style="text-align:right;">−1 </td> <td>2<sup>*</sup>, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ... </td> <td><a href="//oeis.org/A007658" class="extiw" title="oeis:A007658">A007658</a> </td></tr> <tr> <td style="text-align:right;">3 </td> <td style="text-align:right;">−2 </td> <td>3, 4<sup>*</sup>, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ... </td> <td><a href="//oeis.org/A057469" class="extiw" title="oeis:A057469">A057469</a> </td></tr> <tr> <td style="text-align:right;">4 </td> <td style="text-align:right;">3 </td> <td>2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ... </td> <td><a href="//oeis.org/A059801" class="extiw" title="oeis:A059801">A059801</a> </td></tr> <tr> <td style="text-align:right;">4 </td> <td style="text-align:right;">1 </td> <td>2 (no others) </td> <td> </td></tr> <tr> <td style="text-align:right;">4 </td> <td style="text-align:right;">−1 </td> <td>2<sup>*</sup>, 3 (no others) </td> <td> </td></tr> <tr> <td style="text-align:right;">4 </td> <td style="text-align:right;">−3 </td> <td>3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ... </td> <td><a href="//oeis.org/A128066" class="extiw" title="oeis:A128066">A128066</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">4 </td> <td>3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ... </td> <td><a href="//oeis.org/A059802" class="extiw" title="oeis:A059802">A059802</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">3 </td> <td>13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ... </td> <td><a href="//oeis.org/A121877" class="extiw" title="oeis:A121877">A121877</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">2 </td> <td>2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ... </td> <td><a href="//oeis.org/A082182" class="extiw" title="oeis:A082182">A082182</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">1 </td> <td>3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ... </td> <td><a href="//oeis.org/A004061" class="extiw" title="oeis:A004061">A004061</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">−1 </td> <td>5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... </td> <td><a href="//oeis.org/A057171" class="extiw" title="oeis:A057171">A057171</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">−2 </td> <td>2<sup>*</sup>, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ... </td> <td><a href="//oeis.org/A082387" class="extiw" title="oeis:A082387">A082387</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">−3 </td> <td>2<sup>*</sup>, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ... </td> <td><a href="//oeis.org/A122853" class="extiw" title="oeis:A122853">A122853</a> </td></tr> <tr> <td style="text-align:right;">5 </td> <td style="text-align:right;">−4 </td> <td>4<sup>*</sup>, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ... </td> <td><a href="//oeis.org/A128335" class="extiw" title="oeis:A128335">A128335</a> </td></tr> <tr> <td style="text-align:right;">6 </td> <td style="text-align:right;">5 </td> <td>2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ... </td> <td><a href="//oeis.org/A062572" class="extiw" title="oeis:A062572">A062572</a> </td></tr> <tr> <td style="text-align:right;">6 </td> <td style="text-align:right;">1 </td> <td>2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... </td> <td><a href="//oeis.org/A004062" class="extiw" title="oeis:A004062">A004062</a> </td></tr> <tr> <td style="text-align:right;">6 </td> <td style="text-align:right;">−1 </td> <td>2<sup>*</sup>, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... </td> <td><a href="//oeis.org/A057172" class="extiw" title="oeis:A057172">A057172</a> </td></tr> <tr> <td style="text-align:right;">6 </td> <td style="text-align:right;">−5 </td> <td>3, 4<sup>*</sup>, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ... </td> <td><a href="//oeis.org/A128336" class="extiw" title="oeis:A128336">A128336</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">6 </td> <td>2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ... </td> <td><a href="//oeis.org/A062573" class="extiw" title="oeis:A062573">A062573</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">5 </td> <td>3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ... </td> <td><a href="//oeis.org/A128344" class="extiw" title="oeis:A128344">A128344</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">4 </td> <td>2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ... </td> <td><a href="//oeis.org/A213073" class="extiw" title="oeis:A213073">A213073</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">3 </td> <td>3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ... </td> <td><a href="//oeis.org/A128024" class="extiw" title="oeis:A128024">A128024</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">2 </td> <td>3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ... </td> <td><a href="//oeis.org/A215487" class="extiw" title="oeis:A215487">A215487</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">1 </td> <td>5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... </td> <td><a href="//oeis.org/A004063" class="extiw" title="oeis:A004063">A004063</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−1 </td> <td>3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ... </td> <td><a href="//oeis.org/A057173" class="extiw" title="oeis:A057173">A057173</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−2 </td> <td>2<sup>*</sup>, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ... </td> <td><a href="//oeis.org/A125955" class="extiw" title="oeis:A125955">A125955</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−3 </td> <td>3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ... </td> <td><a href="//oeis.org/A128067" class="extiw" title="oeis:A128067">A128067</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−4 </td> <td>2<sup>*</sup>, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ... </td> <td><a href="//oeis.org/A218373" class="extiw" title="oeis:A218373">A218373</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−5 </td> <td>2<sup>*</sup>, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ... </td> <td><a href="//oeis.org/A128337" class="extiw" title="oeis:A128337">A128337</a> </td></tr> <tr> <td style="text-align:right;">7 </td> <td style="text-align:right;">−6 </td> <td>3, 53, 83, 487, 743, ... </td> <td><a href="//oeis.org/A187805" class="extiw" title="oeis:A187805">A187805</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">7 </td> <td>7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ... </td> <td><a href="//oeis.org/A062574" class="extiw" title="oeis:A062574">A062574</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">5 </td> <td>2, 19, 1021, 5077, 34031, 46099, 65707, ... </td> <td><a href="//oeis.org/A128345" class="extiw" title="oeis:A128345">A128345</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">3 </td> <td>2, 3, 7, 19, 31, 67, 89, 9227, 43891, ... </td> <td><a href="//oeis.org/A128025" class="extiw" title="oeis:A128025">A128025</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">1 </td> <td>3 (no others) </td> <td> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">−1 </td> <td>2<sup>*</sup> (no others) </td> <td> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">−3 </td> <td>2<sup>*</sup>, 5, 163, 191, 229, 271, 733, 21059, 25237, ... </td> <td><a href="//oeis.org/A128068" class="extiw" title="oeis:A128068">A128068</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">−5 </td> <td>2<sup>*</sup>, 7, 19, 167, 173, 223, 281, 21647, ... </td> <td><a href="//oeis.org/A128338" class="extiw" title="oeis:A128338">A128338</a> </td></tr> <tr> <td style="text-align:right;">8 </td> <td style="text-align:right;">−7 </td> <td>4<sup>*</sup>, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ... </td> <td><a href="//oeis.org/A181141" class="extiw" title="oeis:A181141">A181141</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">8 </td> <td>2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ... </td> <td><a href="//oeis.org/A059803" class="extiw" title="oeis:A059803">A059803</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">7 </td> <td>3, 5, 7, 4703, 30113, ... </td> <td><a href="//oeis.org/A273010" class="extiw" title="oeis:A273010">A273010</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">5 </td> <td>3, 11, 17, 173, 839, 971, 40867, 45821, ... </td> <td><a href="//oeis.org/A128346" class="extiw" title="oeis:A128346">A128346</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">4 </td> <td>2 (no others) </td> <td> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">2 </td> <td>2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ... </td> <td><a href="//oeis.org/A173718" class="extiw" title="oeis:A173718">A173718</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">1 </td> <td>(none) </td> <td> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−1 </td> <td>3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... </td> <td><a href="//oeis.org/A057175" class="extiw" title="oeis:A057175">A057175</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−2 </td> <td>2<sup>*</sup>, 3, 7, 127, 283, 883, 1523, 4001, ... </td> <td><a href="//oeis.org/A125956" class="extiw" title="oeis:A125956">A125956</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−4 </td> <td>2<sup>*</sup>, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ... </td> <td><a href="//oeis.org/A211409" class="extiw" title="oeis:A211409">A211409</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−5 </td> <td>3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ... </td> <td><a href="//oeis.org/A128339" class="extiw" title="oeis:A128339">A128339</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−7 </td> <td>2<sup>*</sup>, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ... </td> <td><a href="//oeis.org/A301369" class="extiw" title="oeis:A301369">A301369</a> </td></tr> <tr> <td style="text-align:right;">9 </td> <td style="text-align:right;">−8 </td> <td>3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ... </td> <td><a href="//oeis.org/A187819" class="extiw" title="oeis:A187819">A187819</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">9 </td> <td>2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ... </td> <td><a href="//oeis.org/A062576" class="extiw" title="oeis:A062576">A062576</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">7 </td> <td>2, 31, 103, 617, 10253, 10691, ... </td> <td><a href="//oeis.org/A273403" class="extiw" title="oeis:A273403">A273403</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">3 </td> <td>2, 3, 5, 37, 599, 38393, 51431, ... </td> <td><a href="//oeis.org/A128026" class="extiw" title="oeis:A128026">A128026</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">1 </td> <td>2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... </td> <td><a href="//oeis.org/A004023" class="extiw" title="oeis:A004023">A004023</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">−1 </td> <td>5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... </td> <td><a href="//oeis.org/A001562" class="extiw" title="oeis:A001562">A001562</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">−3 </td> <td>2<sup>*</sup>, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ... </td> <td><a href="//oeis.org/A128069" class="extiw" title="oeis:A128069">A128069</a> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">−7 </td> <td>2<sup>*</sup>, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">10 </td> <td style="text-align:right;">−9 </td> <td>4<sup>*</sup>, 7, 67, 73, 1091, 1483, 10937, ... </td> <td><a href="//oeis.org/A217095" class="extiw" title="oeis:A217095">A217095</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">10 </td> <td>3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ... </td> <td><a href="//oeis.org/A062577" class="extiw" title="oeis:A062577">A062577</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">9 </td> <td>5, 31, 271, 929, 2789, 4153, ... </td> <td><a href="//oeis.org/A273601" class="extiw" title="oeis:A273601">A273601</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">8 </td> <td>2, 7, 11, 17, 37, 521, 877, 2423, ... </td> <td><a href="//oeis.org/A273600" class="extiw" title="oeis:A273600">A273600</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">7 </td> <td>5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ... </td> <td><a href="//oeis.org/A273599" class="extiw" title="oeis:A273599">A273599</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">6 </td> <td>2, 3, 11, 163, 191, 269, 1381, 1493, ... </td> <td><a href="//oeis.org/A273598" class="extiw" title="oeis:A273598">A273598</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">5 </td> <td>5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... </td> <td><a href="//oeis.org/A128347" class="extiw" title="oeis:A128347">A128347</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">4 </td> <td>3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ... </td> <td><a href="//oeis.org/A216181" class="extiw" title="oeis:A216181">A216181</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">3 </td> <td>3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ... </td> <td><a href="//oeis.org/A128027" class="extiw" title="oeis:A128027">A128027</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">2 </td> <td>2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ... </td> <td><a href="//oeis.org/A210506" class="extiw" title="oeis:A210506">A210506</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">1 </td> <td>17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... </td> <td><a href="//oeis.org/A005808" class="extiw" title="oeis:A005808">A005808</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−1 </td> <td>5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... </td> <td><a href="//oeis.org/A057177" class="extiw" title="oeis:A057177">A057177</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−2 </td> <td>3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ... </td> <td><a href="//oeis.org/A125957" class="extiw" title="oeis:A125957">A125957</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−3 </td> <td>3, 103, 271, 523, 23087, 69833, ... </td> <td><a href="//oeis.org/A128070" class="extiw" title="oeis:A128070">A128070</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−4 </td> <td>2<sup>*</sup>, 7, 53, 67, 71, 443, 26497, ... </td> <td><a href="//oeis.org/A224501" class="extiw" title="oeis:A224501">A224501</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−5 </td> <td>7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ... </td> <td><a href="//oeis.org/A128340" class="extiw" title="oeis:A128340">A128340</a> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−6 </td> <td>2<sup>*</sup>, 5, 7, 107, 383, 17359, 21929, 26393, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−7 </td> <td>7, 1163, 4007, 10159, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−8 </td> <td>2<sup>*</sup>, 3, 13, 31, 59, 131, 223, 227, 1523, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−9 </td> <td>2<sup>*</sup>, 3, 17, 41, 43, 59, 83, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">11 </td> <td style="text-align:right;">−10 </td> <td>53, 421, 647, 1601, 35527, ... </td> <td><a href="//oeis.org/A185239" class="extiw" title="oeis:A185239">A185239</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">11 </td> <td>2, 3, 7, 89, 101, 293, 4463, 70067, ... </td> <td><a href="//oeis.org/A062578" class="extiw" title="oeis:A062578">A062578</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">7 </td> <td>2, 3, 7, 13, 47, 89, 139, 523, 1051, ... </td> <td><a href="//oeis.org/A273814" class="extiw" title="oeis:A273814">A273814</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">5 </td> <td>2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ... </td> <td><a href="//oeis.org/A128348" class="extiw" title="oeis:A128348">A128348</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">1 </td> <td>2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... </td> <td><a href="//oeis.org/A004064" class="extiw" title="oeis:A004064">A004064</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">−1 </td> <td>2<sup>*</sup>, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... </td> <td><a href="//oeis.org/A057178" class="extiw" title="oeis:A057178">A057178</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">−5 </td> <td>2<sup>*</sup>, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ... </td> <td><a href="//oeis.org/A128341" class="extiw" title="oeis:A128341">A128341</a> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">−7 </td> <td>2<sup>*</sup>, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ... </td> <td> </td></tr> <tr> <td style="text-align:right;">12 </td> <td style="text-align:right;">−11 </td> <td>47, 401, 509, 8609, ... </td> <td><a href="//oeis.org/A213216" class="extiw" title="oeis:A213216">A213216</a> </td></tr></tbody></table> <p><sup>*</sup>Note: if <span class="texhtml"><i>b</i> < 0</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> is even, then the numbers <span class="texhtml mvar" style="font-style:italic;">n</span> are not included in the corresponding OEIS sequence. </p><p>When <span class="texhtml"><i>a</i> = <i>b</i> + 1</span>, it is <span class="texhtml">(<i>b</i> + 1)<sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span>, a difference of two consecutive perfect <span class="texhtml mvar" style="font-style:italic;">n</span>th powers, and if <span class="texhtml"><i>a</i><sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span> is prime, then <span class="texhtml mvar" style="font-style:italic;">a</span> must be <span class="texhtml"><i>b</i> + 1</span>, because it is divisible by <span class="texhtml"><i>a</i> − <i>b</i></span>. </p><p>Least <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml">(<i>b</i> + 1)<sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span> is prime are </p> <dl><dd>2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence <span class="nowrap external"><a href="//oeis.org/A058013" class="extiw" title="oeis:A058013">A058013</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>Least <span class="texhtml mvar" style="font-style:italic;">b</span> such that <span class="texhtml">(<i>b</i> + 1)<sup>prime(<i>n</i>)</sup> − <i>b</i><sup>prime(<i>n</i>)</sup></span> is prime are </p> <dl><dd>1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... (sequence <span class="nowrap external"><a href="//oeis.org/A222119" class="extiw" title="oeis:A222119">A222119</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="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 18em;"> <ul><li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of two</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Borwein_constant" title="Erdős–Borwein constant">Erdős–Borwein constant</a></li> <li><a href="/wiki/Mersenne_conjectures" title="Mersenne conjectures">Mersenne conjectures</a></li> <li><a href="/wiki/Mersenne_twister" class="mw-redirect" title="Mersenne twister">Mersenne twister</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne number</a></li> <li><a href="/wiki/Prime95" title="Prime95">Prime95</a> / MPrime</li> <li><a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a> (GIMPS)</li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known prime number</a></li> <li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich prime</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff prime</a></li> <li><a href="/wiki/Cullen_prime" class="mw-redirect" title="Cullen prime">Cullen prime</a></li> <li><a href="/wiki/Woodall_prime" class="mw-redirect" title="Woodall prime">Woodall prime</a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth prime</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas prime</a></li> <li><a href="/wiki/Gillies%27_conjecture" title="Gillies' conjecture">Gillies' conjecture</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams number</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=18" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">This number is the same as the <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas number</a> <span class="texhtml"><i>U<sub>n</sub></i>(<i>a</i> + <i>b</i>, <i>ab</i>)</span>, since <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are the <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> of the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> <span class="texhtml"><i>x</i><sup>2</sup> − (<i>a</i> + <i>b</i>)<i>x</i> + <i>ab</i> = 0</span>.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Since <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i><sup>4</sup> − <i>b</i><sup>4</sup></span><span class="sr-only">/</span><span class="den"><i>a</i> − <i>b</i></span></span>⁠</span> = (<i>a</i> + <i>b</i>)(<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>)</span>. Thus, in this case the pair <span class="texhtml">(<i>a</i>, <i>b</i>)</span> must be <span class="texhtml">(<i>x</i> + 1, −<i>x</i>)</span> and <span class="texhtml"><i>x</i><sup>2</sup> + (<i>x</i> + 1)<sup>2</sup></span> must be prime. That is, <span class="texhtml mvar" style="font-style:italic;">x</span> must be in <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027861" class="extiw" title="oeis:A027861">A027861</a></span>.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">When <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are both perfect <span class="texhtml mvar" style="font-style:italic;">r</span>th powers for some <span class="texhtml"><i>r</i> > 1</span> or when <span class="texhtml">−4<i>ab</i></span> is a perfect fourth power, it can be shown that there are at most two values of <span class="texhtml mvar" style="font-style:italic;">n</span> with this property: in these cases, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i><sup><i>n</i></sup> − <i>b</i><sup><i>n</i></sup></span><span class="sr-only">/</span><span class="den"><i>a</i> − <i>b</i></span></span>⁠</span></span> can be factored algebraically.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2022)">citation needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-GIMPS-2024-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-GIMPS-2024_1-0">^</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mersenne.org/primes/?press=M136279841">"GIMPS Discovers Largest Known Prime Number: 2<sup>136,279,841</sup> − 1"</a>. <i>Mersenne Research, Inc</i>. 21 October 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">21 October</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mersenne+Research%2C+Inc.&rft.atitle=GIMPS+Discovers+Largest+Known+Prime+Number%3A+2%3Csup%3E136%2C279%2C841%3C%2Fsup%3E+%E2%88%92+1&rft.date=2024-10-21&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2F%3Fpress%3DM136279841&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-GIMPS-2018-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-GIMPS-2018_2-0">^</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://www.mersenne.org/primes/press/M82589933.html">"GIMPS Project Discovers Largest Known Prime Number: 2<sup>82,589,933</sup>-1"</a>. <i>Mersenne Research, Inc</i>. 21 December 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">21 December</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mersenne+Research%2C+Inc.&rft.atitle=GIMPS+Project+Discovers+Largest+Known+Prime+Number%3A+2%3Csup%3E82%2C589%2C933%3C%2Fsup%3E-1&rft.date=2018-12-21&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2Fpress%2FM82589933.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mersenne.org/report_milestones/">"GIMPS Milestones Report"</a>. <i>Mersenne.org</i>. Mersenne Research, Inc<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mersenne.org&rft.atitle=GIMPS+Milestones+Report&rft_id=http%3A%2F%2Fwww.mersenne.org%2Freport_milestones%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-Wagstaff-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wagstaff_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwell" class="citation web cs1">Caldwell, Chris. <a rel="nofollow" class="external text" href="http://primes.utm.edu/mersenne/heuristic.html">"Heuristics: Deriving the Wagstaff Mersenne Conjecture"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Heuristics%3A+Deriving+the+Wagstaff+Mersenne+Conjecture&rft.aulast=Caldwell&rft.aufirst=Chris&rft_id=http%3A%2F%2Fprimes.utm.edu%2Fmersenne%2Fheuristic.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></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">Chris K. Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/mersenne/index.html">Mersenne Primes: History, Theorems and Lists</a></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">The Prime Pages, <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/page.php?sort=MersennesConjecture">Mersenne's conjecture</a>.</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 id="CITEREFHardyWright1959" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (1959). <i>An Introduction to the Theory of Numbers</i> (4th ed.). Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.edition=4th&rft.pub=Oxford+University+Press&rft.date=1959&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" 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 id="CITEREFCole1903" class="citation journal cs1">Cole, F. N. 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Retrieved <span class="nowrap">2018-01-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mersenne.org&rft.atitle=Mersenne+Prime+Discovery+-+2%5E77232917-1+is+Prime%21&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2F%3Fpress%3DM77232917&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</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://christianchronicle.org/largest-known-prime-number-found-church-computer/">"Largest-known prime number found on church computer"</a>. <i>christianchronicle.org</i>. 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January 5, 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Found%3A+A+Special%2C+Mind-Bogglingly+Large+Prime+Number&rft.date=2018-01-05&rft_id=https%3A%2F%2Fwww.atlasobscura.com%2Farticles%2Ffound-special-mind-bogglingly-large-prime-number-mersenne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</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://www.mersenne.org/primes/?press=M82589933">"GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-01-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=GIMPS+Discovers+Largest+Known+Prime+Number%3A+2%5E82%2C589%2C933-1&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2F%3Fpress%3DM82589933&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</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://www.mersenne.org/various/math.php">"GIMPS - The Math - PrimeNet"</a>. <i>www.mersenne.org</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">21 Oct</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mersenne.org&rft.atitle=Mersenne+Prime+Number+discovery+-+2136279841-1+is+Prime%21&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2F%3Fpress%3DM136279841&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.garlic.com/~wedgingt/mersenne.html">Will Edgington's Mersenne Page</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141014102940/http://www.garlic.com/~wedgingt/mersenne.html">Archived</a> 2014-10-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwell" class="citation web cs1">Caldwell, Chris K. <a rel="nofollow" class="external text" href="http://primes.utm.edu/notes/proofs/MerDiv2.html">"Proof of a result of Euler and Lagrange on Mersenne Divisors"</a>. <i><a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Prime+Pages&rft.atitle=Proof+of+a+result+of+Euler+and+Lagrange+on+Mersenne+Divisors&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=http%3A%2F%2Fprimes.utm.edu%2Fnotes%2Fproofs%2FMerDiv2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinjungBosLenstra2014" class="citation book cs1">Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. 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Retrieved <span class="nowrap">2022-09-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PRP+Top+Records&rft.au=Henri+Lifchitz+and+Renaud+Lifchitz&rft_id=http%3A%2F%2Fwww.primenumbers.net%2Fprptop%2Fsearchform.php%3Fform%3D%282%5Ep-1%29%2F%253F%26action%3DSearch&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</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://www.mersenne.ca/exponent/12720787">"M12720787 Mersenne number exponent details"</a>. <i>www.mersenne.ca</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2011-05-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=JPL+Small-Body+Database+Browser&rft.pub=Ssd.jpl.nasa.gov&rft.au=Alan+Chamberlin&rft_id=http%3A%2F%2Fssd.jpl.nasa.gov%2Fsbdb.cgi%3Fsstr%3D8191%2BMersenne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</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="http://oeis.org/A016131">"OEIS A016131"</a>. 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Springer US. pp. <span class="nowrap">509–</span>510. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4419-5906-5_32">10.1007/978-1-4419-5906-5_32</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-5905-8" title="Special:BookSources/978-1-4419-5905-8"><bdi>978-1-4419-5905-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Generalized+Mersenne+Prime&rft.btitle=Encyclopedia+of+Cryptography+and+Security&rft.pages=%3Cspan+class%3D%22nowrap%22%3E509-%3C%2Fspan%3E510&rft.pub=Springer+US&rft.date=2011-01-01&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-5906-5_32&rft.isbn=978-1-4419-5905-8&rft.aulast=Solinas&rft.aufirst=Jerome+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Chris Caldwell: <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/xpage/GaussianMersenne.html">The Prime Glossary: Gaussian Mersenne</a> (part of the <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a>)</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.primenumbers.net/Henri/us/MersFermus.htm"><span class="texhtml">(<i>x</i>, 1)</span> and <span class="texhtml">(<i>x</i>, −1)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 2 to 50</a></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"><span class="texhtml">(<i>x</i>, 1)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 2 to 160</a></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"><span class="texhtml">(<i>x</i>, −1)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 2 to 160</a></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/PrimSerien/PrimPot.txt"><span class="texhtml">(<i>x</i> + 1, <i>x</i>)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 1 to 160</a></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/PrimSerien/PrimPotP.txt"><span class="texhtml">(<i>x</i> + 1, −<i>x</i>)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 1 to 40</a></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/PrimSerien/PrimPot2.txt"><span class="texhtml">(<i>x</i> + 2, <i>x</i>)</span> for odd <span class="texhtml mvar" style="font-style:italic;">x</span> = 1 to 107</a></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf#search=%22107%20811%202819%204817%209601%22"><span class="texhtml">(<i>x</i>, −1)</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> = 2 to 200</a></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.primenumbers.net/prptop/searchform.php?form=%28a%5En-b%5En%29%2Fc&action=Search">PRP records, search for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{n}-b^{n})/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{n}-b^{n})/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027c1d83871b53b9c30a0088376349ee2dd3dcba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.483ex; height:2.843ex;" alt="{\displaystyle (a^{n}-b^{n})/c}"></span>⁠</span>, that is, <span class="texhtml">(<i>a</i>, <i>b</i>)</span></a></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.primenumbers.net/prptop/searchform.php?form=%28a%5En%2Bb%5En%29%2Fc&action=Search">PRP records, search for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{n}+b^{n})/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{n}+b^{n})/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5699168996fb70c686b3e13e7d7a250d7a3de3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.483ex; height:2.843ex;" alt="{\displaystyle (a^{n}+b^{n})/c}"></span>⁠</span>, that is, <span class="texhtml">(<i>a</i>, −<i>b</i>)</span></a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Mersenne_prime" class="extiw" title="wiktionary:Mersenne prime">Mersenne prime</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Mersenne_number">"Mersenne number"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mersenne+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMersenne_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.mersenne.org">GIMPS home page</a></li> <li><a rel="nofollow" class="external text" href="https://www.mersenne.org/report_milestones/">GIMPS Milestones Report</a> – status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of the largest known Mersenne primes</li> <li><a rel="nofollow" class="external text" href="http://www.mersenne.org/report_factors/">GIMPS, known factors of Mersenne numbers</a></li> <li><span class="texhtml"><i>M</i><sub><i>q</i></sub> = (8<i>x</i>)<sup>2</sup> − (3<i>qy</i>)<sup>2</sup></span> <a rel="nofollow" class="external text" href="http://tony.reix.free.fr/Mersenne/Mersenne8x3qy.pdf">Property of Mersenne numbers with prime exponent that are composite</a> (PDF)</li> <li><span class="texhtml"><i>M</i><sub><i>q</i></sub> = <i>x</i><sup>2</sup> + <i>d</i>·<i>y</i><sup>2</sup></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050221194456/http://www.math.leidenuniv.nl/scripties/jansen.ps">math thesis</a> (PS)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrime" class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130531091712/http://numberphile.com/videos/31.html">"31 and Mersenne Primes"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/31.html">the original</a> on 2013-05-31<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=31+and+Mersenne+Primes&rft.aulast=Grime&rft.aufirst=James&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2F31.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.utm.edu/research/primes/mersenne/LukeMirror/biblio.htm">Mersenne prime bibliography</a> with hyperlinks to original publications</li> <li><a rel="nofollow" class="external text" href="http://www.taz.de/pt/2005/03/11/a0355.nf/text">report about Mersenne primes</a> – detection in detail <span class="languageicon">(in German)</span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061205062803/http://mersennewiki.org/index.php/Main_Page">GIMPS wiki</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141014102940/http://www.garlic.com/~wedgingt/mersenne.html">Will Edgington's Mersenne Page</a> – contains factors for small Mersenne numbers</li> <li><a rel="nofollow" class="external text" href="http://www.mersenne.org/report_factors/">Known factors of Mersenne numbers</a></li> <li><a rel="nofollow" class="external text" href="http://www.isthe.com/chongo/tech/math/prime/mersenne.html">Decimal digits and English names of Mersenne primes</a></li> <li><a rel="nofollow" class="external text" href="http://primes.utm.edu/curios/page.php/2305843009213693951.html">Prime curios: 2305843009213693951</a></li> <li><a rel="nofollow" class="external free" href="http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt">http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141105173948/http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt">Archived</a> 2014-11-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external free" href="http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt">http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130502023640/http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt">Archived</a> 2013-05-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A250197">sequence A250197 (Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime)</a> – Factorization of Mersenne numbers <span class="texhtml"><i>M<sub>n</sub></i></span> (<span class="texhtml"><i>n</i></span> up to 1280)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141015022019/http://www.garlic.com/~wedgingt/factoredM.txt">Factorization of completely factored Mersenne numbers</a></li> <li><a rel="nofollow" class="external text" href="https://homes.cerias.purdue.edu/~ssw/cun/index.html">The Cunningham project, factorization of <span class="texhtml"><i>b<sup>n</sup></i> ± 1, <i>b</i> = 2, 3, 5, 6, 7, 10, 11, 12</span></a></li> <li><a rel="nofollow" class="external free" href="http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm">http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304120336/http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external free" href="http://www.leyland.vispa.com/numth/factorization/anbn/main.htm">http://www.leyland.vispa.com/numth/factorization/anbn/main.htm</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160202122746/http://www.leyland.vispa.com/numth/factorization/anbn/main.htm">Archived</a> 2016-02-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="MathWorld_links">MathWorld links</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mersenne_prime&action=edit&section=21" title="Edit section: MathWorld links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Mersenne_number"><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/MersenneNumber.html">"Mersenne number"</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=Mersenne+number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMersenneNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Mersenne_prime"><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/MersennePrime.html">"Mersenne prime"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Mersenne+prime&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMersennePrime.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMersenne+prime" class="Z3988"></span></span></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 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.navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes229" 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 class="mw-selflink selflink">Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>p</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples30" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple"><i>k</i>-tuples</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twin_prime" title="Twin prime">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i> + <i>a·n</i>, <i>n</i> = 0, 1, 2, 3, ...</span>)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (<span class="texhtml texhtml-big" style="font-size:110%;">consecutive <i>p</i> − <i>n</i>, <i>p</i>, <i>p</i> + <i>n</i></span>)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i> ± 1, 2<i>n</i> ± 1, 4<i>n</i> ± 1, …</span>)</a></li> <li><a href="/wiki/Chen_prime" title="Chen prime">Chen</a></li> <li><a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">Sophie Germain/Safe (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> + 1</span>)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> ± 1, 4<i>p</i> ± 3, 8<i>p</i> ± 7, ...</span>)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By size</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Megaprime" title="Megaprime">Mega (1,000,000+ digits)</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known</a> <ul><li><a href="/wiki/List_of_largest_known_primes_and_probable_primes" title="List of largest known primes and probable primes">list</a></li></ul></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Gaussian_integer#Gaussian_primes" title="Gaussian integer">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div></td></tr></tbody></table></div> <div 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="Classes_of_natural_numbers743" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></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="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and 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/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 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_±_1743" 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 class="mw-selflink selflink">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_numbers743" 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_numbers743" 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_numbers743" 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_sums743" 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_numbers743" 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_numbers743" 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="Primes743" 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="Pseudoprimes743" 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_dynamics743" 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_numbers743" 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_numbers743" 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 href="/wiki/Repunit" title="Repunit">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_numbers743" 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_sieve743" 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_related743" 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_related743" 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 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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="Marin_Mersenne18" 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 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title="Mersenne's laws">Mersenne's laws</a></li> <li><a class="mw-selflink selflink">Mersenne prime</a> <ul><li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne number</a></li> <li><a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a></li> <li><a href="/wiki/List_of_Mersenne_primes_and_perfect_numbers" title="List of Mersenne primes and perfect numbers">List</a></li></ul></li> <li><a href="/wiki/Mersenne_Twister" title="Mersenne Twister">Mersenne Twister</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="Large_numbers201" style="padding:3px"><table class="nowraplinks hlist 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:Large_numbers" title="Template:Large numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Large_numbers" title="Template talk:Large numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Large_numbers" title="Special:EditPage/Template:Large numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Large_numbers201" style="font-size:114%;margin:0 4em"><a href="/wiki/Large_numbers" title="Large numbers">Large numbers</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples <br />in<br />numerical <br />order</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/100" title="100">Hundred</a></li> <li><a href="/wiki/1000_(number)" title="1000 (number)">Thousand</a></li> <li><a href="/wiki/10,000" title="10,000">Ten thousand</a></li> <li><a href="/wiki/100,000" title="100,000">Hundred thousand</a></li> <li><a href="/wiki/1,000,000" title="1,000,000">Million</a></li> <li><a href="/wiki/1,000,000,000" title="1,000,000,000">Billion</a></li> <li><a href="/wiki/Trillion" title="Trillion">Trillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1015" title="Orders of magnitude (numbers)">Quadrillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1018" title="Orders of magnitude (numbers)">Quintillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1021" title="Orders of magnitude (numbers)">Sextillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1024" title="Orders of magnitude (numbers)">Septillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1027" title="Orders of magnitude (numbers)">Octillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1030" title="Orders of magnitude (numbers)">Nonillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1033" title="Orders of magnitude (numbers)">Decillion</a></li> <li><a href="/wiki/Eddington_number" title="Eddington number">Eddington number</a></li> <li><a href="/wiki/Googol" title="Googol">Googol</a></li> <li><a href="/wiki/Shannon_number" title="Shannon number">Shannon number</a></li> <li><a href="/wiki/Googolplex" title="Googolplex">Googolplex</a></li> <li><a href="/wiki/Skewes%27s_number" title="Skewes's number">Skewes's number</a></li> <li><a href="/wiki/Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Moser's number</a></li> <li><a href="/wiki/Graham%27s_number" title="Graham's number">Graham's number</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">TREE(3)</a></li> <li><a href="/wiki/Friedman%27s_SSCG_function" title="Friedman's SSCG function">SSCG(3)</a></li> <li><a href="/wiki/Buchholz_hydra#BH(n)" title="Buchholz hydra">BH(3)</a></li> <li><a href="/wiki/Rayo%27s_number" title="Rayo's number">Rayo's number</a></li> <li><a href="/wiki/Infinity" title="Infinity">Infinity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Expression<br />methods</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%">Notations</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/Scientific_notation" title="Scientific notation">Scientific notation</a></li> <li><a href="/wiki/Knuth%27s_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li> <li><a href="/wiki/Conway_chained_arrow_notation" title="Conway chained arrow notation">Conway chained arrow notation</a></li> <li><a href="/wiki/Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Steinhaus–Moser notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</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/Hyperoperation" title="Hyperoperation">Hyperoperation</a> <ul><li><a href="/wiki/Tetration" title="Tetration">Tetration</a></li> <li><a href="/wiki/Pentation" title="Pentation">Pentation</a></li></ul></li> <li><a href="/wiki/Ackermann_function" title="Ackermann function">Ackermann function</a></li> <li><a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li> <li><a href="/wiki/Fast-growing_hierarchy" title="Fast-growing hierarchy">Fast-growing hierarchy</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related <br />articles<br />(alphabetical <br />order)<br /></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/Busy_beaver" title="Busy beaver">Busy beaver</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real number line</a></li> <li><a href="/wiki/Indefinite_and_fictitious_numbers" title="Indefinite and fictitious numbers">Indefinite and fictitious numbers</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known prime number</a></li> <li><a href="/wiki/List_of_numbers" title="List of numbers">List of numbers</a></li> <li><a href="/wiki/Long_and_short_scales" title="Long and short scales">Long and short scales</a></li> <li><a href="/wiki/Number" title="Number">Number systems</a></li> <li><a href="/wiki/Numeral_(linguistics)" title="Numeral (linguistics)">Number names</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)" title="Orders of magnitude (numbers)">Orders of magnitude</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Carl_Sagan#Sagan_units" title="Carl Sagan">Sagan Unit</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="font-weight:bold;"><div> <ul><li><a href="/wiki/Names_of_large_numbers" title="Names of large numbers">Names</a></li> <li><a href="/wiki/History_of_large_numbers" title="History of large numbers">History</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mersenne_prime&oldid=1273109529">https://en.wikipedia.org/w/index.php?title=Mersenne_prime&oldid=1273109529</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Classes_of_prime_numbers" title="Category:Classes of prime numbers">Classes of prime numbers</a></li><li><a href="/wiki/Category:Eponymous_numbers_in_mathematics" 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