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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heuristic_arguments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Heuristic arguments</span> </div> </a> <ul id="toc-Heuristic_arguments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Equivalent conditions</span> </div> </a> <ul id="toc-Equivalent_conditions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Factorization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Factorization</span> </div> </a> <ul id="toc-Factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudoprimes_and_Fermat_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pseudoprimes_and_Fermat_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Pseudoprimes and Fermat numbers</span> </div> </a> <ul id="toc-Pseudoprimes_and_Fermat_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_theorems_about_Fermat_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_theorems_about_Fermat_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other theorems about Fermat numbers</span> </div> </a> <ul id="toc-Other_theorems_about_Fermat_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_constructible_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_constructible_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Relationship to constructible polygons</span> </div> </a> <ul id="toc-Relationship_to_constructible_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_of_Fermat_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_of_Fermat_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications of Fermat numbers</span> </div> </a> <button aria-controls="toc-Applications_of_Fermat_numbers-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 Applications of Fermat numbers subsection</span> </button> <ul id="toc-Applications_of_Fermat_numbers-sublist" class="vector-toc-list"> <li id="toc-Pseudorandom_number_generation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudorandom_number_generation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Pseudorandom number generation</span> </div> </a> <ul id="toc-Pseudorandom_number_generation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalized_Fermat_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalized_Fermat_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Generalized Fermat numbers</span> </div> </a> <button aria-controls="toc-Generalized_Fermat_numbers-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 Generalized Fermat numbers subsection</span> </button> <ul id="toc-Generalized_Fermat_numbers-sublist" class="vector-toc-list"> <li id="toc-Generalized_Fermat_primes_of_the_form_Fn(a)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_Fermat_primes_of_the_form_Fn(a)"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Generalized Fermat primes of the form F<sub>n</sub>(<i>a</i>)</span> </div> </a> <ul id="toc-Generalized_Fermat_primes_of_the_form_Fn(a)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_Fermat_primes_of_the_form_Fn(a,_b)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_Fermat_primes_of_the_form_Fn(a,_b)"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Generalized Fermat primes of the form F<sub>n</sub>(<i>a</i>, <i>b</i>)</span> </div> </a> <ul id="toc-Generalized_Fermat_primes_of_the_form_Fn(a,_b)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Largest_known_generalized_Fermat_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Largest_known_generalized_Fermat_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Largest known generalized Fermat primes</span> </div> </a> <ul id="toc-Largest_known_generalized_Fermat_primes-sublist" class="vector-toc-list"> </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">9</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">10</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">11</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">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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">Fermat number</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 41 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-41" 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">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/Fermat_t%C3%A6l" title="Fermat tæl – Old English" lang="ang" hreflang="ang" data-title="Fermat tæl" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%81%D9%8A%D8%B1%D9%85%D8%A7" title="عدد فيرما – Arabic" lang="ar" hreflang="ar" data-title="عدد فيرما" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ferma_%C9%99d%C9%99dl%C9%99ri" title="Ferma ədədləri – Azerbaijani" lang="az" hreflang="az" data-title="Ferma ədədləri" 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%AB%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%AE%E0%A6%BE_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="ফার্মা সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="ফার্মা সংখ্যা" 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%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число на Ферма – Bulgarian" lang="bg" hreflang="bg" data-title="Число на Ферма" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Fermat" title="Nombre de Fermat – Catalan" lang="ca" hreflang="ca" data-title="Nombre de Fermat" 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/Fermatovo_%C4%8D%C3%ADslo" title="Fermatovo číslo – Czech" lang="cs" hreflang="cs" data-title="Fermatovo číslo" 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/Fermatprimtal" title="Fermatprimtal – Danish" lang="da" hreflang="da" data-title="Fermatprimtal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fermat-Zahl" title="Fermat-Zahl – German" lang="de" hreflang="de" data-title="Fermat-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%A6%CE%B5%CF%81%CE%BC%CE%AC" title="Αριθμός Φερμά – Greek" lang="el" hreflang="el" data-title="Αριθμός Φερμά" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – Spanish" lang="es" hreflang="es" data-title="Número de Fermat" 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/Nombro_de_Fermat" title="Nombro de Fermat – Esperanto" lang="eo" hreflang="eo" data-title="Nombro de Fermat" 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%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%81%D8%B1%D9%85%D8%A7" title="اعداد فرما – Persian" lang="fa" hreflang="fa" data-title="اعداد فرما" 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_Fermat" title="Nombre de Fermat – French" lang="fr" hreflang="fr" data-title="Nombre de Fermat" 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_de_Fermat" title="Número de Fermat – Galician" lang="gl" hreflang="gl" data-title="Número de Fermat" 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/%ED%8E%98%EB%A5%B4%EB%A7%88_%EC%88%98" title="페르마 수 – Korean" lang="ko" hreflang="ko" data-title="페르마 수" 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%96%D5%A5%D6%80%D5%B4%D5%A1%D5%B5%D5%AB_%D5%A9%D5%AB%D5%BE" title="Ֆերմայի թիվ – Armenian" lang="hy" hreflang="hy" data-title="Ֆերմայի թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Fermat" title="Numero di Fermat – Italian" lang="it" hreflang="it" data-title="Numero di Fermat" 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%A4%D7%A8%D7%9E%D7%94" title="מספר פרמה – Hebrew" lang="he" hreflang="he" data-title="מספר פרמה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ferma_skai%C4%8Dius" title="Ferma skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Ferma skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fermat-sz%C3%A1mok" title="Fermat-számok – Hungarian" lang="hu" hreflang="hu" data-title="Fermat-számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fermatgetal" title="Fermatgetal – Dutch" lang="nl" hreflang="nl" data-title="Fermatgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9E%E3%83%BC%E6%95%B0" title="フェルマー数 – Japanese" lang="ja" hreflang="ja" data-title="フェルマー数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fermat-tallene" title="Fermat-tallene – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fermat-tallene" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fermattal" title="Fermattal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fermattal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_%C3%ABd_Fermat" title="Nùmer ëd Fermat – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer ëd Fermat" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Fermata" title="Liczby Fermata – Polish" lang="pl" hreflang="pl" data-title="Liczby Fermata" 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/N%C3%BAmero_de_Fermat" title="Número de Fermat – Portuguese" lang="pt" hreflang="pt" data-title="Número de Fermat" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число Ферма – Russian" lang="ru" hreflang="ru" data-title="Число Ферма" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fermat_number" title="Fermat number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fermat number" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fermatovo_pra%C5%A1tevilo" title="Fermatovo praštevilo – Slovenian" lang="sl" hreflang="sl" data-title="Fermatovo praštevilo" 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_%D9%81%DB%8E%D8%B1%D9%85%D8%A7" title="ژمارەی فێرما – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی فێرما" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fermat%E2%80%99n_luku" title="Fermat’n luku – Finnish" lang="fi" hreflang="fi" data-title="Fermat’n luku" 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/Fermattal" title="Fermattal – Swedish" lang="sv" hreflang="sv" data-title="Fermattal" 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%83%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="ஃபெர்மா எண் – Tamil" lang="ta" hreflang="ta" data-title="ஃபெர்மா எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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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">Positive integer of the form (2^(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">Fermat prime</caption><tbody><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></td></tr><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">5</td></tr><tr><th scope="row" class="infobox-label">Conjectured <abbr title="number">no.</abbr> of terms</th><td class="infobox-data">5</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Subsequence" title="Subsequence">Subsequence</a> of</th><td class="infobox-data">Fermat 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/5_(number)" class="mw-redirect" title="5 (number)">5</a>, <a href="/wiki/17_(number)" title="17 (number)">17</a>, <a href="/wiki/257_(number)" title="257 (number)">257</a>, <a href="/wiki/65537_(number)" class="mw-redirect" title="65537 (number)">65537</a></td></tr><tr><th scope="row" class="infobox-label">Largest known term</th><td class="infobox-data">65537</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"><a rel="nofollow" class="external text" href="//oeis.org/A019434">A019434</a></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Fermat number</b>, named after <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> (1607–1665), the first known to have studied them, is a <a href="/wiki/Natural_number" title="Natural number">positive integer</a> of the form:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c168e5c8ec2740a7e6eab3f171c979166f1c96d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.643ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1,}"></span> where <i>n</i> is a <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> integer. The first few Fermat numbers are: <a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a>, <a href="/wiki/5_(number)" class="mw-redirect" title="5 (number)">5</a>, <a href="/wiki/17_(number)" title="17 (number)">17</a>, <a href="/wiki/257_(number)" title="257 (number)">257</a>, <a href="/wiki/65537_(number)" class="mw-redirect" title="65537 (number)">65537</a>, 4294967297, 18446744073709551617, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000215" class="extiw" title="oeis:A000215">A000215</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>If 2<sup><i>k</i></sup> + 1 is <a href="/wiki/Prime_number" title="Prime number">prime</a> and <span class="nowrap"><i>k</i> > 0</span>, then <i>k</i> itself must be a power of 2,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> so <span class="nowrap">2<sup><i>k</i></sup> + 1</span> is a Fermat number; such primes are called <b>Fermat primes</b>. As of 2023<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fermat_number&action=edit">[update]</a></sup>, the only known Fermat primes are <span class="nowrap"><i>F</i><sub>0</sub> = 3</span>, <span class="nowrap"><i>F</i><sub>1</sub> = 5</span>, <span class="nowrap"><i>F</i><sub>2</sub> = 17</span>, <span class="nowrap"><i>F</i><sub>3</sub> = 257</span>, and <span class="nowrap"><i>F</i><sub>4</sub> = 65537</span> (sequence <span class="nowrap external"><a href="//oeis.org/A019434" class="extiw" title="oeis:A019434">A019434</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=1" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Fermat numbers satisfy the following <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relations</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ff1c52a7a38d3afcce05117270671017d1b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.494ex; height:3.176ex;" alt="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"></span></dd></dl> <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 F_{n}=F_{0}\cdots F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42e25a368baa3cee4a5f1e1db0a00d38e197a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.674ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}"></span></dd></dl> <p>for <i>n</i> ≥ 1, </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 F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99e131e1d0c14da739b86e0e145409caae43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.212ex; height:3.343ex;" alt="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}"></span></dd></dl> <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 F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d330d839625d9bec0ae45d2b7fc591e2a62065e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.308ex; height:3.509ex;" alt="{\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}"></span></dd></dl> <p>for <span class="nowrap"><i>n</i> ≥ 2</span>. Each of these relations can be proved by <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. From the second equation, we can deduce <b>Goldbach's theorem</b> (named after <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a>): no two Fermat numbers <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">share a common integer factor greater than 1</a>. To see this, suppose that <span class="nowrap">0 ≤ <i>i</i> < <i>j</i></span> and <i>F</i><sub><i>i</i></sub> and <i>F</i><sub><i>j</i></sub> have a common factor <span class="nowrap"><i>a</i> > 1</span>. Then <i>a</i> divides both </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 F_{0}\cdots F_{j-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}\cdots F_{j-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc56e472eceb60b01dd528a3a2682d20be5275b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.551ex; height:2.843ex;" alt="{\displaystyle F_{0}\cdots F_{j-1}}"></span></dd></dl> <p>and <i>F</i><sub><i>j</i></sub>; hence <i>a</i> divides their difference, 2. Since <span class="nowrap"><i>a</i> > 1</span>, this forces <span class="nowrap"><i>a</i> = 2</span>. This is a <a href="/wiki/Contradiction" title="Contradiction">contradiction</a>, because each Fermat number is clearly odd. As a <a href="/wiki/Corollary" title="Corollary">corollary</a>, we obtain another proof of the <a href="/wiki/Infinity" title="Infinity">infinitude</a> of the prime numbers: for each <i>F</i><sub><i>n</i></sub>, choose a prime factor <i>p</i><sub><i>n</i></sub>; then the sequence {<i>p</i><sub><i>n</i></sub>} is an infinite sequence of distinct primes. </p> <div class="mw-heading mw-heading3"><h3 id="Further_properties">Further properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=2" title="Edit section: Further properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>No Fermat prime can be expressed as the difference of two <i>p</i>th powers, where <i>p</i> is an odd prime.</li> <li>With the exception of <i>F</i><sub>0</sub> and <i>F</i><sub>1</sub>, the last decimal digit of a Fermat number is 7.</li> <li>The <a href="/wiki/Sums_of_reciprocals" class="mw-redirect" title="Sums of reciprocals">sum of the reciprocals</a> of all the Fermat numbers (sequence <span class="nowrap external"><a href="//oeis.org/A051158" class="extiw" title="oeis:A051158">A051158</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>. (<a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a>, 1963)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Primality">Primality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=3" title="Edit section: Primality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who <a href="/wiki/Conjecture" title="Conjecture">conjectured</a> that all Fermat numbers are prime. Indeed, the first five Fermat numbers <i>F</i><sub>0</sub>, ..., <i>F</i><sub>4</sub> are easily shown to be prime. Fermat's conjecture was refuted by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1732 when he showed that </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 F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>4294967297</mn> <mo>=</mo> <mn>641</mn> <mo>×</mo> <mn>6700417.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b673f5ccb1c12d72056f32ac51dc742faa375031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.772ex; height:3.343ex;" alt="{\displaystyle F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.}"></span></dd></dl> <p>Euler proved that every factor of <i>F</i><sub><i>n</i></sub> must have the form <span class="nowrap"><i>k</i><span class="nowrap"> </span>2<sup><i>n</i>+1</sup> + 1</span> (later improved to <span class="nowrap"><i>k</i><span class="nowrap"> </span>2<sup><i>n</i>+2</sup> + 1</span> by <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Lucas</a>) for <span class="nowrap"><i>n</i> ≥ 2</span>. </p><p>That 641 is a factor of <i>F</i><sub>5</sub> can be deduced from the equalities 641 = 2<sup>7</sup> × 5 + 1 and 641 = 2<sup>4</sup> + 5<sup>4</sup>. It follows from the first equality that 2<sup>7</sup> × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2<sup>28</sup> × 5<sup>4</sup> ≡ 1 (mod 641). On the other hand, the second equality implies that 5<sup>4</sup> ≡ −2<sup>4</sup> (mod 641). These <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">congruences</a> imply that 2<sup>32</sup> ≡ −1 (mod 641). </p><p>Fermat was probably aware of the form of the factors later proved by Euler, so it seems curious that he failed to follow through on the straightforward calculation to find the factor.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> One common explanation is that Fermat made a computational mistake. </p><p>There are no other known Fermat primes <i>F</i><sub><i>n</i></sub> with <span class="nowrap"><i>n</i> > 4</span>, but little is known about Fermat numbers for large <i>n</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In fact, each of the following is an open problem: </p> <ul><li>Is <i>F</i><sub><i>n</i></sub> <a href="/wiki/Composite_number" title="Composite number">composite</a> <a href="/wiki/For_all" class="mw-redirect" title="For all">for all</a> <span class="nowrap"><i>n</i> > 4</span>?</li> <li>Are there infinitely many Fermat primes? (<a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a> 1844<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>)</li> <li>Are there infinitely many composite Fermat numbers?</li> <li>Does a Fermat number exist that is not <a href="/wiki/Square-free_number" class="mw-redirect" title="Square-free number">square-free</a>?</li></ul> <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=Fermat_number&action=edit">[update]</a></sup>, it is known that <i>F</i><sub><i>n</i></sub> is composite for <span class="nowrap">5 ≤ <i>n</i> ≤ 32</span>, although of these, complete factorizations of <i>F</i><sub><i>n</i></sub> are known only for <span class="nowrap">0 ≤ <i>n</i> ≤ 11</span>, and there are no known prime factors for <span class="nowrap"><i>n</i> = 20</span> and <span class="nowrap"><i>n</i> = 24</span>.<sup id="cite_ref-Keller_5-0" class="reference"><a href="#cite_note-Keller-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The largest Fermat number known to be composite is <i>F</i><sub>18233954</sub>, and its prime factor <span class="nowrap">7 × 2<sup>18233956</sup> + 1</span> was discovered in October 2020. </p> <div class="mw-heading mw-heading3"><h3 id="Heuristic_arguments">Heuristic arguments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=4" title="Edit section: Heuristic arguments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Heuristics suggest that <i>F</i><sub>4</sub> is the last Fermat prime. </p><p>The <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> implies that a random integer in a suitable interval around <i>N</i> is prime with probability 1<span class="nowrap"> </span>/<span class="nowrap"> </span>ln <i>N</i>. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that <i>F</i><sub>5</sub>, ..., <i>F</i><sub>32</sub> are composite, then the expected number of Fermat primes beyond <i>F</i><sub>4</sub> (or equivalently, beyond <i>F</i><sub>32</sub>) should be </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 \sum _{n\geq 33}{\frac {1}{\ln F_{n}}}<{\frac {1}{\ln 2}}\sum _{n\geq 33}{\frac {1}{\log _{2}(2^{2^{n}})}}={\frac {1}{\ln 2}}2^{-32}<3.36\times 10^{-10}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>33</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>33</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>32</mn> </mrow> </msup> <mo><</mo> <mn>3.36</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\geq 33}{\frac {1}{\ln F_{n}}}<{\frac {1}{\ln 2}}\sum _{n\geq 33}{\frac {1}{\log _{2}(2^{2^{n}})}}={\frac {1}{\ln 2}}2^{-32}<3.36\times 10^{-10}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee9ebfa9161b66e7a9a625263a648aa2384a99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.071ex; height:6.509ex;" alt="{\displaystyle \sum _{n\geq 33}{\frac {1}{\ln F_{n}}}<{\frac {1}{\ln 2}}\sum _{n\geq 33}{\frac {1}{\log _{2}(2^{2^{n}})}}={\frac {1}{\ln 2}}2^{-32}<3.36\times 10^{-10}.}"></span></dd></dl> <p>One may interpret this number as an upper bound for the probability that a Fermat prime beyond <i>F</i><sub>4</sub> exists. </p><p>This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. Boklan and <a href="/wiki/John_H._Conway" class="mw-redirect" title="John H. Conway">Conway</a> published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.<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>Anders Bjorn and <a href="/wiki/Hans_Riesel" title="Hans Riesel">Hans Riesel</a> estimated the number of square factors of Fermat numbers from <i>F</i><sub>5</sub> onward as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\geq 5}\sum _{k\geq 1}{\frac {1}{k(k2^{n}+1)\ln(k2^{n})}}<{\frac {\pi ^{2}}{6\ln 2}}\sum _{n\geq 5}{\frac {1}{n2^{n}}}\approx 0.02576;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.02576</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\geq 5}\sum _{k\geq 1}{\frac {1}{k(k2^{n}+1)\ln(k2^{n})}}<{\frac {\pi ^{2}}{6\ln 2}}\sum _{n\geq 5}{\frac {1}{n2^{n}}}\approx 0.02576;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad98784e203ea58f18e86db661047957bfc256bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.745ex; height:7.176ex;" alt="{\displaystyle \sum _{n\geq 5}\sum _{k\geq 1}{\frac {1}{k(k2^{n}+1)\ln(k2^{n})}}<{\frac {\pi ^{2}}{6\ln 2}}\sum _{n\geq 5}{\frac {1}{n2^{n}}}\approx 0.02576;}"></span></dd></dl> <p>in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{n}}+b^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{n}}+b^{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f53aa4dffb27beff2fb1199a8daead8ca553c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.106ex; height:2.843ex;" alt="{\displaystyle a^{2^{n}}+b^{2^{n}}}"></span> are very rare for large <i>n</i>.<sup id="cite_ref-bjorn_7-0" class="reference"><a href="#cite_note-bjorn-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_conditions">Equivalent conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=5" title="Edit section: Equivalent conditions"><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/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's test</a></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"></span> be the <i>n</i>th Fermat number. Pépin's test states that for <span class="nowrap"><i>n</i> > 0</span>, </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 F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> is prime if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{(F_{n}-1)/2}\equiv -1{\pmod {F_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{(F_{n}-1)/2}\equiv -1{\pmod {F_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7214b8f696e493461cb9ee82b8f815bf883c5e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.553ex; height:3.343ex;" alt="{\displaystyle 3^{(F_{n}-1)/2}\equiv -1{\pmod {F_{n}}}.}"></span></dd></dl> <p>The expression <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 3^{(F_{n}-1)/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{(F_{n}-1)/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c078731c97efa9348e3407f1f71526509a282d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.44ex; height:2.843ex;" alt="{\displaystyle 3^{(F_{n}-1)/2}}"></span> can be evaluated modulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> by <a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">repeated squaring</a>. This makes the test a fast <a href="/wiki/Polynomial-time" class="mw-redirect" title="Polynomial-time">polynomial-time</a> algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space. </p><p>There are some tests for numbers of the form <span class="nowrap"><i>k</i><span class="nowrap"> </span>2<sup><i>m</i></sup> + 1</span>, such as factors of Fermat numbers, for primality. </p> <dl><dd><b><a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a></b> (1878). Let <span class="nowrap"><i>N</i> = <i>k</i><span class="nowrap"> </span>2<sup><i>m</i></sup> + 1</span> with odd <span class="nowrap"><i>k</i> < 2<sup><i>m</i></sup></span>. If there is an integer <i>a</i> such that <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{(N-1)/2}\equiv -1{\pmod {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{(N-1)/2}\equiv -1{\pmod {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6adf05fa7b48a596fbd57f26ad82f0bb1243914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.762ex; height:3.343ex;" alt="{\displaystyle a^{(N-1)/2}\equiv -1{\pmod {N}}}"></span></dd></dl></dd> <dd>then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is prime. Conversely, if the above congruence does not hold, and in addition <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{N}}\right)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>N</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{N}}\right)=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff862fa65a1e21a7e0fed28567ae13d53c074db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.744ex; height:4.843ex;" alt="{\displaystyle \left({\frac {a}{N}}\right)=-1}"></span> (See <a href="/wiki/Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a>)</dd></dl></dd> <dd>then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is composite.</dd></dl> <p>If <span class="nowrap"><i>N</i> = <i>F</i><sub><i>n</i></sub> > 3</span>, then the above Jacobi symbol is always equal to −1 for <span class="nowrap"><i>a</i> = 3</span>, and this special case of Proth's theorem is known as <a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's test</a>. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for <span class="nowrap"><i>n</i> = 20</span> and 24. </p> <div class="mw-heading mw-heading2"><h2 id="Factorization">Factorization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=6" title="Edit section: Factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because of Fermat numbers' size, it is difficult to factorize or even to check primality. <a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's test</a> gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The <a href="/wiki/Elliptic_curve_method" class="mw-redirect" title="Elliptic curve method">elliptic curve method</a> is a fast method for finding small prime divisors of numbers. Distributed computing project <i>Fermatsearch</i> has found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span>, with <i>n</i> at least 2, is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times 2^{n+2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times 2^{n+2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52aae1ff75feaa9e60899e85fd17c567e01dda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.536ex; height:2.843ex;" alt="{\displaystyle k\times 2^{n+2}+1}"></span> (see <a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth number</a>), where <i>k</i> is a positive integer. By itself, this makes it easy to prove the primality of the known Fermat primes. </p><p>Factorizations of the first 12 Fermat numbers are: </p> <dl><dd><table> <tbody><tr valign="top"> <td><i>F</i><sub>0</sub></td> <td>=</td> <td>2<sup>1</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a> is prime</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>1</sub></td> <td>=</td> <td>2<sup>2</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/5_(number)" class="mw-redirect" title="5 (number)">5</a> is prime</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>2</sub></td> <td>=</td> <td>2<sup>4</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/17_(number)" title="17 (number)">17</a> is prime</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>3</sub></td> <td>=</td> <td>2<sup>8</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/257_(number)" title="257 (number)">257</a> is prime</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>4</sub></td> <td>=</td> <td>2<sup>16</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/65537_(number)" class="mw-redirect" title="65537 (number)">65,537</a> is the largest known Fermat prime</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>5</sub></td> <td>=</td> <td>2<sup>32</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>4,294,967,297</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>641 × 6,700,417 (fully factored 1732<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>) </td></tr> <tr valign="top"> <td><i>F</i><sub>6</sub></td> <td>=</td> <td>2<sup>64</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>18,446,744,073,709,551,617 (20 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>274,177 × 67,280,421,310,721 (14 digits) (fully factored 1855) </td></tr> <tr valign="top"> <td><i>F</i><sub>7</sub></td> <td>=</td> <td>2<sup>128</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits) (fully factored 1970) </td></tr> <tr valign="top"> <td><i>F</i><sub>8</sub></td> <td>=</td> <td>2<sup>256</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,<br />639,937 (78 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>1,238,926,361,552,897 (16 digits) × <br />93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 digits) (fully factored 1980) </td></tr> <tr valign="top"> <td><i>F</i><sub>9</sub></td> <td>=</td> <td>2<sup>512</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0<br />30,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,6<br />49,006,084,097 (155 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 digits) × <br />741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759,<br />504,705,008,092,818,711,693,940,737 (99 digits) (fully factored 1990) </td></tr> <tr valign="top"> <td><i>F</i><sub>10</sub></td> <td>=</td> <td>2<sup>1024</sup></td> <td>+</td> <td>1 </td> <td>=</td> <td>179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 digits) × <br />130,439,874,405,488,189,727,484...806,217,820,753,127,014,424,577 (252 digits) (fully factored 1995) </td></tr> <tr valign="top"> <td><i>F</i><sub>11</sub></td> <td>=</td> <td>2<sup>2048</sup></td> <td>+</td> <td>1 </td> <td>=</td> <td>32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 digits)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>319,489 × 974,849 × 167,988,556,341,760,475,137 (21 digits) × 3,560,841,906,445,833,920,513 (22 digits) × <br />173,462,447,179,147,555,430,258...491,382,441,723,306,598,834,177 (564 digits) (fully factored 1988) </td></tr> </tbody></table></dd></dl> <p>As of April 2023<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fermat_number&action=edit">[update]</a></sup>, only <i>F</i><sub>0</sub> to <i>F</i><sub>11</sub> have been completely <a href="/wiki/Integer_factorization" title="Integer factorization">factored</a>.<sup id="cite_ref-Keller_5-1" class="reference"><a href="#cite_note-Keller-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project Fermat Search is searching for new factors of Fermat numbers.<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> The set of all Fermat factors is <a href="//oeis.org/A050922" class="extiw" title="oeis:A050922">A050922</a> (or, sorted, <a href="//oeis.org/A023394" class="extiw" title="oeis:A023394">A023394</a>) in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>. </p><p>The following factors of Fermat numbers were known before 1950 (since then, digital computers have helped find more factors): </p> <table class="wikitable"> <tbody><tr> <th>Year </th> <th>Finder </th> <th>Fermat number </th> <th>Factor </th></tr> <tr> <td>1732 </td> <td><a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64647603e8358bf2b07099963d5ac2d8b75ee9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{5}}"></span> </td> <td><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 5\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{7}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d3e7e954a01c1d2f1b8e5a117b566434f54ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.061ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{7}+1}"></span> </td></tr> <tr> <td>1732 </td> <td>Euler </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64647603e8358bf2b07099963d5ac2d8b75ee9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{5}}"></span> (fully factored) </td> <td><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 52347\cdot 2^{7}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>52347</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 52347\cdot 2^{7}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e28bcad1647d7ddca3c12b69d033dfc51a28d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.711ex; height:2.843ex;" alt="{\displaystyle 52347\cdot 2^{7}+1}"></span> </td></tr> <tr> <td>1855 </td> <td><a href="/wiki/Thomas_Clausen_(mathematician)" title="Thomas Clausen (mathematician)">Clausen</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f760637659fa525945b3fdb906c673089642ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{6}}"></span> </td> <td><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 1071\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1071</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1071\cdot 2^{8}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1a689078f22c1298e81ad2aff61c1858170dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle 1071\cdot 2^{8}+1}"></span> </td></tr> <tr> <td>1855 </td> <td>Clausen </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f760637659fa525945b3fdb906c673089642ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{6}}"></span> (fully factored) </td> <td><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 262814145745\cdot 2^{8}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>262814145745</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 262814145745\cdot 2^{8}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e867c688a29ba92d2c6bc5377132cf19acb21d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.848ex; height:2.843ex;" alt="{\displaystyle 262814145745\cdot 2^{8}+1}"></span> </td></tr> <tr> <td>1877 </td> <td><a href="/wiki/Ivan_Pervushin" title="Ivan Pervushin">Pervushin</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35bd8bf2dd739f74f6c888ae839d364d7ca8cf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{12}}"></span> </td> <td><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 7\cdot 2^{14}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7\cdot 2^{14}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07da03276872015654c82e79f49da89e3765fb51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.883ex; height:2.843ex;" alt="{\displaystyle 7\cdot 2^{14}+1}"></span> </td></tr> <tr> <td>1878 </td> <td>Pervushin </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{23}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c260adffae2bd1576ed9ac412679f9e276066e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{23}}"></span> </td> <td><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 5\cdot 2^{25}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{25}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93383aab93f8a6b182c87e80826b524783e40bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.883ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{25}+1}"></span> </td></tr> <tr> <td>1886 </td> <td><a href="/w/index.php?title=Paul_Peter_Heinrich_Seelhoff&action=edit&redlink=1" class="new" title="Paul Peter Heinrich Seelhoff (page does not exist)">Seelhoff</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{36}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{36}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36984931f009d93c3d766103bc97b9dc8a6c32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{36}}"></span> </td> <td><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 5\cdot 2^{39}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>39</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{39}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6688ec0248d6f373d736da9dc7ece06126ce87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.883ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{39}+1}"></span> </td></tr> <tr> <td>1899 </td> <td><a href="/wiki/Allan_Joseph_Champneys_Cunningham" class="mw-redirect" title="Allan Joseph Champneys Cunningham">Cunningham</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{11}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e69bd87c448e9e231b992ff374a23d8aa110c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{11}}"></span> </td> <td><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 39\cdot 2^{13}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>39</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 39\cdot 2^{13}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d52084307a3b515662ba2f2d59a056bcbe31743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.046ex; height:2.843ex;" alt="{\displaystyle 39\cdot 2^{13}+1}"></span> </td></tr> <tr> <td>1899 </td> <td>Cunningham </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{11}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e69bd87c448e9e231b992ff374a23d8aa110c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{11}}"></span> </td> <td><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 119\cdot 2^{13}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>119</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 119\cdot 2^{13}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68370c63df22f77322e1c27ad4c6f377eae3606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.208ex; height:2.843ex;" alt="{\displaystyle 119\cdot 2^{13}+1}"></span> </td></tr> <tr> <td>1903 </td> <td><a href="/w/index.php?title=Alfred_Western&action=edit&redlink=1" class="new" title="Alfred Western (page does not exist)">Western</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a4659f9c70614686b3d4df437538cb048d70dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{9}}"></span> </td> <td><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 37\cdot 2^{16}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>37</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 37\cdot 2^{16}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856546f1f872fc9ed4ef4fa1f4b15985ab4cab70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.046ex; height:2.843ex;" alt="{\displaystyle 37\cdot 2^{16}+1}"></span> </td></tr> <tr> <td>1903 </td> <td>Western </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35bd8bf2dd739f74f6c888ae839d364d7ca8cf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{12}}"></span> </td> <td><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 397\cdot 2^{16}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>397</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 397\cdot 2^{16}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8a284827c2b6c4e877742c5ad5417d874ceab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.208ex; height:2.843ex;" alt="{\displaystyle 397\cdot 2^{16}+1}"></span> </td></tr> <tr> <td>1903 </td> <td>Western </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35bd8bf2dd739f74f6c888ae839d364d7ca8cf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{12}}"></span> </td> <td><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 973\cdot 2^{16}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>973</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 973\cdot 2^{16}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4960cb8d88fe338ca21fa43b6f94046684d45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.208ex; height:2.843ex;" alt="{\displaystyle 973\cdot 2^{16}+1}"></span> </td></tr> <tr> <td>1903 </td> <td>Western </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{18}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{18}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ede394dd18c605447ac155c9f6464d51a88ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{18}}"></span> </td> <td><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 13\cdot 2^{20}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13\cdot 2^{20}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090c5d5b6a9fc8268805178bbcbe482246bf8d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.046ex; height:2.843ex;" alt="{\displaystyle 13\cdot 2^{20}+1}"></span> </td></tr> <tr> <td>1903 </td> <td><a href="/wiki/James_Cullen_(mathematician)" title="James Cullen (mathematician)">Cullen</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{38}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>38</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{38}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3100232f2f2ed4d64a1089af4e2a825a4b10a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{38}}"></span> </td> <td><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 3\cdot 2^{41}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{41}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197580c18ae9186896c883ff0f98d711fbb5de1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.883ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{41}+1}"></span> </td></tr> <tr> <td>1906 </td> <td><a href="/w/index.php?title=James_C._Morehead&action=edit&redlink=1" class="new" title="James C. Morehead (page does not exist)">Morehead</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{73}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>73</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{73}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97bca85afd01503cef2946063289025ee1977c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{73}}"></span> </td> <td><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 5\cdot 2^{75}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>75</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2^{75}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f98c6714e2529e8be4b1d5b5313718751394eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.883ex; height:2.843ex;" alt="{\displaystyle 5\cdot 2^{75}+1}"></span> </td></tr> <tr> <td>1925 </td> <td><a href="/wiki/Maurice_Kraitchik" title="Maurice Kraitchik">Kraitchik</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{15}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34de01ce00b6f81d7f42bd501e436fbfd99ca21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{15}}"></span> </td> <td><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 579\cdot 2^{21}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>579</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 579\cdot 2^{21}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a34b6de3c6d8798dd95d01271f336b8061786992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.208ex; height:2.843ex;" alt="{\displaystyle 579\cdot 2^{21}+1}"></span> </td></tr> </tbody></table> <p>As of July 2023<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fermat_number&action=edit">[update]</a></sup>, 368 prime factors of Fermat numbers are known, and 324 Fermat numbers are known to be composite.<sup id="cite_ref-Keller_5-2" class="reference"><a href="#cite_note-Keller-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Several new Fermat factors are found each year.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Pseudoprimes_and_Fermat_numbers">Pseudoprimes and Fermat numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=7" title="Edit section: Pseudoprimes and Fermat numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like <a href="/wiki/Composite_number" title="Composite number">composite numbers</a> of the form 2<sup><i>p</i></sup> − 1, every composite Fermat number is a <a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">strong pseudoprime</a> to base 2. This is because all strong pseudoprimes to base 2 are also <a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprimes</a> – i.e., </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 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa84381b4d951079e94a544ef8dcb5f67f97b437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.175ex; height:3.176ex;" alt="{\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}"></span></dd></dl> <p>for all Fermat numbers.<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> </p><p>In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a}F_{b}\dots F_{s},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a}F_{b}\dots F_{s},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add1888e055fcd5d34f15599cc6c071ec39dc64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.671ex; height:2.509ex;" alt="{\displaystyle F_{a}F_{b}\dots F_{s},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b>\dots >s>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>></mo> <mo>⋯</mo> <mo>></mo> <mi>s</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b>\dots >s>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a27158d4eda33db27b55b5125be41bd829e673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.597ex; height:2.176ex;" alt="{\displaystyle a>b>\dots >s>1}"></span> will be a Fermat pseudoprime to base 2 if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{s}>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{s}>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999f69741b2ff85d9d73bbc6df58e912a12bbdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle 2^{s}>a}"></span>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Other_theorems_about_Fermat_numbers">Other theorems about Fermat numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=8" title="Edit section: Other theorems about Fermat numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Lemma.</strong><span class="theoreme-tiret"> — </span>If <i>n</i> is a positive integer, </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28935739b4a233d60f327c2396df91c28791ea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.026ex; height:7.509ex;" alt="{\displaystyle a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.}"></span></dd></dl></dd></dl> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p><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 {\begin{aligned}(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}&=\sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum _{k=0}^{n-1}a^{k}b^{n-k}\\&=a^{n}+\sum _{k=1}^{n-1}a^{k}b^{n-k}-\sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}\\&=a^{n}-b^{n}\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> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}&=\sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum _{k=0}^{n-1}a^{k}b^{n-k}\\&=a^{n}+\sum _{k=1}^{n-1}a^{k}b^{n-k}-\sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}\\&=a^{n}-b^{n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32b9981f174f1fa512da48acd3610f7356909f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:57.635ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}&=\sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum _{k=0}^{n-1}a^{k}b^{n-k}\\&=a^{n}+\sum _{k=1}^{n-1}a^{k}b^{n-k}-\sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}\\&=a^{n}-b^{n}\end{aligned}}}"></span> </p> </div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"></span> is an odd prime, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is a power of 2. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is a positive integer but not a power of 2, it must have an odd prime factor <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 s>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c013be86f6c987991f8bac51367391f9f5549411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s>2}"></span>, and we may write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=rs}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>r</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=rs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b82f85aa0d76d2f7045ff91d3128e12a8e948b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.449ex; height:2.176ex;" alt="{\displaystyle k=rs}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq r<k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq r<k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3f18d9f9047cc3beb1675fded776ace942dbe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.619ex; height:2.343ex;" alt="{\displaystyle 1\leq r<k}"></span>. </p><p>By the preceding lemma, for positive integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a-b)\mid (a^{m}-b^{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-b)\mid (a^{m}-b^{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b5a2030406c936a69f3c60076682705a07c44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.041ex; height:2.843ex;" alt="{\displaystyle (a-b)\mid (a^{m}-b^{m})}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mid }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∣</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mid }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7b2136e276c4aec285a6c40b91180c16432b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.647ex; height:2.843ex;" alt="{\displaystyle \mid }"></span> means "evenly divides". Substituting <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=2^{r},b=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2^{r},b=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2292990b62992a10229acd8c025670fe1d77902d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.565ex; height:2.676ex;" alt="{\displaystyle a=2^{r},b=-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 m=s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3870bcccd4dbc94bb45a403c5d4e0d4dcb207b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.229ex; height:1.676ex;" alt="{\displaystyle m=s}"></span> and using that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is odd, </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 (2^{r}+1)\mid (2^{rs}+1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{r}+1)\mid (2^{rs}+1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed276cebac63e8fb36b282a8f2d596feeec765a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.252ex; height:2.843ex;" alt="{\displaystyle (2^{r}+1)\mid (2^{rs}+1),}"></span></dd></dl> <p>and thus </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 (2^{r}+1)\mid (2^{k}+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{r}+1)\mid (2^{k}+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d53208a022de5c55fdecb6f98f20013ae54d597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.596ex; height:3.176ex;" alt="{\displaystyle (2^{r}+1)\mid (2^{k}+1).}"></span></dd></dl> <p>Because <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 1<2^{r}+1<2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<2^{r}+1<2^{k}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a84b0f91636f96d3a013d817ff4bfb66522d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.752ex; height:2.843ex;" alt="{\displaystyle 1<2^{r}+1<2^{k}+1}"></span>, it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37106849bec61399f55dd477a364a2b2bfbb8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle 2^{k}+1}"></span> is not prime. Therefore, by <a href="/wiki/Contraposition" title="Contraposition">contraposition</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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> must be a power of 2. </p> </div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span> A Fermat prime cannot be a <a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich prime</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>We show if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2^{m}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2^{m}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872b67adaa665953ba1f002627d1927ccef85cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.198ex; height:2.676ex;" alt="{\displaystyle p=2^{m}+1}"></span> is a Fermat prime (and hence by the above, <i>m</i> is a power of 2), then the congruence <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}\equiv 1{\bmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p-1}\equiv 1{\bmod {p^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a716856b5e0551de0b81ab83f460e066e61deb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.488ex; height:3.009ex;" alt="{\displaystyle 2^{p-1}\equiv 1{\bmod {p^{2}}}}"></span> does not hold. </p><p>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 2m|p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m|p-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12815ece13b1f4e157aa0b65e8e184623e6804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.022ex; height:2.843ex;" alt="{\displaystyle 2m|p-1}"></span> we may write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1=2m\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-1=2m\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e75ca894ea379db8d51ab8e0e5d461fff5500be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.918ex; height:2.509ex;" alt="{\displaystyle p-1=2m\lambda }"></span>. If the given congruence holds, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}|2^{2m\lambda }-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> <mi>λ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}|2^{2m\lambda }-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a31c1ac1e9782b790f5d976f62d5469cfa724ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.581ex; height:3.176ex;" alt="{\displaystyle p^{2}|2^{2m\lambda }-1}"></span>, and therefore </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 0\equiv {\frac {2^{2m\lambda }-1}{2^{m}+1}}=(2^{m}-1)\left(1+2^{2m}+2^{4m}+\cdots +2^{2(\lambda -1)m}\right)\equiv -2\lambda {\pmod {2^{m}+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> <mi>λ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mo>−</mo> <mn>2</mn> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\equiv {\frac {2^{2m\lambda }-1}{2^{m}+1}}=(2^{m}-1)\left(1+2^{2m}+2^{4m}+\cdots +2^{2(\lambda -1)m}\right)\equiv -2\lambda {\pmod {2^{m}+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db311e7f0a02e3a2def62e5478bbfccd757e8d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:83.788ex; height:6.009ex;" alt="{\displaystyle 0\equiv {\frac {2^{2m\lambda }-1}{2^{m}+1}}=(2^{m}-1)\left(1+2^{2m}+2^{4m}+\cdots +2^{2(\lambda -1)m}\right)\equiv -2\lambda {\pmod {2^{m}+1}}.}"></span></dd></dl> <p>Hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m}+1|2\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}+1|2\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ecf1766862e36d8079702e568a6601267008f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.005ex; height:2.843ex;" alt="{\displaystyle 2^{m}+1|2\lambda }"></span>, and therefore <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\lambda \geq 2^{m}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>λ</mi> <mo>≥</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\lambda \geq 2^{m}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9118b8185106c2568a10991020ada6ddffa97398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.456ex; height:2.509ex;" alt="{\displaystyle 2\lambda \geq 2^{m}+1}"></span>. This leads to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1\geq m(2^{m}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>≥</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-1\geq m(2^{m}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284ebdc04d3289c931366dc518f26c9e4b09c4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.05ex; height:2.843ex;" alt="{\displaystyle p-1\geq m(2^{m}+1)}"></span>, which is impossible 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 m\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca2e437e89ef4565a87f1a6d90ed37eef1d8ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 2}"></span>. </p> </div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong> <span class="theorem-note">(<a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>)</span><span class="theoreme-tiret"> — </span> Any prime divisor <i>p</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"></span> is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k2^{n+2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k2^{n+2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8be628c5669f7f93180fb7d3f6fe20a63a6c751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.696ex; height:2.843ex;" alt="{\displaystyle k2^{n+2}+1}"></span> whenever <span class="nowrap"><i>n</i> > 1</span>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong><i>Sketch of proof</i></strong> <p>Let <i>G</i><sub><i>p</i></sub> denote the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">group of non-zero integers modulo <i>p</i> under multiplication</a>, which has order <span class="nowrap"><i>p</i> − 1</span>. Notice that 2 (strictly speaking, its image modulo <i>p</i>) has multiplicative order equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec7da881b9da391c503b68a69a46ba3c43e189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n+1}}"></span> in <i>G</i><sub><i>p</i></sub> (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 2^{2^{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a4e2f66425dc5d47e9c23935e2b99e0d29f6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.887ex; height:3.009ex;" alt="{\displaystyle 2^{2^{n+1}}}"></span> is the square of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc9417763ad5d68870290ddaa2ca025ffdaf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{2^{n}}}"></span> which is −1 modulo <i>F</i><sub><i>n</i></sub>), so that, by <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>, <span class="nowrap"><i>p</i> − 1</span> is divisible by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec7da881b9da391c503b68a69a46ba3c43e189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n+1}}"></span> and <i>p</i> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k2^{n+1}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k2^{n+1}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09688c41e536b741eca232e49c0f379c83fa2f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.696ex; height:2.843ex;" alt="{\displaystyle k2^{n+1}+1}"></span> for some integer <i>k</i>, as <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> knew. Édouard Lucas went further. Since <span class="nowrap"><i>n</i> > 1</span>, the prime <i>p</i> above is congruent to 1 modulo 8. Hence (as was known to <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>), 2 is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> modulo <i>p</i>, that is, there is integer <i>a</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p|a^{2}-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p|a^{2}-2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c383d975d135d741a501784e0e6a51703de582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.839ex; height:3.176ex;" alt="{\displaystyle p|a^{2}-2.}"></span> Then the image of <i>a</i> has order <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^{n+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e77677f73b1374495dccc5e0e8292e593460471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n+2}}"></span> in the group <i>G</i><sub><i>p</i></sub> and (using Lagrange's theorem again), <span class="nowrap"><i>p</i> − 1</span> is divisible by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e77677f73b1374495dccc5e0e8292e593460471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n+2}}"></span> and <i>p</i> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s2^{n+2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s2^{n+2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc77b3fa7cb9dccb6a056d221571dc6ef654723f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.575ex; height:2.843ex;" alt="{\displaystyle s2^{n+2}+1}"></span> for some integer <i>s</i>. </p><p>In fact, it can be seen directly that 2 is a quadratic residue modulo <i>p</i>, since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1+2^{2^{n-1}}\right)^{2}\equiv 2^{1+2^{n-1}}{\pmod {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+2^{2^{n-1}}\right)^{2}\equiv 2^{1+2^{n-1}}{\pmod {p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18352c8db89ed4eaddecc91fad10032b489db409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.306ex; height:5.176ex;" alt="{\displaystyle \left(1+2^{2^{n-1}}\right)^{2}\equiv 2^{1+2^{n-1}}{\pmod {p}}.}"></span></dd></dl> <p>Since an odd power of 2 is a quadratic residue modulo <i>p</i>, so is 2 itself. </p> </div> </div> <p>A Fermat number cannot be a perfect number or part of a pair of <a href="/wiki/Amicable_numbers" title="Amicable numbers">amicable numbers</a>. (<a href="#CITEREFLuca2000">Luca 2000</a>) </p><p>The series of reciprocals of all prime divisors of Fermat numbers is <a href="/wiki/Convergent_series" title="Convergent series">convergent</a>. (<a href="#CITEREFKřížekLucaSomer2002">Křížek, Luca & Somer 2002</a>) </p><p>If <span class="nowrap"><i>n</i><sup><i>n</i></sup> + 1</span> is prime, there exists an integer <i>m</i> such that <span class="nowrap"><i>n</i> = 2<sup>2<sup><i>m</i></sup></sup></span>. The equation <span class="nowrap"><i>n</i><sup><i>n</i></sup> + 1 = <i>F</i><sub>(2<sup><i>m</i></sup>+<i>m</i>)</sub></span> holds in that case.<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><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>Let the largest prime factor of the Fermat number <i>F</i><sub><i>n</i></sub> be <i>P</i>(<i>F</i><sub><i>n</i></sub>). Then, </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 P(F_{n})\geq 2^{n+2}(4n+9)+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(F_{n})\geq 2^{n+2}(4n+9)+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec30b46249e7852632e5fd28583e21d459ecd428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.867ex; height:3.176ex;" alt="{\displaystyle P(F_{n})\geq 2^{n+2}(4n+9)+1.}"></span> (<a href="#CITEREFGrytczukLucaWójtowicz2001">Grytczuk, Luca & Wójtowicz 2001</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_constructible_polygons">Relationship to constructible polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=9" title="Edit section: Relationship to constructible polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Constructible_polygon_set.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Constructible_polygon_set.svg/300px-Constructible_polygon_set.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Constructible_polygon_set.svg/450px-Constructible_polygon_set.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Constructible_polygon_set.svg/600px-Constructible_polygon_set.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Constructible_polygon" title="Constructible polygon">Constructible polygon</a></div> <p><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> developed the theory of <a href="/wiki/Gaussian_period" title="Gaussian period">Gaussian periods</a> in his <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i> and formulated a <a href="/wiki/Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient condition</a> for the constructibility of regular polygons. Gauss stated that this condition was also <a href="/wiki/Necessary_condition" class="mw-redirect" title="Necessary condition">necessary</a>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> but never published a proof. <a href="/wiki/Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> gave a full proof of necessity in 1837. The result is known as the <b>Gauss–Wantzel theorem</b>: </p> <dl><dd>An <i>n</i>-sided regular polygon can be constructed with <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a> if and only if <i>n</i> is either a power of 2 or the product of a power of 2 and distinct Fermat primes: in other words, if and only if <i>n</i> is of the form <span class="nowrap"><i>n</i> = 2<sup><i>k</i></sup></span> or <span class="nowrap"><i>n</i> = 2<sup><i>k</i></sup><i>p</i><sub>1</sub><i>p</i><sub>2</sub>...<i>p</i><sub><i>s</i></sub></span>, where <i>k</i>, <i>s</i> are nonnegative integers and the <i>p</i><sub><i>i</i></sub> are distinct Fermat primes.</dd></dl> <p>A positive integer <i>n</i> is of the above form if and only if its <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">totient</a> <i>φ</i>(<i>n</i>) is a power of 2. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_of_Fermat_numbers">Applications of Fermat numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=10" title="Edit section: Applications of Fermat numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Pseudorandom_number_generation">Pseudorandom number generation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=11" title="Edit section: Pseudorandom number generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1, ..., <i>N</i>, where <i>N</i> is a power of 2. The most common method used is to take any seed value between 1 and <span class="nowrap"><i>P</i> − 1</span>, where <i>P</i> is a Fermat prime. Now multiply this by a number <i>A</i>, which is greater than the <a href="/wiki/Square_root" title="Square root">square root</a> of <i>P</i> and is a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive root</a> modulo <i>P</i> (i.e., it is not a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a>). Then take the result modulo <i>P</i>. The result is the new value for the RNG. </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 V_{j+1}=(A\times V_{j}){\bmod {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>×</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{j+1}=(A\times V_{j}){\bmod {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96fcd0968a926725f5ffc308b1b19572af5ad664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.548ex; height:3.009ex;" alt="{\displaystyle V_{j+1}=(A\times V_{j}){\bmod {P}}}"></span> (see <a href="/wiki/Linear_congruential_generator" title="Linear congruential generator">linear congruential generator</a>)</dd></dl> <p>This is useful in computer science, since most data structures have members with 2<sup><i>X</i></sup> possible values. For example, a byte has 256 (2<sup>8</sup>) possible values (0–255). Therefore, to fill a byte or bytes with random values, a random number generator that produces values 1–256 can be used, the byte taking the output value −1. Very large Fermat primes are of particular interest in data encryption for this reason. This method produces only <a href="/wiki/Pseudorandom" class="mw-redirect" title="Pseudorandom">pseudorandom</a> values, as after <span class="nowrap"><i>P</i> − 1</span> repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than <span class="nowrap"><i>P</i> − 1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalized_Fermat_numbers">Generalized Fermat numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=12" title="Edit section: Generalized Fermat numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Numbers of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{\overset {n}{}}}\!\!+b^{2^{\overset {n}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mi>n</mi> </mover> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mi>n</mi> </mover> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{\overset {n}{}}}\!\!+b^{2^{\overset {n}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df0e1623aa02606c5ab251c9f156d510d18344e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.332ex; height:3.343ex;" alt="{\displaystyle a^{2^{\overset {n}{}}}\!\!+b^{2^{\overset {n}{}}}}"></span> with <i>a</i>, <i>b</i> any <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> integers, <span class="nowrap"><i>a</i> > <i>b</i> > 0</span>, are called <b>generalized Fermat numbers</b>. An odd prime <i>p</i> is a generalized Fermat number if and only if <i>p</i> is congruent to <a href="/wiki/Pythagorean_prime" title="Pythagorean prime">1 (mod 4)</a>. (Here we consider only the case <span class="nowrap"><i>n</i> > 0</span>, so <span class="nowrap">3 = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{0}}\!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{0}}\!+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cbb922e5788cd5fd9216e3af384deca390078c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.664ex; height:3.176ex;" alt="{\displaystyle 2^{2^{0}}\!+1}"></span></span> is not a counterexample.) </p><p>An example of a <a href="/wiki/Probable_prime" title="Probable prime">probable prime</a> of this form is 1215<sup>131072</sup> + 242<sup>131072</sup> (found by Kellen Shenton).<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> </p><p>By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mi>n</mi> </mover> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8994207f00c1fce41b50af047951914123a2083a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.478ex; height:3.343ex;" alt="{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}"></span> as <i>F</i><sub><i>n</i></sub>(<i>a</i>). In this notation, for instance, the number 100,000,001 would be written as <i>F</i><sub>3</sub>(10). In the following we shall restrict ourselves to primes of this form, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mi>n</mi> </mover> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8994207f00c1fce41b50af047951914123a2083a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.478ex; height:3.343ex;" alt="{\displaystyle a^{2^{\overset {n}{}}}\!\!+1}"></span>, such primes are called "Fermat primes base <i>a</i>". Of course, these primes exist only if <i>a</i> is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a>. </p><p>If we require <span class="nowrap"><i>n</i> > 0</span>, then <a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's fourth problem</a> asks if there are infinitely many generalized Fermat primes <i>F<sub>n</sub></i>(<i>a</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_Fermat_primes_of_the_form_Fn(a)"><span id="Generalized_Fermat_primes_of_the_form_Fn.28a.29"></span>Generalized Fermat primes of the form F<sub>n</sub>(<i>a</i>)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=13" title="Edit section: Generalized Fermat primes of the form Fn(a)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because of the ease of proving their primality, generalized Fermat primes have become in recent years a topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes. </p><p>Generalized Fermat numbers can be prime only for even <span class="texhtml mvar" style="font-style:italic;">a</span>, because if <span class="texhtml mvar" style="font-style:italic;">a</span> is odd then every generalized Fermat number will be divisible by 2. The smallest prime number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b13dc8b113121cdaf76a723a61aa4f8be1468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>4}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}(30)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>30</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}(30)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81aa7021973af26d06bbce838c0436487e03787c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.683ex; height:2.843ex;" alt="{\displaystyle F_{5}(30)}"></span>, or 30<sup>32</sup> + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base <i>a</i> (for odd <i>a</i>) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a^{2^{n}}\!+1}{2}}}"> <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"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a^{2^{n}}\!+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22175f600177f9af549f5d5483de5e767cc4e741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.701ex; height:5.676ex;" alt="{\displaystyle {\frac {a^{2^{n}}\!+1}{2}}}"></span>, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base. </p><p>In this list, the generalized Fermat numbers (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span>) to an even <span class="texhtml mvar" style="font-style:italic;">a</span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{n}}\!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{n}}\!+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f33b763749f887bda23003c16d58ced23dff6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.865ex; height:2.843ex;" alt="{\displaystyle a^{2^{n}}\!+1}"></span>, for odd <span class="texhtml mvar" style="font-style:italic;">a</span>, they are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a^{2^{n}}\!\!+1}{2}}}"> <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"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a^{2^{n}}\!\!+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac701bd6e8db2dca8dbb7a6588e9110afede6da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.314ex; height:5.676ex;" alt="{\displaystyle {\frac {a^{2^{n}}\!\!+1}{2}}}"></span>. If <span class="texhtml mvar" style="font-style:italic;">a</span> is a perfect power with an odd exponent (sequence <span class="nowrap external"><a href="//oeis.org/A070265" class="extiw" title="oeis:A070265">A070265</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), then all generalized Fermat number can be algebraic factored, so they cannot be prime. </p><p>See<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><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> for even bases up to 1000, and<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> for odd bases. For the smallest number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime, see <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/A253242" class="extiw" title="oeis:A253242">A253242</a></span>. </p> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th>numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><br />such that<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th>numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><br />such that<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th>numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><br />such that<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th>numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><br />such that<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime </th></tr> <tr> <td>2 </td> <td>0, 1, 2, 3, 4, ... </td> <td>18 </td> <td>0, ... </td> <td>34 </td> <td>2, ... </td> <td>50 </td> <td>... </td></tr> <tr> <td>3 </td> <td>0, 1, 2, 4, 5, 6, ... </td> <td>19 </td> <td>1, ... </td> <td>35 </td> <td>1, 2, 6, ... </td> <td>51 </td> <td>1, 3, 6, ... </td></tr> <tr> <td>4 </td> <td>0, 1, 2, 3, ... </td> <td>20 </td> <td>1, 2, ... </td> <td>36 </td> <td>0, 1, ... </td> <td>52 </td> <td>0, ... </td></tr> <tr> <td>5 </td> <td>0, 1, 2, ... </td> <td>21 </td> <td>0, 2, 5, ... </td> <td>37 </td> <td>0, ... </td> <td>53 </td> <td>3, ... </td></tr> <tr> <td>6 </td> <td>0, 1, 2, ... </td> <td>22 </td> <td>0, ... </td> <td>38 </td> <td>... </td> <td>54 </td> <td>1, 2, 5, ... </td></tr> <tr> <td>7 </td> <td>2, ... </td> <td>23 </td> <td>2, ... </td> <td>39 </td> <td>1, 2, ... </td> <td>55 </td> <td>... </td></tr> <tr> <td>8 </td> <td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">(none) </td> <td>24 </td> <td>1, 2, ... </td> <td>40 </td> <td>0, 1, ... </td> <td>56 </td> <td>1, 2, ... </td></tr> <tr> <td>9 </td> <td>0, 1, 3, 4, 5, ... </td> <td>25 </td> <td>0, 1, ... </td> <td>41 </td> <td>4, ... </td> <td>57 </td> <td>0, 2, ... </td></tr> <tr> <td>10 </td> <td>0, 1, ... </td> <td>26 </td> <td>1, ... </td> <td>42 </td> <td>0, ... </td> <td>58 </td> <td>0, ... </td></tr> <tr> <td>11 </td> <td>1, 2, ... </td> <td>27 </td> <td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">(none) </td> <td>43 </td> <td>3, ... </td> <td>59 </td> <td>1, ... </td></tr> <tr> <td>12 </td> <td>0, ... </td> <td>28 </td> <td>0, 2, ... </td> <td>44 </td> <td>4, ... </td> <td>60 </td> <td>0, ... </td></tr> <tr> <td>13 </td> <td>0, 2, 3, ... </td> <td>29 </td> <td>1, 2, 4, ... </td> <td>45 </td> <td>0, 1, ... </td> <td>61 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>14 </td> <td>1, ... </td> <td>30 </td> <td>0, 5, ... </td> <td>46 </td> <td>0, 2, 9, ... </td> <td>62 </td> <td>... </td></tr> <tr> <td>15 </td> <td>1, ... </td> <td>31 </td> <td>... </td> <td>47 </td> <td>3, ... </td> <td>63 </td> <td>... </td></tr> <tr> <td>16 </td> <td>0, 1, 2, ... </td> <td>32 </td> <td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">(none) </td> <td>48 </td> <td>2, ... </td> <td>64 </td> <td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">(none) </td></tr> <tr> <td>17 </td> <td>2, ... </td> <td>33 </td> <td>0, 3, ... </td> <td>49 </td> <td>1, ... </td> <td>65 </td> <td>1, 2, 5, ... </td></tr></tbody></table> <p>For the smallest even base <span class="texhtml mvar" style="font-style:italic;">a</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime, see <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/A056993" class="extiw" title="oeis:A056993">A056993</a></span>. </p> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> </th> <th>bases <span class="texhtml mvar" style="font-style:italic;">a</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> is prime (only consider even <span class="texhtml mvar" style="font-style:italic;">a</span>) </th> <th><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence </th></tr> <tr> <td>0 </td> <td>2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, ... </td> <td><a href="//oeis.org/A006093" class="extiw" title="oeis:A006093">A006093</a> </td></tr> <tr> <td>1 </td> <td>2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, ... </td> <td><a href="//oeis.org/A005574" class="extiw" title="oeis:A005574">A005574</a> </td></tr> <tr> <td>2 </td> <td>2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, ... </td> <td><a href="//oeis.org/A000068" class="extiw" title="oeis:A000068">A000068</a> </td></tr> <tr> <td>3 </td> <td>2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, ... </td> <td><a href="//oeis.org/A006314" class="extiw" title="oeis:A006314">A006314</a> </td></tr> <tr> <td>4 </td> <td>2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, ... </td> <td><a href="//oeis.org/A006313" class="extiw" title="oeis:A006313">A006313</a> </td></tr> <tr> <td>5 </td> <td>30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, ... </td> <td><a href="//oeis.org/A006315" class="extiw" title="oeis:A006315">A006315</a> </td></tr> <tr> <td>6 </td> <td>102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, ... </td> <td><a href="//oeis.org/A006316" class="extiw" title="oeis:A006316">A006316</a> </td></tr> <tr> <td>7 </td> <td>120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, ... </td> <td><a href="//oeis.org/A056994" class="extiw" title="oeis:A056994">A056994</a> </td></tr> <tr> <td>8 </td> <td>278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, ... </td> <td><a href="//oeis.org/A056995" class="extiw" title="oeis:A056995">A056995</a> </td></tr> <tr> <td>9 </td> <td>46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, ... </td> <td><a href="//oeis.org/A057465" class="extiw" title="oeis:A057465">A057465</a> </td></tr> <tr> <td>10 </td> <td>824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, ... </td> <td><a href="//oeis.org/A057002" class="extiw" title="oeis:A057002">A057002</a> </td></tr> <tr> <td>11 </td> <td>150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, ... </td> <td><a href="//oeis.org/A088361" class="extiw" title="oeis:A088361">A088361</a> </td></tr> <tr> <td>12 </td> <td>1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, ... </td> <td><a href="//oeis.org/A088362" class="extiw" title="oeis:A088362">A088362</a> </td></tr> <tr> <td>13 </td> <td>30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, ... </td> <td><a href="//oeis.org/A226528" class="extiw" title="oeis:A226528">A226528</a> </td></tr> <tr> <td>14 </td> <td>67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, ... </td> <td><a href="//oeis.org/A226529" class="extiw" title="oeis:A226529">A226529</a> </td></tr> <tr> <td>15 </td> <td>70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, ... </td> <td><a href="//oeis.org/A226530" class="extiw" title="oeis:A226530">A226530</a> </td></tr> <tr> <td>16 </td> <td>48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, ... </td> <td><a href="//oeis.org/A251597" class="extiw" title="oeis:A251597">A251597</a> </td></tr> <tr> <td>17 </td> <td>62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, ... </td> <td><a href="//oeis.org/A253854" class="extiw" title="oeis:A253854">A253854</a> </td></tr> <tr> <td>18 </td> <td>24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, ... </td> <td><a href="//oeis.org/A244150" class="extiw" title="oeis:A244150">A244150</a> </td></tr> <tr> <td>19 </td> <td>75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, ... </td> <td><a href="//oeis.org/A243959" class="extiw" title="oeis:A243959">A243959</a> </td></tr> <tr> <td>20 </td> <td>919444, 1059094, 1951734, 1963736, ... </td> <td><a href="//oeis.org/A321323" class="extiw" title="oeis:A321323">A321323</a> </td></tr></tbody></table> <p>The smallest bases <i>b=b(n)</i> such that <i>b</i><sup>2<sup><i>n</i></sup></sup> + 1 (for given <i>n= 0,1,2, ...</i>) is prime are </p> <dl><dd>2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, ... (sequence <span class="nowrap external"><a href="//oeis.org/A056993" class="extiw" title="oeis:A056993">A056993</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>Conversely, the smallest <i>k=k(n)</i> such that (2<i>n</i>)<sup><i>k</i></sup> + 1 (for given <i>n</i>) is prime are </p> <dl><dd>1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) (sequence <span class="nowrap external"><a href="//oeis.org/A079706" class="extiw" title="oeis:A079706">A079706</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) (also see <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/A228101" class="extiw" title="oeis:A228101">A228101</a></span> and <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/A084712" class="extiw" title="oeis:A084712">A084712</a></span>)</dd></dl> <p>A more elaborate theory can be used to predict the number of bases 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 F_{n}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b609432f821e07b0377567bf60763a04c207045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.752ex; height:2.843ex;" alt="{\displaystyle F_{n}(a)}"></span> will be prime for fixed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. The number of generalized Fermat primes can be roughly expected to halve as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is increased by 1. </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_Fermat_primes_of_the_form_Fn(a,_b)"><span id="Generalized_Fermat_primes_of_the_form_Fn.28a.2C_b.29"></span>Generalized Fermat primes of the form F<sub>n</sub>(<i>a</i>, <i>b</i>)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=14" title="Edit section: Generalized Fermat primes of the form Fn(a, b)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is also possible to construct generalized Fermat primes of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2^{n}}+b^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2^{n}}+b^{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f53aa4dffb27beff2fb1199a8daead8ca553c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.106ex; height:2.843ex;" alt="{\displaystyle a^{2^{n}}+b^{2^{n}}}"></span>. As in the case where <i>b</i>=1, numbers of this form will always be divisible by 2 if <i>a+b</i> is even, but it is still possible to define generalized half-Fermat primes of this type. For the smallest prime of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f406d93132a524540cb55d826726a9cbab5920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.784ex; height:2.843ex;" alt="{\displaystyle F_{n}(a,b)}"></span> (for odd <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+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span>), see also <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/A111635" class="extiw" title="oeis:A111635">A111635</a></span>. </p> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th> <th>numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> such that<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(a,b)={\frac {a^{2^{n}}+b^{2^{n}}}{\gcd(a+b,2)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(a,b)={\frac {a^{2^{n}}+b^{2^{n}}}{\gcd(a+b,2)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8997951e81861e6513ce8e1ec54b80c2ba6b627e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.279ex; height:6.509ex;" alt="{\displaystyle F_{n}(a,b)={\frac {a^{2^{n}}+b^{2^{n}}}{\gcd(a+b,2)}}}"></span><br />is prime<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-bjorn_7-1" class="reference"><a href="#cite_note-bjorn-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>2 </td> <td>1 </td> <td>0, 1, 2, 3, 4, ... </td></tr> <tr> <td>3 </td> <td>1 </td> <td>0, 1, 2, 4, 5, 6, ... </td></tr> <tr> <td>3 </td> <td>2 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>4 </td> <td>1 </td> <td>0, 1, 2, 3, ... (equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(2,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(2,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c9641efd12abd54074767d0c3f400ce46f39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.881ex; height:2.843ex;" alt="{\displaystyle F_{n}(2,1)}"></span>) </td></tr> <tr> <td>4 </td> <td>3 </td> <td>0, 2, 4, ... </td></tr> <tr> <td>5 </td> <td>1 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>5 </td> <td>2 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>5 </td> <td>3 </td> <td>1, 2, 3, ... </td></tr> <tr> <td>5 </td> <td>4 </td> <td>1, 2, ... </td></tr> <tr> <td>6 </td> <td>1 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>6 </td> <td>5 </td> <td>0, 1, 3, 4, ... </td></tr> <tr> <td>7 </td> <td>1 </td> <td>2, ... </td></tr> <tr> <td>7 </td> <td>2 </td> <td>1, 2, ... </td></tr> <tr> <td>7 </td> <td>3 </td> <td>0, 1, 8, ... </td></tr> <tr> <td>7 </td> <td>4 </td> <td>0, 2, ... </td></tr> <tr> <td>7 </td> <td>5 </td> <td>1, 4, </td></tr> <tr> <td>7 </td> <td>6 </td> <td>0, 2, 4, ... </td></tr> <tr> <td>8 </td> <td>1 </td> <td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">(none) </td></tr> <tr> <td>8 </td> <td>3 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>8 </td> <td>5 </td> <td>0, 1, 2, </td></tr> <tr> <td>8 </td> <td>7 </td> <td>1, 4, ... </td></tr> <tr> <td>9 </td> <td>1 </td> <td>0, 1, 3, 4, 5, ... (equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(3,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(3,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f76fd9a263a0668c9bb767b9522972507ac0b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.881ex; height:2.843ex;" alt="{\displaystyle F_{n}(3,1)}"></span>) </td></tr> <tr> <td>9 </td> <td>2 </td> <td>0, 2, ... </td></tr> <tr> <td>9 </td> <td>4 </td> <td>0, 1, ... (equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(3,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(3,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d473d2cc734eb32673419558052b8e95d21a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.881ex; height:2.843ex;" alt="{\displaystyle F_{n}(3,2)}"></span>) </td></tr> <tr> <td>9 </td> <td>5 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>9 </td> <td>7 </td> <td>2, ... </td></tr> <tr> <td>9 </td> <td>8 </td> <td>0, 2, 5, ... </td></tr> <tr> <td>10 </td> <td>1 </td> <td>0, 1, ... </td></tr> <tr> <td>10 </td> <td>3 </td> <td>0, 1, 3, ... </td></tr> <tr> <td>10 </td> <td>7 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>10 </td> <td>9 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>11 </td> <td>1 </td> <td>1, 2, ... </td></tr> <tr> <td>11 </td> <td>2 </td> <td>0, 2, ... </td></tr> <tr> <td>11 </td> <td>3 </td> <td>0, 3, ... </td></tr> <tr> <td>11 </td> <td>4 </td> <td>1, 2, ... </td></tr> <tr> <td>11 </td> <td>5 </td> <td>1, ... </td></tr> <tr> <td>11 </td> <td>6 </td> <td>0, 1, 2, ... </td></tr> <tr> <td>11 </td> <td>7 </td> <td>2, 4, 5, ... </td></tr> <tr> <td>11 </td> <td>8 </td> <td>0, 6, ... </td></tr> <tr> <td>11 </td> <td>9 </td> <td>1, 2, ... </td></tr> <tr> <td>11 </td> <td>10 </td> <td>5, ... </td></tr> <tr> <td>12 </td> <td>1 </td> <td>0, ... </td></tr> <tr> <td>12 </td> <td>5 </td> <td>0, 4, ... </td></tr> <tr> <td>12 </td> <td>7 </td> <td>0, 1, 3, ... </td></tr> <tr> <td>12 </td> <td>11 </td> <td>0, ... </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Largest_known_generalized_Fermat_primes">Largest known generalized Fermat primes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=15" title="Edit section: Largest known generalized Fermat primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following is a list of the five largest known generalized Fermat primes.<sup id="cite_ref-Top_Twenty's_Generalized_Fermat_Primes_21-0" class="reference"><a href="#cite_note-Top_Twenty's_Generalized_Fermat_Primes-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The whole top-5 is discovered by participants in the <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> project. </p> <table class="wikitable" style="text-align:right"> <tbody><tr> <th>Rank </th> <th>Prime number </th> <th>Generalized Fermat notation </th> <th>Number of digits </th> <th>Discovery date </th> <th>ref. </th></tr> <tr> <td>1 </td> <td>4×5<sup>11786358</sup> + 1 </td> <td><i>F</i><sub>1</sub>(2×5<sup>5893179</sup>) </td> <td>8,238,312 </td> <td style="text-align:left">Oct 2024 </td> <td><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> </td></tr> <tr> <td>2 </td> <td>1963736<sup>1048576</sup> + 1 </td> <td><i>F</i><sub>20</sub>(1963736) </td> <td>6,598,776 </td> <td style="text-align:left">Sep 2022 </td> <td><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> </td></tr> <tr> <td>3 </td> <td>1951734<sup>1048576</sup> + 1 </td> <td><i>F</i><sub>20</sub>(1951734) </td> <td>6,595,985 </td> <td style="text-align:left">Aug 2022 </td> <td><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> </td></tr> <tr> <td>4 </td> <td>1059094<sup>1048576</sup> + 1 </td> <td><i>F</i><sub>20</sub>(1059094) </td> <td>6,317,602 </td> <td style="text-align:left">Nov 2018 </td> <td><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> </td></tr> <tr> <td>5 </td> <td>919444<sup>1048576</sup> + 1 </td> <td><i>F</i><sub>20</sub>(919444) </td> <td>6,253,210 </td> <td style="text-align:left">Sep 2017 </td> <td><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> </td></tr></tbody></table> <p>On the <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a> one can find the <a rel="nofollow" class="external text" href="http://primes.utm.edu/primes/search.php?Comment=Generalized+Fermat&OnList=yes&Number=100&Style=HTML">current top 100 generalized Fermat primes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Constructible_polygon" title="Constructible polygon">Constructible polygon</a>: which regular polygons are constructible partially depends on Fermat primes.</li> <li><a href="/wiki/Double_exponential_function" title="Double exponential function">Double exponential function</a></li> <li><a href="/wiki/Lucas%27_theorem" class="mw-redirect" title="Lucas' theorem">Lucas' theorem</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne prime</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont prime</a></li> <li><a href="/wiki/Primality_test" title="Primality test">Primality test</a></li> <li><a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a></li> <li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński number</a></li> <li><a href="/wiki/Sylvester%27s_sequence" title="Sylvester's sequence">Sylvester's sequence</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat_number&action=edit&section=17" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">For any positive odd number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{k}m}+1=(a+1)(a^{m-1}-a^{m-2}+\ldots -a+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{k}m}+1=(a+1)(a^{m-1}-a^{m-2}+\ldots -a+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3fa1e103e2d642510de00493155d5ce1460cec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.959ex; height:3.509ex;" alt="{\displaystyle 2^{2^{k}m}+1=(a+1)(a^{m-1}-a^{m-2}+\ldots -a+1)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2^{2^{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2^{2^{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ff467c319501356a963a3b32d920073e8a44d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.405ex; height:3.009ex;" alt="{\displaystyle a=2^{2^{k}}}"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFKřížekLucaSomer2001">Křížek, Luca & Somer 2001</a>, p. 38, Remark 4.15</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">Chris Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/links/theory/special_forms/">"Prime Links++: special forms"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131224224552/http://primes.utm.edu/links/theory/special_forms/">Archived</a> 2013-12-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at The <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFRibenboim1996">Ribenboim 1996</a>, p. 88.</span> </li> <li id="cite_note-Keller-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Keller_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Keller_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Keller_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKeller2021" class="citation cs2">Keller, Wilfrid (January 18, 2021), <a rel="nofollow" class="external text" href="http://www.prothsearch.com/fermat.html#Summary">"Prime Factors of Fermat Numbers"</a>, <i>ProthSearch.com</i><span class="reference-accessdate">, retrieved <span class="nowrap">January 19,</span> 2021</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ProthSearch.com&rft.atitle=Prime+Factors+of+Fermat+Numbers&rft.date=2021-01-18&rft.aulast=Keller&rft.aufirst=Wilfrid&rft_id=http%3A%2F%2Fwww.prothsearch.com%2Ffermat.html%23Summary&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoklanConway2017" class="citation journal cs1">Boklan, Kent D.; Conway, John H. (2017). "Expect at most one billionth of a new Fermat Prime!". <i>The Mathematical Intelligencer</i>. <b>39</b> (1): 3–5. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1605.01371">1605.01371</a></span>. <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%2Fs00283-016-9644-3">10.1007/s00283-016-9644-3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119165671">119165671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=Expect+at+most+one+billionth+of+a+new+Fermat+Prime%21&rft.volume=39&rft.issue=1&rft.pages=3-5&rft.date=2017&rft_id=info%3Aarxiv%2F1605.01371&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119165671%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00283-016-9644-3&rft.aulast=Boklan&rft.aufirst=Kent+D.&rft.au=Conway%2C+John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-bjorn-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-bjorn_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bjorn_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBjörnRiesel1998" class="citation journal cs1">Björn, Anders; Riesel, Hans (1998). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/">"Factors of generalized Fermat numbers"</a>. <i>Mathematics of Computation</i>. <b>67</b> (221): 441–446. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-98-00891-6">10.1090/S0025-5718-98-00891-6</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5718">0025-5718</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Factors+of+generalized+Fermat+numbers&rft.volume=67&rft.issue=221&rft.pages=441-446&rft.date=1998&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-98-00891-6&rft.issn=0025-5718&rft.aulast=Bj%C3%B6rn&rft.aufirst=Anders&rft.au=Riesel%2C+Hans&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1998-67-221%2FS0025-5718-98-00891-6%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" 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="CITEREFSandifer" class="citation web cs1">Sandifer, Ed. <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/hedi/HEDI-2007-03.pdf">"How Euler Did it"</a> <span class="cs1-format">(PDF)</span>. <i>MAA Online</i>. Mathematical Association of America. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://eulerarchive.maa.org/hedi/HEDI-2007-03.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-06-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MAA+Online&rft.atitle=How+Euler+Did+it&rft.aulast=Sandifer&rft.aufirst=Ed&rft_id=http%3A%2F%2Feulerarchive.maa.org%2Fhedi%2FHEDI-2007-03.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.fermatsearch.org/">":: F E R M A T S E A R C H . O R G :: Home page"</a>. <i>www.fermatsearch.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</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=www.fermatsearch.org&rft.atitle=%3A%3A+F+E+R+M+A+T+S+E+A+R+C+H+.+O+R+G+%3A%3A+Home+page&rft_id=http%3A%2F%2Fwww.fermatsearch.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><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.fermatsearch.org/news.html">"::FERMATSEARCH.ORG:: News"</a>. <i>www.fermatsearch.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</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=www.fermatsearch.org&rft.atitle=%3A%3AFERMATSEARCH.ORG%3A%3A+News&rft_id=http%3A%2F%2Fwww.fermatsearch.org%2Fnews.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchroeder2006" class="citation book cs1">Schroeder, M. R. (2006). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/ocm61430240"><i>Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity</i></a>. Springer series in information sciences (4th ed.). Berlin ; New York: Springer. p. 216. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-26596-2" title="Special:BookSources/978-3-540-26596-2"><bdi>978-3-540-26596-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/61430240">61430240</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+theory+in+science+and+communication%3A+with+applications+in+cryptography%2C+physics%2C+digital+information%2C+computing%2C+and+self-similarity&rft.place=Berlin+%3B+New+York&rft.series=Springer+series+in+information+sciences&rft.pages=216&rft.edition=4th&rft.pub=Springer&rft.date=2006&rft_id=info%3Aoclcnum%2Focm61430240&rft.isbn=978-3-540-26596-2&rft.aulast=Schroeder&rft.aufirst=M.+R.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2Focm61430240&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrizekLucaSomer2013" class="citation book cs1">Krizek, Michal; Luca, Florian; Somer, Lawrence (14 March 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hgfSBwAAQBAJ&q=cipolla+fermat+1904&pg=PA132"><i>17 Lectures on Fermat Numbers: From Number Theory to Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387218502" title="Special:BookSources/9780387218502"><bdi>9780387218502</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</span> 2018</span> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-03-14&rft.isbn=9780387218502&rft.aulast=Krizek&rft.aufirst=Michal&rft.au=Luca%2C+Florian&rft.au=Somer%2C+Lawrence&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhgfSBwAAQBAJ%26q%3Dcipolla%2Bfermat%2B1904%26pg%3DPA132&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Jeppe Stig Nielsen, <a rel="nofollow" class="external text" href="http://jeppesn.dk/nton.html">"S(n) = n^n + 1"</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Sierpiński_Number_of_the_First_Kind"><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/SierpinskiNumberoftheFirstKind.html">"Sierpiński Number of the First Kind"</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=Sierpi%C5%84ski+Number+of+the+First+Kind&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSierpinskiNumberoftheFirstKind.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1966" class="citation book cs1">Gauss, Carl Friedrich (1966). <a rel="nofollow" class="external text" href="https://archive.org/details/disquisitionesar0000carl/"><i>Disquisitiones arithmeticae</i></a>. New Haven and London: Yale University Press. pp. 458–460<span class="reference-accessdate">. Retrieved <span class="nowrap">25 January</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Disquisitiones+arithmeticae&rft.place=New+Haven+and+London&rft.pages=458-460&rft.pub=Yale+University+Press&rft.date=1966&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdisquisitionesar0000carl%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.primenumbers.net/prptop/searchform.php?form=x%5E131072%2By%5E131072&action=Search">PRP Top Records, search for x^131072+y^131072</a>, by Henri & Renaud Lifchitz.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</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://jeppesn.dk/generalized-fermat.html">"Generalized Fermat Primes"</a>. <i>jeppesn.dk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</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=jeppesn.dk&rft.atitle=Generalized+Fermat+Primes&rft_id=http%3A%2F%2Fjeppesn.dk%2Fgeneralized-fermat.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</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.noprimeleftbehind.net/crus/GFN-primes.htm">"Generalized Fermat primes for bases up to 1030"</a>. <i>noprimeleftbehind.net</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</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=noprimeleftbehind.net&rft.atitle=Generalized+Fermat+primes+for+bases+up+to+1030&rft_id=http%3A%2F%2Fwww.noprimeleftbehind.net%2Fcrus%2FGFN-primes.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</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.fermatquotient.com/PrimSerien/GenFermOdd.txt">"Generalized Fermat primes in odd bases"</a>. <i>fermatquotient.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</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=fermatquotient.com&rft.atitle=Generalized+Fermat+primes+in+odd+bases&rft_id=http%3A%2F%2Fwww.fermatquotient.com%2FPrimSerien%2FGenFermOdd.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" 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="http://www.prothsearch.com/OriginalGFNs.html">"Original GFN factors"</a>. <i>www.prothsearch.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.prothsearch.com&rft.atitle=Original+GFN+factors&rft_id=http%3A%2F%2Fwww.prothsearch.com%2FOriginalGFNs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span> </li> <li id="cite_note-Top_Twenty's_Generalized_Fermat_Primes-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Top_Twenty's_Generalized_Fermat_Primes_21-0">^</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 K. <a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=12">"The Top Twenty: Generalized Fermat"</a>. <i>The Prime Pages</i><span class="reference-accessdate">. Retrieved <span class="nowrap">5 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=The+Prime+Pages&rft.atitle=The+Top+Twenty%3A+Generalized+Fermat&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=http%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" 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"><a rel="nofollow" class="external text" href="https://t5k.org/primes/page.php?id=138596">4×5<sup>11786358</sup> + 1</a></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"><a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=134423">1963736<sup>1048576</sup> + 1</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=134298">1951734<sup>1048576</sup> + 1</a></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="https://primes.utm.edu/primes/page.php?id=125753">1059094<sup>1048576</sup> + 1</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"><a rel="nofollow" class="external text" href="https://primes.utm.edu/primes/page.php?id=123875">919444<sup>1048576</sup> + 1</a></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=Fermat_number&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolomb1963" class="citation cs2">Golomb, S. W. (January 1, 1963), "On the sum of the reciprocals of the Fermat numbers and related irrationalities", <i>Canadian Journal of Mathematics</i>, <b>15</b>: 475–478, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1963-051-0">10.4153/CJM-1963-051-0</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123138118">123138118</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=On+the+sum+of+the+reciprocals+of+the+Fermat+numbers+and+related+irrationalities&rft.volume=15&rft.pages=475-478&rft.date=1963-01-01&rft_id=info%3Adoi%2F10.4153%2FCJM-1963-051-0&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123138118%23id-name%3DS2CID&rft.aulast=Golomb&rft.aufirst=S.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrytczukLucaWójtowicz2001" class="citation cs2">Grytczuk, A.; Luca, F. & Wójtowicz, M. (2001), "Another note on the greatest prime factors of Fermat numbers", <i>Southeast Asian Bulletin of Mathematics</i>, <b>25</b> (1): 111–115, <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%2Fs10012-001-0111-4">10.1007/s10012-001-0111-4</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122332537">122332537</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Southeast+Asian+Bulletin+of+Mathematics&rft.atitle=Another+note+on+the+greatest+prime+factors+of+Fermat+numbers&rft.volume=25&rft.issue=1&rft.pages=111-115&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2Fs10012-001-0111-4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122332537%23id-name%3DS2CID&rft.aulast=Grytczuk&rft.aufirst=A.&rft.au=Luca%2C+F.&rft.au=W%C3%B3jtowicz%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation cs2"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004), <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/numbers/book/978-0-387-20860-2?otherVersion=978-0-387-26677-0"><i>Unsolved Problems in Number Theory</i></a>, Problem Books in Mathematics, vol. 1 (3rd ed.), New York: <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>, pp. A3, A12, B21, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20860-2" title="Special:BookSources/978-0-387-20860-2"><bdi>978-0-387-20860-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.place=New+York&rft.series=Problem+Books+in+Mathematics&rft.pages=A3%2C+A12%2C+B21&rft.edition=3rd&rft.pub=Springer+Verlag&rft.date=2004&rft.isbn=978-0-387-20860-2&rft.aulast=Guy&rft.aufirst=Richard+K.&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fnumbers%2Fbook%2F978-0-387-20860-2%3FotherVersion%3D978-0-387-26677-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKřížekLucaSomer2001" class="citation cs2">Křížek, Michal; Luca, Florian & Somer, Lawrence (2001), <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/numbers/book/978-0-387-95332-8"><i>17 Lectures on Fermat Numbers: From Number Theory to Geometry</i></a>, CMS books in mathematics, vol. 10, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95332-8" title="Special:BookSources/978-0-387-95332-8"><bdi>978-0-387-95332-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=17+Lectures+on+Fermat+Numbers%3A+From+Number+Theory+to+Geometry&rft.place=New+York&rft.series=CMS+books+in+mathematics&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-387-95332-8&rft.aulast=K%C5%99%C3%AD%C5%BEek&rft.aufirst=Michal&rft.au=Luca%2C+Florian&rft.au=Somer%2C+Lawrence&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fnumbers%2Fbook%2F978-0-387-95332-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span> - This book contains an extensive list of references.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKřížekLucaSomer2002" class="citation cs2">Křížek, Michal; Luca, Florian & Somer, Lawrence (2002), "On the convergence of series of reciprocals of primes related to the Fermat numbers", <i>Journal of Number Theory</i>, <b>97</b> (1): 95–112, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.2002.2782">10.1006/jnth.2002.2782</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=On+the+convergence+of+series+of+reciprocals+of+primes+related+to+the+Fermat+numbers&rft.volume=97&rft.issue=1&rft.pages=95-112&rft.date=2002&rft_id=info%3Adoi%2F10.1006%2Fjnth.2002.2782&rft.aulast=K%C5%99%C3%AD%C5%BEek&rft.aufirst=Michal&rft.au=Luca%2C+Florian&rft.au=Somer%2C+Lawrence&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuca2000" class="citation cs2">Luca, Florian (2000), <a rel="nofollow" class="external text" href="http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-february-2000">"The anti-social Fermat number"</a>, <i>American Mathematical Monthly</i>, <b>107</b> (2): 171–173, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2589441">10.2307/2589441</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2589441">2589441</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+anti-social+Fermat+number&rft.volume=107&rft.issue=2&rft.pages=171-173&rft.date=2000&rft_id=info%3Adoi%2F10.2307%2F2589441&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589441%23id-name%3DJSTOR&rft.aulast=Luca&rft.aufirst=Florian&rft_id=http%3A%2F%2Fwww.maa.org%2Fpublications%2Fperiodicals%2Famerican-mathematical-monthly%2Famerican-mathematical-monthly-february-2000&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996), <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/numbers/book/978-0-387-94457-9"><i>The New Book of Prime Number Records</i></a> (3rd ed.), New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94457-9" title="Special:BookSources/978-0-387-94457-9"><bdi>978-0-387-94457-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.edition=3rd&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-94457-9&rft.aulast=Ribenboim&rft.aufirst=Paulo&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fnumbers%2Fbook%2F978-0-387-94457-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinson1954" class="citation cs2">Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", <i>Proceedings of the American Mathematical Society</i>, <b>5</b> (5): 842–846, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2031878">10.2307/2031878</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2031878">2031878</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Mersenne+and+Fermat+Numbers&rft.volume=5&rft.issue=5&rft.pages=842-846&rft.date=1954&rft_id=info%3Adoi%2F10.2307%2F2031878&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2031878%23id-name%3DJSTOR&rft.aulast=Robinson&rft.aufirst=Raphael+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYabuta2001" class="citation cs2">Yabuta, M. (2001), <a rel="nofollow" class="external text" href="http://www.fq.math.ca/Scanned/39-5/yabuta.pdf">"A simple proof of Carmichael's theorem on primitive divisors"</a> <span class="cs1-format">(PDF)</span>, <i>Fibonacci Quarterly</i>, <b>39</b> (5): 439–443, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.2001.12428701">10.1080/00150517.2001.12428701</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.fq.math.ca/Scanned/39-5/yabuta.pdf">archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fibonacci+Quarterly&rft.atitle=A+simple+proof+of+Carmichael%27s+theorem+on+primitive+divisors&rft.volume=39&rft.issue=5&rft.pages=439-443&rft.date=2001&rft_id=info%3Adoi%2F10.1080%2F00150517.2001.12428701&rft.aulast=Yabuta&rft.aufirst=M.&rft_id=http%3A%2F%2Fwww.fq.math.ca%2FScanned%2F39-5%2Fyabuta.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></li></ul> <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=Fermat_number&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chris Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/page.php?sort=FermatNumber">The Prime Glossary: Fermat number</a> at The <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a>.</li> <li>Luigi Morelli, <a rel="nofollow" class="external text" href="http://www.fermatsearch.org/history.html">History of Fermat Numbers</a></li> <li>John Cosgrave, <a rel="nofollow" class="external text" href="http://johnbcosgrave.com/archive/fermat6.htm">Unification of Mersenne and Fermat Numbers</a></li> <li>Wilfrid Keller, <a rel="nofollow" class="external text" href="http://www.prothsearch.com/fermat.html">Prime Factors of Fermat Numbers</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Fermat_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/FermatNumber.html">"Fermat 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=Fermat+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFermatNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Fermat_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/FermatPrime.html">"Fermat 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=Fermat+Prime&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFermatPrime.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Generalized_Fermat_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/GeneralizedFermatNumber.html">"Generalized Fermat 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=Generalized+Fermat+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGeneralizedFermatNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermat+number" class="Z3988"></span></span></li> <li>Yves Gallot, <a rel="nofollow" class="external text" href="http://pagesperso-orange.fr/yves.gallot/primes/index.html">Generalized Fermat Prime Search</a></li> <li>Mark S. Manasse, <a rel="nofollow" class="external text" href="https://groups.google.com/forum/#!topic/sci.math/7usZOcN2_zc">Complete factorization of the ninth Fermat number</a> (original announcement)</li> <li>Peyton Hayslette, <a rel="nofollow" class="external text" href="http://www.primegrid.com/download/GFN-341112_524288.pdf">Largest Known Generalized Fermat Prime Announcement</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output 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texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple"><i>k</i>-tuples</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twin_prime" title="Twin prime">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a 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_numbers" 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_numbers" 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_numbers" 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_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a class="mw-selflink selflink">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a 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_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Pierre_de_Fermat" 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:Pierre_de_Fermat" title="Template:Pierre de Fermat"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Pierre_de_Fermat&action=edit&redlink=1" class="new" title="Template talk:Pierre de Fermat (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Pierre_de_Fermat" title="Special:EditPage/Template:Pierre de Fermat"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Pierre_de_Fermat" style="font-size:114%;margin:0 4em"><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Work</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/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a></li> <li><a class="mw-selflink selflink">Fermat number</a></li> <li><a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a></li> <li><a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a></li> <li><a href="/wiki/Fermat_polygonal_number_theorem" title="Fermat polygonal number theorem">Fermat polygonal number theorem</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Fermat_point" title="Fermat point">Fermat point</a></li> <li><a href="/wiki/Fermat%27s_theorem_(stationary_points)" title="Fermat's theorem (stationary points)">Fermat's theorem (stationary points)</a></li> <li><a href="/wiki/Fermat%27s_theorem_on_sums_of_two_squares" title="Fermat's theorem on sums of two squares">Fermat's theorem on sums of two squares</a></li> <li><a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's spiral</a></li> <li><a href="/wiki/Fermat%27s_right_triangle_theorem" title="Fermat's right triangle theorem">Fermat's right triangle theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">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/List_of_things_named_after_Pierre_de_Fermat" title="List of things named after Pierre de Fermat">List of things named after Pierre de Fermat</a></li> <li><a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a></li> <li><a href="/wiki/Fermat%27s_Last_Theorem_in_fiction" title="Fermat's Last Theorem in fiction">Fermat's Last Theorem in fiction</a></li> <li><a href="/wiki/Fermat_Prize" title="Fermat Prize">Fermat Prize</a></li> <li><i><a href="/wiki/Fermat%27s_Last_Tango" title="Fermat's Last Tango">Fermat's Last Tango</a></i> (2000 musical)</li> <li><i><a href="/wiki/Fermat%27s_Last_Theorem_(book)" title="Fermat's Last Theorem (book)">Fermat's Last Theorem</a></i> (popular science book)</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"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" 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