Pell number - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Pell number - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"c5809d77-6d2a-4df0-a3b1-60a17e84c8e5","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Pell_number","wgTitle":"Pell number","wgCurRevisionId":1256792511,"wgRevisionId":1256792511,"wgArticleId":1355482,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Integer sequences","Recurrence relations","Unsolved problems in mathematics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Pell_number","wgRelevantArticleId":1355482,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false ,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1386491","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth", "ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/1200px-Silver_spiral_approximation.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="505"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/800px-Silver_spiral_approximation.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="337"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/640px-Silver_spiral_approximation.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="269"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Pell number - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Pell_number"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Pell_number&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Pell_number"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Pell_number rootpage-Pell_number skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Pell+number" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Pell+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Pell+number" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Pell+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Pell_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pell_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Pell numbers</span> </div> </a> <ul id="toc-Pell_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximation_to_the_square_root_of_two" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Approximation_to_the_square_root_of_two"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Approximation to the square root of two</span> </div> </a> <ul id="toc-Approximation_to_the_square_root_of_two-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primes_and_squares" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primes_and_squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Primes and squares</span> </div> </a> <ul id="toc-Primes_and_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagorean_triples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pythagorean_triples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Pythagorean triples</span> </div> </a> <ul id="toc-Pythagorean_triples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pell–Lucas_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pell–Lucas_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Pell–Lucas numbers</span> </div> </a> <ul id="toc-Pell–Lucas_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computations_and_connections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computations_and_connections"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Computations and connections</span> </div> </a> <button aria-controls="toc-Computations_and_connections-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 Computations and connections subsection</span> </button> <ul id="toc-Computations_and_connections-sublist" class="vector-toc-list"> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Raising_to_powers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Raising_to_powers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Raising to powers</span> </div> </a> <ul id="toc-Raising_to_powers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paired_recurrences" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Paired_recurrences"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Paired recurrences</span> </div> </a> <ul id="toc-Paired_recurrences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocal_recurrence_formulas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reciprocal_recurrence_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.3</span> <span>Reciprocal recurrence formulas</span> </div> </a> <ul id="toc-Reciprocal_recurrence_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_formulations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Matrix_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.4</span> <span>Matrix formulations</span> </div> </a> <ul id="toc-Matrix_formulations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Approximations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Approximations</span> </div> </a> <ul id="toc-Approximations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-H_2_−_2P_2_=_±1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#H_2_−_2P_2_=_±1"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span><i>H</i><sup> 2</sup> − 2<i>P</i><sup> 2</sup> = ±1</span> </div> </a> <ul id="toc-H_2_−_2P_2_=_±1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_triangular_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_triangular_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Square triangular numbers</span> </div> </a> <ul id="toc-Square_triangular_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagorean_triples_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_triples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Pythagorean triples</span> </div> </a> <ul id="toc-Pythagorean_triples_2-sublist" class="vector-toc-list"> </ul> </li> </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">7</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">8</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">9</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">Pell 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 20 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-20" 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">20 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%82%D9%85_%D8%A8%D9%8A%D9%84" title="رقم بيل – Arabic" lang="ar" hreflang="ar" data-title="رقم بيل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%87%E0%A6%B2_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF%E0%A6%AE%E0%A6%BE%E0%A6%B2%E0%A6%BE" title="পেল রাশিমালা – Bangla" lang="bn" hreflang="bn" data-title="পেল রাশিমালা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Pell" title="Nombre de Pell – Catalan" lang="ca" hreflang="ca" data-title="Nombre de Pell" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pell-Folge" title="Pell-Folge – German" lang="de" hreflang="de" data-title="Pell-Folge" 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%CE%BF%CE%AF_%CF%84%CE%BF%CF%85_%CE%A0%CE%B5%CE%BB" title="Αριθμοί του Πελ – Greek" lang="el" hreflang="el" data-title="Αριθμοί του Πελ" 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_Pell" title="Número de Pell – Spanish" lang="es" hreflang="es" data-title="Número de Pell" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Suite_de_Pell" title="Suite de Pell – French" lang="fr" hreflang="fr" data-title="Suite de Pell" 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_Pell" title="Número de Pell – Galician" lang="gl" hreflang="gl" data-title="Número de Pell" 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%A0_%EC%88%98" title="펠 수 – Korean" lang="ko" hreflang="ko" data-title="펠 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%8A%D5%A5%D5%AC%D5%AB_%D5%A9%D5%AB%D5%BE" title="Պելի թիվ – Armenian" lang="hy" hreflang="hy" data-title="Պելի թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A4%D7%9C" title="סדרת פל – Hebrew" lang="he" hreflang="he" data-title="סדרת פל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pellgetal" title="Pellgetal – Dutch" lang="nl" hreflang="nl" data-title="Pellgetal" 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%9A%E3%83%AB%E6%95%B0" title="ペル数 – Japanese" lang="ja" hreflang="ja" data-title="ペル数" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Pella" title="Liczby Pella – Polish" lang="pl" hreflang="pl" data-title="Liczby Pella" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_Pell" title="Număr Pell – Romanian" lang="ro" hreflang="ro" data-title="Număr Pell" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D0%B5%D0%BB%D0%BB%D1%8F" title="Число Пелля – Russian" lang="ru" hreflang="ru" data-title="Число Пелля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B5%D0%BB_%D0%B1%D1%80%D0%BE%D1%98" title="Пел број – Serbian" lang="sr" hreflang="sr" data-title="Пел број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%9F%D0%B5%D0%BB%D0%BB%D1%8F" title="Число Пелля – Ukrainian" lang="uk" hreflang="uk" data-title="Число Пелля" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BD%A9%E7%88%BE%E6%95%B8" title="佩爾數 – Cantonese" lang="yue" hreflang="yue" data-title="佩爾數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BD%A9%E5%B0%94%E6%95%B0" title="佩尔数 – Chinese" lang="zh" hreflang="zh" data-title="佩尔数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1386491#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Pell_number" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Pell_number" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Pell_number"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pell_number&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pell_number&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Pell_number"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Pell_number&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Pell_number&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Pell_number" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Pell_number" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Pell_number&oldid=1256792511" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Pell_number&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Pell_number&id=1256792511&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPell_number"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPell_number"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Pell_number&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Pell_number&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1386491" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Number used to approximate the square root of 2</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Bell_number" title="Bell number">Bell number</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Silver_spiral_approximation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/220px-Silver_spiral_approximation.svg.png" decoding="async" width="220" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/330px-Silver_spiral_approximation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Silver_spiral_approximation.svg/440px-Silver_spiral_approximation.svg.png 2x" data-file-width="855" data-file-height="360" /></a><figcaption>The sides of the <a href="/wiki/Square" title="Square">squares</a> used to construct a silver spiral are the Pell numbers</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Pell numbers</b> are an infinite <a href="/wiki/Integer_sequence" title="Integer sequence">sequence of integers</a>, known since ancient times, that comprise the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominators</a> of the <a href="/wiki/Closest_rational_approximation" class="mw-redirect" title="Closest rational approximation">closest rational approximations</a> to the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>. This <a href="/wiki/Sequence" title="Sequence">sequence</a> of approximations begins <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">17</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">41</span><span class="sr-only">/</span><span class="den">29</span></span>⁠</span>, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the <b>companion Pell numbers</b> or <b>Pell–Lucas numbers</b>; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. </p><p>Both the Pell numbers and the companion Pell numbers may be calculated by means of a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> similar to that for the <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a>, and both sequences of numbers <a href="/wiki/Exponential_growth" title="Exponential growth">grow exponentially</a>, proportionally to powers of the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> 1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. As well as being used to approximate the square root of two, Pell numbers can be used to find <a href="/wiki/Square_triangular_number" title="Square triangular number">square triangular numbers</a>, to construct <a href="/wiki/Integer" title="Integer">integer</a> approximations to the <a href="/wiki/Right_isosceles_triangle" class="mw-redirect" title="Right isosceles triangle">right isosceles triangle</a>, and to solve certain <a href="/wiki/Combinatorial_enumeration" class="mw-redirect" title="Combinatorial enumeration">combinatorial enumeration</a> problems.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>As with <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a>, the name of the Pell numbers stems from <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler's</a> mistaken attribution of the equation and the numbers derived from it to <a href="/wiki/John_Pell_(mathematician)" title="John Pell (mathematician)">John Pell</a>. The Pell–Lucas numbers are also named after <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas sequences</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pell_numbers">Pell numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=1" title="Edit section: Pell numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pell numbers are defined by the <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</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 P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\2P_{n-1}+P_{n-2}&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\2P_{n-1}+P_{n-2}&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18486111734f3e70484363b82252c23c8b5016ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:34.799ex; height:8.509ex;" alt="{\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\2P_{n-1}+P_{n-2}&{\mbox{otherwise.}}\end{cases}}}"></span></dd></dl> <p>In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number, plus the Pell number before that. The first few terms of the sequence are </p> <dl><dd>0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, … (sequence <span class="nowrap external"><a href="//oeis.org/A000129" class="extiw" title="oeis:A000129">A000129</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>Analogously to the <a href="/wiki/Binet_formula" class="mw-redirect" title="Binet formula">Binet formula</a>, the Pell numbers can also be expressed by the closed form formula </p><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 P_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2{\sqrt {2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2{\sqrt {2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d79d61042808601ee27b68656a1e25945d8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.032ex; height:7.343ex;" alt="{\displaystyle P_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2{\sqrt {2}}}}.}"></span> For large values of <i>n</i>, the <span class="nowrap">(1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>)<sup><i>n</i></sup></span> term dominates this expression, so the Pell numbers are approximately proportional to powers of the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> <span class="nowrap">1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></span>, analogous to the growth rate of Fibonacci numbers as powers of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>. </p><p>A third definition is possible, from the <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> formula </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 {\begin{pmatrix}P_{n+1}&P_{n}\\P_{n}&P_{n-1}\end{pmatrix}}={\begin{pmatrix}2&1\\1&0\end{pmatrix}}^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}P_{n+1}&P_{n}\\P_{n}&P_{n-1}\end{pmatrix}}={\begin{pmatrix}2&1\\1&0\end{pmatrix}}^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef0cb0cdb7c7fc4a64ede65c289a229964d904f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.227ex; margin-bottom: -0.278ex; width:29.902ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}P_{n+1}&P_{n}\\P_{n}&P_{n-1}\end{pmatrix}}={\begin{pmatrix}2&1\\1&0\end{pmatrix}}^{n}.}"></span></dd></dl> <p>Many <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a> can be derived or <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a> from these definitions; for instance an identity analogous to <a href="/wiki/Cassini%27s_identity" class="mw-redirect" title="Cassini's identity">Cassini's identity</a> for Fibonacci numbers, </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_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd0772ebce706cf52c120c56807c2a2cd9f67e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.082ex; height:3.009ex;" alt="{\displaystyle P_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},}"></span></dd></dl> <p>is an immediate consequence of the matrix formula (found by considering the <a href="/wiki/Determinant" title="Determinant">determinants</a> of the matrices on the left and right sides of the matrix formula).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Approximation_to_the_square_root_of_two">Approximation to the square root of two</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=2" title="Edit section: Approximation to the square root of two"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pell_octagons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pell_octagons.svg/300px-Pell_octagons.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pell_octagons.svg/450px-Pell_octagons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pell_octagons.svg/600px-Pell_octagons.svg.png 2x" data-file-width="324" data-file-height="324" /></a><figcaption>Rational approximations to <a href="/wiki/Regular_polygon" title="Regular polygon">regular</a> <a href="/wiki/Octagon" title="Octagon">octagons</a>, with coordinates derived from the Pell numbers.</figcaption></figure> <p>Pell numbers arise historically and most notably in the <a href="/wiki/Diophantine_approximation" title="Diophantine approximation">rational approximation</a> to <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. If two large integers <i>x</i> and <i>y</i> form a solution to the <a href="/wiki/Pell_equation" class="mw-redirect" title="Pell equation">Pell equation</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 x^{2}-2y^{2}=\pm 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-2y^{2}=\pm 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc19867e9442dde835ad9bcba6a259e3b0ff065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.317ex; height:3.009ex;" alt="{\displaystyle x^{2}-2y^{2}=\pm 1,}"></span></dd></dl> <p>then their ratio <i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">x</span><span class="sr-only">/</span><span class="den">y</span></span>⁠</span></i> provides a close approximation to <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. The sequence of approximations of this form is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1}},{\frac {3}{2}},{\frac {7}{5}},{\frac {17}{12}},{\frac {41}{29}},{\frac {99}{70}},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>41</mn> <mn>29</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>99</mn> <mn>70</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1}},{\frac {3}{2}},{\frac {7}{5}},{\frac {17}{12}},{\frac {41}{29}},{\frac {99}{70}},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd581889ba9b6ac1079ca91749a8eb04a4b25c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.406ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{1}},{\frac {3}{2}},{\frac {7}{5}},{\frac {17}{12}},{\frac {41}{29}},{\frac {99}{70}},\dots }"></span></dd></dl> <p>where the denominator of each <a href="/wiki/Fraction" title="Fraction">fraction</a> is a Pell number and the numerator is the sum of a Pell number and its predecessor in the sequence. That is, the solutions have the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P_{n-1}+P_{n}}{P_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P_{n-1}+P_{n}}{P_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad70a8338fe6f73d6a02e3f157e0b496fc5aeb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.845ex; height:5.843ex;" alt="{\displaystyle {\frac {P_{n-1}+P_{n}}{P_{n}}}.}"></span></dd></dl> <p>The approximation </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 {\sqrt {2}}\approx {\frac {577}{408}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>577</mn> <mn>408</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx {\frac {577}{408}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3300f82dab3620758cf6d37a6a6eeae2297e174e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.52ex; height:5.343ex;" alt="{\displaystyle {\sqrt {2}}\approx {\frac {577}{408}}}"></span></dd></dl> <p>of this type was known to Indian mathematicians in the third or fourth century BCE.<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> The Greek mathematicians of the fifth century BCE also knew of this sequence of approximations:<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> Plato refers to the numerators as <b>rational diameters</b>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In the second century CE <a href="/wiki/Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a> used the term the <b>side and diameter numbers</b> to describe the denominators and numerators of this sequence.<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>These approximations can be derived from the <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">continued fraction expansion</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></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 {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots \,}}}}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mo>⋱</mo> <mspace width="thinmathspace" /> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots \,}}}}}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7060806006034ff9d42a7c889a998baf4c88e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:38.407ex; height:23.009ex;" alt="{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots \,}}}}}}}}}}.}"></span></dd></dl> <p>Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {577}{408}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2}}}}}}}}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>577</mn> <mn>408</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {577}{408}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2}}}}}}}}}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb66dfe15f23a4c0d89843e3ace728114d0da0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.505ex; width:43.104ex; height:28.676ex;" alt="{\displaystyle {\frac {577}{408}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2}}}}}}}}}}}}}}.}"></span></dd></dl> <p>As <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth</a> (1994) describes, the fact that Pell numbers approximate <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates <span class="nowrap">(± <i>P<sub>i</sub></i>, ± <i>P</i><sub><i>i</i> +1</sub>)</span> and <span class="nowrap">(± <i>P</i><sub><i>i</i> +1</sub>, ± <i>P<sub>i</sub></i> )</span>. All vertices are equally distant from the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, and form nearly uniform <a href="/wiki/Angle" title="Angle">angles</a> around the origin. Alternatively, the points <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 (\pm (P_{i}+P_{i-1}),0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pm (P_{i}+P_{i-1}),0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462056f02d2e4433e2dfc68c3634d90698338760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.148ex; height:2.843ex;" alt="{\displaystyle (\pm (P_{i}+P_{i-1}),0)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\pm (P_{i}+P_{i-1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\pm (P_{i}+P_{i-1}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717ab1a3045eae105929183babb5e86afcccca21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.148ex; height:2.843ex;" alt="{\displaystyle (0,\pm (P_{i}+P_{i-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 (\pm P_{i},\pm P_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>±</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>±</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pm P_{i},\pm P_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63e7f71b9672df5c84d6c3ac077b09cf959ed18c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.043ex; height:2.843ex;" alt="{\displaystyle (\pm P_{i},\pm P_{i})}"></span> form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles. </p> <div class="mw-heading mw-heading2"><h2 id="Primes_and_squares">Primes and squares</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=3" title="Edit section: Primes and squares"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Pell prime</b> is a Pell number that is <a href="/wiki/Prime_number" title="Prime number">prime</a>. The first few Pell primes are </p> <dl><dd>2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, ... (sequence <span class="nowrap external"><a href="//oeis.org/A086383" class="extiw" title="oeis:A086383">A086383</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>The indices of these primes within the sequence of all Pell numbers are </p> <dl><dd>2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, ... (sequence <span class="nowrap external"><a href="//oeis.org/A096650" class="extiw" title="oeis:A096650">A096650</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>These indices are all themselves prime. As with the Fibonacci numbers, a Pell number <i>P</i><sub><i>n</i></sub> can only be prime if <i>n</i> itself is prime, because if <i>d</i> is a <a href="/wiki/Divisor" title="Divisor">divisor</a> of <i>n</i> then <i>P</i><sub><i>d</i></sub> is a divisor of <i>P</i><sub><i>n</i></sub>. </p><p>The only Pell numbers that are <a href="/wiki/Square_number" title="Square number">squares</a>, <a href="/wiki/Perfect_cube" class="mw-redirect" title="Perfect cube">cubes</a>, or any higher <a href="/wiki/Perfect_power" title="Perfect power">power of an integer</a> are 0, 1, and 169 = 13<sup>2</sup>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>However, despite having so few squares or other powers, Pell numbers have a close connection to <a href="/wiki/Square_triangular_number" title="Square triangular number">square triangular numbers</a>.<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> Specifically, these numbers arise from the following identity of Pell numbers: </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 {\bigl (}\left(P_{k-1}+P_{k}\right)\cdot P_{k}{\bigr )}^{2}={\frac {\left(P_{k-1}+P_{k}\right)^{2}\cdot \left(\left(P_{k-1}+P_{k}\right)^{2}-(-1)^{k}\right)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}\left(P_{k-1}+P_{k}\right)\cdot P_{k}{\bigr )}^{2}={\frac {\left(P_{k-1}+P_{k}\right)^{2}\cdot \left(\left(P_{k-1}+P_{k}\right)^{2}-(-1)^{k}\right)}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8432c1d6816a203958c60b146f186c5359193ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.421ex; height:7.509ex;" alt="{\displaystyle {\bigl (}\left(P_{k-1}+P_{k}\right)\cdot P_{k}{\bigr )}^{2}={\frac {\left(P_{k-1}+P_{k}\right)^{2}\cdot \left(\left(P_{k-1}+P_{k}\right)^{2}-(-1)^{k}\right)}{2}}.}"></span></dd></dl> <p>The left side of this identity describes a square number, while the right side describes a <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a>, so the result is a square triangular number. </p><p>Falcón and Díaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to <i>P</i><sub>4<i>n</i> +1</sub> is always a square: </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 _{i=0}^{4n+1}P_{i}=\left(\sum _{r=0}^{n}2^{r}{2n+1 \choose 2r}\right)^{\!2}=\left(P_{2n}+P_{2n+1}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{4n+1}P_{i}=\left(\sum _{r=0}^{n}2^{r}{2n+1 \choose 2r}\right)^{\!2}=\left(P_{2n}+P_{2n+1}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3e5ef7ca040920638141bd1695ad1574beec7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.509ex; height:8.009ex;" alt="{\displaystyle \sum _{i=0}^{4n+1}P_{i}=\left(\sum _{r=0}^{n}2^{r}{2n+1 \choose 2r}\right)^{\!2}=\left(P_{2n}+P_{2n+1}\right)^{2}.}"></span></dd></dl> <p>For instance, the sum of the Pell numbers up to <i>P</i><sub>5</sub>, <span class="nowrap">0 + 1 + 2 + 5 + 12 + 29 = 49</span>, is the square of <span class="nowrap"><i>P</i><sub>2</sub> + <i>P</i><sub>3</sub> = 2 + 5 = 7</span>. The numbers <span class="nowrap"><i>P</i><sub>2<i>n</i></sub> + <i>P</i><sub>2<i>n</i> +1</sub></span> forming the square roots of these sums, </p> <dl><dd>1, 7, 41, 239, 1393, 8119, 47321, … (sequence <span class="nowrap external"><a href="//oeis.org/A002315" class="extiw" title="oeis:A002315">A002315</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>are known as the <a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_number" class="mw-redirect" title="Newman–Shanks–Williams number">Newman–Shanks–Williams (NSW) numbers</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Pythagorean_triples">Pythagorean triples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=4" title="Edit section: Pythagorean triples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pell_right_triangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Pell_right_triangles.svg/300px-Pell_right_triangles.svg.png" decoding="async" width="300" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Pell_right_triangles.svg/450px-Pell_right_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Pell_right_triangles.svg/600px-Pell_right_triangles.svg.png 2x" data-file-width="252" data-file-height="243" /></a><figcaption>Integer right triangles with nearly equal legs, derived from the Pell numbers.</figcaption></figure> <p>If a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> has integer side lengths <i>a</i>, <i>b</i>, <i>c</i> (necessarily satisfying the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> <span class="nowrap"><i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> = <i>c</i><sup>2</sup></span>), then (<i>a</i>,<i>b</i>,<i>c</i>) is known as a <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a>. As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which <i>a</i> and <i>b</i> are one unit apart, corresponding to right triangles that are nearly isosceles. Each such triple has the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2}=P_{2n+1}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msubsup> <mi>P</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> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>P</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> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2}=P_{2n+1}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104873443cc8e978abe6d95de57f943b3eb36eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.703ex; height:3.343ex;" alt="{\displaystyle \left(2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2}=P_{2n+1}\right).}"></span></dd></dl> <p>The sequence of Pythagorean triples formed in this way is </p> <dl><dd>(4,3,5), (20,21,29), (120,119,169), (696,697,985), …</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Pell–Lucas_numbers"><span id="Pell.E2.80.93Lucas_numbers"></span>Pell–Lucas numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=5" title="Edit section: Pell–Lucas numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>companion Pell numbers</b> or <b>Pell–Lucas numbers</b> are defined by the <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</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 Q_{n}={\begin{cases}2&{\mbox{if }}n=0;\\2&{\mbox{if }}n=1;\\2Q_{n-1}+Q_{n-2}&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}={\begin{cases}2&{\mbox{if }}n=0;\\2&{\mbox{if }}n=1;\\2Q_{n-1}+Q_{n-2}&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e274516111782cbe04e8f189a5e4cc9555cfc285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:35.837ex; height:8.509ex;" alt="{\displaystyle Q_{n}={\begin{cases}2&{\mbox{if }}n=0;\\2&{\mbox{if }}n=1;\\2Q_{n-1}+Q_{n-2}&{\mbox{otherwise.}}\end{cases}}}"></span></dd></dl> <p>In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and <span class="nowrap">82 = 2 × 34 + 14 = 70 + 12.</span> The first few terms of the sequence are (sequence <span class="nowrap external"><a href="//oeis.org/A002203" class="extiw" title="oeis:A002203">A002203</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, 2, <a href="/wiki/6_(number)" class="mw-redirect" title="6 (number)">6</a>, <a href="/wiki/14_(number)" title="14 (number)">14</a>, <a href="/wiki/34_(number)" title="34 (number)">34</a>, <a href="/wiki/82_(number)" title="82 (number)">82</a>, 198, <a href="/wiki/478_(number)" class="mw-redirect" title="478 (number)">478</a>, … </p><p>Like the relationship between <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> and <a href="/wiki/Lucas_number" title="Lucas number">Lucas numbers</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 Q_{n}={\frac {P_{2n}}{P_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}={\frac {P_{2n}}{P_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971b83a2c67cc33c00c82dcb1f25b7bcc006f122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.524ex; height:5.676ex;" alt="{\displaystyle Q_{n}={\frac {P_{2n}}{P_{n}}}}"></span></dd></dl> <p>for all <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <i>n</i>. </p><p>The companion Pell numbers can be expressed by the closed form formula </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 Q_{n}=\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}=\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1706a574fdedd23fb4084a5fee5f7f7991f7d953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.541ex; height:3.343ex;" alt="{\displaystyle Q_{n}=\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}.}"></span></dd></dl> <p>These numbers are all <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a>; each such number is twice the numerator in one of the rational approximations 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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> discussed above. </p><p>Like the Lucas sequence, if a Pell–Lucas number <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>Q<sub>n</sub></i> is prime, it is necessary that <i>n</i> be either prime or a <a href="/wiki/Power_of_2" class="mw-redirect" title="Power of 2">power of 2</a>. The Pell–Lucas primes are </p> <dl><dd>3, 7, 17, 41, 239, 577, … (sequence <span class="nowrap external"><a href="//oeis.org/A086395" class="extiw" title="oeis:A086395">A086395</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>For these <i>n</i> are </p> <dl><dd>2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, … (sequence <span class="nowrap external"><a href="//oeis.org/A099088" class="extiw" title="oeis:A099088">A099088</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Computations_and_connections">Computations and connections</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=6" title="Edit section: Computations and connections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following table gives the first few powers of the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> <i>δ</i> = <i>δ</i><sub> S</sub> = 1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> and its <a href="/wiki/Conjugate_(square_roots)" title="Conjugate (square roots)">conjugate</a> <span style="text-decoration:overline;"><i>δ</i></span> = 1 − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <th><i>n</i> </th> <th>(1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>)<sup><i>n</i></sup> </th> <th>(1 − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>)<sup><i>n</i></sup> </th></tr> <tr> <th>0 </th> <td>1 + 0<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 1 </td> <td>1 − 0<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 1 </td></tr> <tr> <th>1 </th> <td>1 + 1<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 2.41421… </td> <td>1 − 1<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.41421… </td></tr> <tr> <th>2 </th> <td>3 + 2<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 5.82842… </td> <td>3 − 2<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.17157… </td></tr> <tr> <th>3 </th> <td>7 + 5<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 14.07106… </td> <td>7 − 5<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.07106… </td></tr> <tr> <th>4 </th> <td>17 + 12<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 33.97056… </td> <td>17 − 12<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.02943… </td></tr> <tr> <th>5 </th> <td>41 + 29<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 82.01219… </td> <td>41 − 29<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.01219… </td></tr> <tr> <th>6 </th> <td>99 + 70<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 197.9949… </td> <td>99 − 70<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.0050… </td></tr> <tr> <th>7 </th> <td>239 + 169<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 478.00209… </td> <td>239 − 169<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.00209… </td></tr> <tr> <th>8 </th> <td>577 + 408<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 1153.99913… </td> <td>577 − 408<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.00086… </td></tr> <tr> <th>9 </th> <td>1393 + 985<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 2786.00035… </td> <td>1393 − 985<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.00035… </td></tr> <tr> <th>10 </th> <td>3363 + 2378<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 6725.99985… </td> <td>3363 − 2378<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.00014… </td></tr> <tr> <th>11 </th> <td>8119 + 5741<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 16238.00006… </td> <td>8119 − 5741<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = −0.00006… </td></tr> <tr> <th>12 </th> <td>19601 + 13860<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 39201.99997… </td> <td>19601 − 13860<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 0.00002… </td></tr></tbody></table></dd></dl> <p>The <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> are the half-companion Pell numbers <i>H<sub>n</sub></i> and the Pell numbers <i>P<sub>n</sub></i> which are the (non-negative) solutions to <span class="nowrap"><i>H</i><sup>  2</sup> − 2<i>P</i><sup>  2</sup> = ±1</span>. A <a href="/wiki/Square_triangular_number" title="Square triangular number">square triangular number</a> is a number </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 N={\frac {t(t+1)}{2}}=s^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\frac {t(t+1)}{2}}=s^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6897293bdf096db4f06aab2d28961a5740ccc640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.38ex; height:5.676ex;" alt="{\displaystyle N={\frac {t(t+1)}{2}}=s^{2},}"></span></dd></dl> <p>which is both the <i>t</i>-th triangular number and the <i>s</i>-th square number. A <i>near-isosceles Pythagorean triple</i> is an integer solution to <span class="nowrap"><i>a</i><sup> 2</sup> + <i>b</i><sup> 2</sup> = <i>c</i><sup> 2</sup></span> where <span class="nowrap"><i>a</i> + 1 = <i>b</i></span>. </p><p>The next table shows that splitting the <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> number <i>H<sub>n</sub></i> into nearly equal halves gives a square triangular number when <i>n</i> is even and a near isosceles Pythagorean triple when <i>n</i> is odd. All solutions arise in this manner. </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <th><i>n</i> </th> <th><i>H<sub>n</sub></i> </th> <th><i>P<sub>n</sub></i> </th> <th><i>t</i> </th> <th><i>t</i> + 1 </th> <th><i>s</i> </th> <th><i>a</i> </th> <th><i>b</i> </th> <th><i>c</i> </th></tr> <tr> <th>0 </th> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>1 </th> <td>1 </td> <td>1 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>0 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>2 </th> <td>3 </td> <td>2 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>3 </th> <td>7 </td> <td>5 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>3 </td> <td>4 </td> <td>5 </td></tr> <tr> <th>4 </th> <td>17 </td> <td>12 </td> <td>8 </td> <td>9 </td> <td>6 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>5 </th> <td>41 </td> <td>29 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>20 </td> <td>21 </td> <td>29 </td></tr> <tr> <th>6 </th> <td>99 </td> <td>70 </td> <td>49 </td> <td>50 </td> <td>35 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>7 </th> <td>239 </td> <td>169 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>119 </td> <td>120 </td> <td>169 </td></tr> <tr> <th>8 </th> <td>577 </td> <td>408 </td> <td>288 </td> <td>289 </td> <td>204 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>9 </th> <td>1393 </td> <td>985 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>696 </td> <td>697 </td> <td>985 </td></tr> <tr> <th>10 </th> <td>3363 </td> <td>2378 </td> <td>1681 </td> <td>1682 </td> <td>1189 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>11 </th> <td>8119 </td> <td>5741 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td>4059 </td> <td>4060 </td> <td>5741 </td></tr> <tr> <th>12 </th> <td>19601 </td> <td>13860 </td> <td>9800 </td> <td>9801 </td> <td>6930 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=7" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The half-companion Pell numbers <i>H<sub>n</sub></i> and the Pell numbers <i>P<sub>n</sub></i> can be derived in a number of easily equivalent ways. </p> <div class="mw-heading mw-heading4"><h4 id="Raising_to_powers">Raising to powers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=8" title="Edit section: Raising to powers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824bd67e5988cd241b02fb774e899cbd7a1d8f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.347ex; height:3.343ex;" alt="{\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {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 \left(1-{\sqrt {2}}\right)^{n}=H_{n}-P_{n}{\sqrt {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\sqrt {2}}\right)^{n}=H_{n}-P_{n}{\sqrt {2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf676f28066d7f496a8ca5be706c8f34fead40d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.994ex; height:3.343ex;" alt="{\displaystyle \left(1-{\sqrt {2}}\right)^{n}=H_{n}-P_{n}{\sqrt {2}}.}"></span></dd></dl> <p>From this it follows that there are <i>closed forms</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c113d73d68a6df1777dc352f352c9712af05d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.47ex; height:6.343ex;" alt="{\displaystyle H_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}}{2}}.}"></span></dd></dl> <p>and </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_{n}{\sqrt {2}}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}{\sqrt {2}}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e825f96c6c65b4ebd04dca04817207b9d1142a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.13ex; height:6.343ex;" alt="{\displaystyle P_{n}{\sqrt {2}}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2}}.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Paired_recurrences">Paired recurrences</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=9" title="Edit section: Paired recurrences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}={\begin{cases}1&{\mbox{if }}n=0;\\H_{n-1}+2P_{n-1}&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}={\begin{cases}1&{\mbox{if }}n=0;\\H_{n-1}+2P_{n-1}&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a71bc0c61f10822b04872e4734d0b25d283172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.354ex; height:6.176ex;" alt="{\displaystyle H_{n}={\begin{cases}1&{\mbox{if }}n=0;\\H_{n-1}+2P_{n-1}&{\mbox{otherwise.}}\end{cases}}}"></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 P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\H_{n-1}+P_{n-1}&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\H_{n-1}+P_{n-1}&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60585b87126216dd8c1d4c2a80cd1ec9485ede5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.752ex; height:6.176ex;" alt="{\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\H_{n-1}+P_{n-1}&{\mbox{otherwise.}}\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Reciprocal_recurrence_formulas">Reciprocal recurrence formulas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=10" title="Edit section: Reciprocal recurrence formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>n</i> be at least 2. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}=(3P_{n}-P_{n-2})/2=3P_{n-1}+P_{n-2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <mn>3</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}=(3P_{n}-P_{n-2})/2=3P_{n-1}+P_{n-2};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e4709b9062e267c16114a0c65390dda105181f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.278ex; height:2.843ex;" alt="{\displaystyle H_{n}=(3P_{n}-P_{n-2})/2=3P_{n-1}+P_{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 P_{n}=(3H_{n}-H_{n-2})/4=(3H_{n-1}+H_{n-2})/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=(3H_{n}-H_{n-2})/4=(3H_{n-1}+H_{n-2})/2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5080cf3c26d259ce6452816b32b0e5bb005570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.729ex; height:2.843ex;" alt="{\displaystyle P_{n}=(3H_{n}-H_{n-2})/4=(3H_{n-1}+H_{n-2})/2.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Matrix_formulations">Matrix formulations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=11" title="Edit section: Matrix formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\begin{pmatrix}H_{n}\\P_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}{\begin{pmatrix}H_{n-1}\\P_{n-1}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}{\begin{pmatrix}1\\0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}H_{n}\\P_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}{\begin{pmatrix}H_{n-1}\\P_{n-1}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}{\begin{pmatrix}1\\0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a331b8afdffe73674933ec6991468356a61d3658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.227ex; margin-bottom: -0.278ex; width:47.783ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}H_{n}\\P_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}{\begin{pmatrix}H_{n-1}\\P_{n-1}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}{\begin{pmatrix}1\\0\end{pmatrix}}.}"></span></dd></dl> <p>So </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 {\begin{pmatrix}H_{n}&2P_{n}\\P_{n}&H_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}H_{n}&2P_{n}\\P_{n}&H_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff42d34b40351f1a1539636a35a60d740895c92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.302ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}H_{n}&2P_{n}\\P_{n}&H_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Approximations">Approximations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=12" title="Edit section: Approximations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The difference between <i>H<sub>n</sub></i> and <i>P<sub>n</sub></i><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\sqrt {2}}\right)^{n}\approx (-0.41421)^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>≈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>0.41421</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\sqrt {2}}\right)^{n}\approx (-0.41421)^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540df6f98c57f80477f15541cc8534950ac818fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.652ex; height:3.343ex;" alt="{\displaystyle \left(1-{\sqrt {2}}\right)^{n}\approx (-0.41421)^{n},}"></span></dd></dl> <p>which goes rapidly to zero. So </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+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824bd67e5988cd241b02fb774e899cbd7a1d8f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.347ex; height:3.343ex;" alt="{\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}"></span></dd></dl> <p>is extremely close to 2<i>H<sub>n</sub></i>. </p><p>From this last observation it follows that the integer ratios <i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">H<sub>n</sub></span><span class="sr-only">/</span><span class="den">P<sub>n</sub></span></span>⁠</span></i> rapidly approach <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>; and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>H<sub>n</sub></i></span><span class="sr-only">/</span><span class="den"><i>H</i><sub><i>n</i> −1</sub></span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>P<sub>n</sub></i></span><span class="sr-only">/</span><span class="den"><i>P</i><sub><i>n</i> −1</sub></span></span>⁠</span> rapidly approach 1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="H_2_−_2P_2_=_±1"><span id="H_2_.E2.88.92_2P_2_.3D_.C2.B11"></span><i>H</i><sup>  2</sup> − 2<i>P</i><sup>  2</sup> = ±1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=13" title="Edit section: H 2 − 2P 2 = ±1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> is irrational, we cannot have <i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">H</span><span class="sr-only">/</span><span class="den">P</span></span>⁠</span></i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>, 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 {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}}{P^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}}{P^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6009696a781b94c8300637685b8006892f22674c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.613ex; height:5.843ex;" alt="{\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}}{P^{2}}}.}"></span></dd></dl> <p>The best we can achieve is either </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}-1}{P^{2}}}\quad {\mbox{or}}\quad {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}+1}{P^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>or</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}-1}{P^{2}}}\quad {\mbox{or}}\quad {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}+1}{P^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ba34097cbd130a8d90b1c0c12baec0d3adb740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.305ex; height:5.843ex;" alt="{\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}-1}{P^{2}}}\quad {\mbox{or}}\quad {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}+1}{P^{2}}}.}"></span></dd></dl> <p>The (non-negative) solutions to <span class="nowrap"><i>H</i><sup>  2</sup> − 2<i>P</i><sup>  2</sup> = 1</span> are exactly the pairs <span class="nowrap">(<i>H<sub>n</sub></i>, <i>P<sub>n</sub></i>)</span> with <i>n</i> even, and the solutions to <span class="nowrap"><i>H</i><sup>  2</sup> − 2<i>P</i><sup>  2</sup> = −1</span> are exactly the pairs <span class="nowrap">(<i>H<sub>n</sub></i>, <i>P<sub>n</sub></i>)</span> with <i>n</i> odd. To see this, note first 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 H_{n+1}^{2}-2P_{n+1}^{2}=\left(H_{n}+2P_{n}\right)^{2}-2\left(H_{n}+P_{n}\right)^{2}=-\left(H_{n}^{2}-2P_{n}^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</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> <msubsup> <mi>P</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> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n+1}^{2}-2P_{n+1}^{2}=\left(H_{n}+2P_{n}\right)^{2}-2\left(H_{n}+P_{n}\right)^{2}=-\left(H_{n}^{2}-2P_{n}^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900b7c67b5e8690a17ad2f14ef07cded47790573" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:63.951ex; height:3.676ex;" alt="{\displaystyle H_{n+1}^{2}-2P_{n+1}^{2}=\left(H_{n}+2P_{n}\right)^{2}-2\left(H_{n}+P_{n}\right)^{2}=-\left(H_{n}^{2}-2P_{n}^{2}\right),}"></span></dd></dl> <p>so that these differences, starting with <span class="nowrap"><i>H</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">  2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">  0</sub></span></span> − 2<i>P</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">  2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">  0</sub></span></span> = 1</span>, are alternately 1 and −1. Then note that every positive solution comes in this way from a solution with smaller integers 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 (2P-H)^{2}-2(H-P)^{2}=-\left(H^{2}-2P^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>P</mi> <mo>−</mo> <mi>H</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>H</mi> <mo>−</mo> <mi>P</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2P-H)^{2}-2(H-P)^{2}=-\left(H^{2}-2P^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4789ae391a191c7018facf59c6ae27cb6a5d167b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.685ex; height:3.343ex;" alt="{\displaystyle (2P-H)^{2}-2(H-P)^{2}=-\left(H^{2}-2P^{2}\right).}"></span></dd></dl> <p>The smaller solution also has positive integers, with the one exception: <span class="nowrap"><i>H</i> = <i>P</i> = 1</span> which comes from <i>H</i><sub>0</sub> = 1 and <i>P</i><sub>0</sub> = 0. </p> <div class="mw-heading mw-heading3"><h3 id="Square_triangular_numbers">Square triangular numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=14" title="Edit section: Square triangular numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular number</a></div> <p>The required equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {t(t+1)}{2}}=s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {t(t+1)}{2}}=s^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b459fd38d9ffbaf5d3c401afa739919073ae6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.571ex; height:5.676ex;" alt="{\displaystyle {\frac {t(t+1)}{2}}=s^{2}}"></span></dd></dl> <p>is 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 4t^{2}+4t+1=8s^{2}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4t^{2}+4t+1=8s^{2}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef1013562d021c01ab5dd7f718c1400fb1e6be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.957ex; height:3.009ex;" alt="{\displaystyle 4t^{2}+4t+1=8s^{2}+1,}"></span> which becomes <span class="nowrap"><i>H</i><sup>  2</sup> = 2<i>P</i><sup>  2</sup> + 1</span> with the substitutions <i>H</i> = 2<i>t</i> + 1 and <i>P</i> = 2<i>s</i>. Hence the <i>n</i>-th solution is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}={\frac {H_{2n}-1}{2}}\quad {\mbox{and}}\quad s_{n}={\frac {P_{2n}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>and</mtext> </mstyle> </mrow> <mspace width="1em" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}={\frac {H_{2n}-1}{2}}\quad {\mbox{and}}\quad s_{n}={\frac {P_{2n}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4baca4b6e4cccef4f7552edb2ed1227518bbcbc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.783ex; height:5.176ex;" alt="{\displaystyle t_{n}={\frac {H_{2n}-1}{2}}\quad {\mbox{and}}\quad s_{n}={\frac {P_{2n}}{2}}.}"></span></dd></dl> <p>Observe that <i>t</i> and <i>t</i> + 1 are relatively prime, so that <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>t</i> (<i>t</i> + 1)</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> = <i>s</i><sup> 2</sup> happens exactly when they are adjacent integers, one a square <i>H</i><sup>  2</sup> and the other twice a square 2<i>P</i><sup>  2</sup>. Since we know all solutions of that equation, we also have </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 t_{n}={\begin{cases}2P_{n}^{2}&{\mbox{if }}n{\mbox{ is even}};\\H_{n}^{2}&{\mbox{if }}n{\mbox{ is odd.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> is even</mtext> </mstyle> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> is odd.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}={\begin{cases}2P_{n}^{2}&{\mbox{if }}n{\mbox{ is even}};\\H_{n}^{2}&{\mbox{if }}n{\mbox{ is odd.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a0989816222d5375f23ec6be2448dc62f012a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.302ex; height:6.176ex;" alt="{\displaystyle t_{n}={\begin{cases}2P_{n}^{2}&{\mbox{if }}n{\mbox{ is even}};\\H_{n}^{2}&{\mbox{if }}n{\mbox{ is odd.}}\end{cases}}}"></span></dd></dl> <p>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 s_{n}=H_{n}P_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=H_{n}P_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5833fb0f49bfe59f51439941f8fbc3895a9625ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle s_{n}=H_{n}P_{n}.}"></span> </p><p>This alternate expression is seen in the next table. </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <th><i>n</i> </th> <th><i>H<sub>n</sub></i> </th> <th><i>P<sub>n</sub></i> </th> <th><i>t</i> </th> <th><i>t</i> + 1 </th> <th><i>s</i> </th> <th><i>a</i> </th> <th><i>b</i> </th> <th><i>c</i> </th></tr> <tr> <th>0 </th> <td>1 </td> <td>0 </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td> <td style="background: grey;">  </td></tr> <tr> <th>1 </th> <td>1 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>3 </td> <td>4 </td> <td>5 </td></tr> <tr> <th>2 </th> <td>3 </td> <td>2 </td> <td>8 </td> <td>9 </td> <td>6 </td> <td>20 </td> <td>21 </td> <td>29 </td></tr> <tr> <th>3 </th> <td>7 </td> <td>5 </td> <td>49 </td> <td>50 </td> <td>35 </td> <td>119 </td> <td>120 </td> <td>169 </td></tr> <tr> <th>4 </th> <td>17 </td> <td>12 </td> <td>288 </td> <td>289 </td> <td>204 </td> <td>696 </td> <td>697 </td> <td>985 </td></tr> <tr> <th>5 </th> <td>41 </td> <td>29 </td> <td>1681 </td> <td>1682 </td> <td>1189 </td> <td>4059 </td> <td>4060 </td> <td>5741 </td></tr> <tr> <th>6 </th> <td>99 </td> <td>70 </td> <td>9800 </td> <td>9801 </td> <td>6930 </td> <td>23660 </td> <td>23661 </td> <td>33461 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_triples_2">Pythagorean triples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=15" title="Edit section: Pythagorean triples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equality <span class="nowrap"><i>c</i><sup> 2</sup> = <i>a</i><sup> 2</sup> + (<i>a</i> + 1)<sup> 2</sup> = 2<i>a</i><sup> 2</sup> + 2<i>a</i> + 1</span> occurs exactly when <span class="nowrap">2<i>c</i><sup> 2</sup> = 4<i>a</i><sup> 2</sup> + 4<i>a</i> + 2</span> which becomes <span class="nowrap">2<i>P</i><sup>  2</sup> = <i>H</i><sup>  2</sup> + 1</span> with the substitutions <span class="nowrap"><i>H</i> = 2<i>a</i> + 1</span> and <span class="nowrap"><i>P</i> = <i>c</i></span>. Hence the <i>n</i>-th solution is <span class="nowrap"><i>a<sub>n</sub></i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>H</i><sub>2<i>n</i> +1</sub> − 1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and <span class="nowrap"><i>c<sub>n</sub></i> = <i>P</i><sub>2<i>n</i> +1</sub></span>. </p><p>The table above shows that, in one order or the other, <i>a<sub>n</sub></i> and <span class="nowrap"><i>b<sub>n</sub></i> = <i>a<sub>n</sub></i> + 1</span> are <span class="nowrap"><i>H<sub>n</sub></i> <i>H</i><sub><i>n</i> +1</sub></span> and <span class="nowrap">2<i>P<sub>n</sub></i> <i>P</i><sub><i>n</i> +1</sub></span> while <span class="nowrap"><i>c<sub>n</sub></i> = <i>H</i><sub><i>n</i> +1</sub> <i>P<sub>n</sub></i> + <i>P</i><sub><i>n</i> +1</sub> <i>H<sub>n</sub></i></span>. </p> <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=Pell_number&action=edit&section=16" 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"><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 instance, Sellers (2002) proves that the number of <a href="/wiki/Perfect_matching" title="Perfect matching">perfect matchings</a> in the <a href="/wiki/Cartesian_product_of_graphs" title="Cartesian product of graphs">Cartesian product</a> of a <a href="/wiki/Path_graph" title="Path graph">path graph</a> and the <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <i>K</i><sub>4</sub> − <i>e</i> can be calculated as the product of a Pell number with the corresponding Fibonacci number.</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">For the matrix formula and its consequences see Ercolano (1979) and Kilic and Tasci (2005). Additional identities for the Pell numbers are listed by Horadam (1971) and Bicknell (1975).</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">As recorded in the <a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a>; see e.g. Dutka (1986), who cites Thibaut (1875) for this information.</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">See Knorr (1976) for the fifth century date, which matches <a href="/wiki/Proclus" title="Proclus">Proclus</a>' claim that the side and diameter numbers were discovered by the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a>. For more detailed exploration of later Greek knowledge of these numbers see Thompson (1929), Vedova (1951), Ridenhour (1986), Knorr (1998), and Filep (1999).</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">For instance, as several of the references from the previous note observe, in <a href="/wiki/Plato%27s_Republic" class="mw-redirect" title="Plato's Republic">Plato's Republic</a> there is a reference to the "rational diameter of 5", by which <a href="/wiki/Plato" title="Plato">Plato</a> means 7, the numerator of the approximation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> of which 5 is the denominator.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHeath1921" class="citation cs2"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Sir Thomas Little</a> (1921), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=drnY3Vjix3kC&pg=PA112"><i>History of Greek Mathematics: From Thales to Euclid</i></a>, Courier Dover Publications, p. 112, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486240732" title="Special:BookSources/9780486240732"><bdi>9780486240732</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Greek+Mathematics%3A+From+Thales+to+Euclid&rft.pages=112&rft.pub=Courier+Dover+Publications&rft.date=1921&rft.isbn=9780486240732&rft.aulast=Heath&rft.aufirst=Sir+Thomas+Little&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdrnY3Vjix3kC%26pg%3DPA112&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Pethő (1992); Cohn (1996). Although the Fibonacci numbers are defined by a very similar recurrence to the Pell numbers, Cohn writes that an analogous result for the Fibonacci numbers seems much more difficult to prove. (However, this was proven in 2006 by Bugeaud et al.)</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">Sesskin (1962). See the <a href="/wiki/Square_triangular_number" title="Square triangular number">square triangular number</a> article for a more detailed derivation.</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=Pell_number&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBicknell,_Marjorie1975" class="citation journal cs1">Bicknell, Marjorie (1975). "A primer on the Pell sequence and related sequences". <i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>. <b>13</b> (4): 345–349. <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%2F00150517.1975.12430627">10.1080/00150517.1975.12430627</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0387173">0387173</a>.</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+primer+on+the+Pell+sequence+and+related+sequences&rft.volume=13&rft.issue=4&rft.pages=345-349&rft.date=1975&rft_id=info%3Adoi%2F10.1080%2F00150517.1975.12430627&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0387173%23id-name%3DMR&rft.au=Bicknell%2C+Marjorie&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn,_J._H._E.1996" class="citation journal cs1">Cohn, J. H. E. (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0017089500031207">"Perfect Pell powers"</a>. <i><a href="/wiki/Glasgow_Mathematical_Journal" title="Glasgow Mathematical Journal">Glasgow Mathematical Journal</a></i>. <b>38</b> (1): 19–20. <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/10.1017%2FS0017089500031207">10.1017/S0017089500031207</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1373953">1373953</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Glasgow+Mathematical+Journal&rft.atitle=Perfect+Pell+powers&rft.volume=38&rft.issue=1&rft.pages=19-20&rft.date=1996&rft_id=info%3Adoi%2F10.1017%2FS0017089500031207&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1373953%23id-name%3DMR&rft.au=Cohn%2C+J.+H.+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252FS0017089500031207&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDutka,_Jacques1986" class="citation journal cs1">Dutka, Jacques (1986). "On square roots and their representations". <i><a href="/wiki/Archive_for_History_of_Exact_Sciences" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a></i>. <b>36</b> (1): 21–39. <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%2FBF00357439">10.1007/BF00357439</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0863340">0863340</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:122277481">122277481</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=On+square+roots+and+their+representations&rft.volume=36&rft.issue=1&rft.pages=21-39&rft.date=1986&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0863340%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122277481%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00357439&rft.au=Dutka%2C+Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErcolano,_Joseph1979" class="citation journal cs1">Ercolano, Joseph (1979). "Matrix generators of Pell sequences". <i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>. <b>17</b> (1): 71–77. <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%2F00150517.1979.12430264">10.1080/00150517.1979.12430264</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0525602">0525602</a>.</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=Matrix+generators+of+Pell+sequences&rft.volume=17&rft.issue=1&rft.pages=71-77&rft.date=1979&rft_id=info%3Adoi%2F10.1080%2F00150517.1979.12430264&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0525602%23id-name%3DMR&rft.au=Ercolano%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFilep,_László1999" class="citation journal cs1"><a href="/wiki/L%C3%A1szl%C3%B3_Filep" title="László Filep">Filep, László</a> (1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200706013018/https://www.emis.de/journals/AMAPN/vol15/filep.pdf">"Pythagorean side and diagonal numbers"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/w/index.php?title=Acta_Mathematica_Academiae_Paedagogicae_Ny%C3%ADregyh%C3%A1ziensis&action=edit&redlink=1" class="new" title="Acta Mathematica Academiae Paedagogicae Nyíregyháziensis (page does not exist)">Acta Mathematica Academiae Paedagogicae Nyíregyháziensis</a></i>. <b>15</b>: 1–7. Archived from <a rel="nofollow" class="external text" href="http://www.emis.de/journals/AMAPN/vol15/filep.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-07-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-01-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica+Academiae+Paedagogicae+Ny%C3%ADregyh%C3%A1ziensis&rft.atitle=Pythagorean+side+and+diagonal+numbers&rft.volume=15&rft.pages=1-7&rft.date=1999&rft.au=Filep%2C+L%C3%A1szl%C3%B3&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FAMAPN%2Fvol15%2Ffilep.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoradam,_A._F.1971" class="citation journal cs1">Horadam, A. F. (1971). "Pell identities". <i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>. <b>9</b> (3): 245–252, 263. <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%2F00150517.1971.12431004">10.1080/00150517.1971.12431004</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0308029">0308029</a>.</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=Pell+identities&rft.volume=9&rft.issue=3&rft.pages=245-252%2C+263&rft.date=1971&rft_id=info%3Adoi%2F10.1080%2F00150517.1971.12431004&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0308029%23id-name%3DMR&rft.au=Horadam%2C+A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKilic,_EmrahTasci,_Dursun2005" class="citation journal cs1">Kilic, Emrah; Tasci, Dursun (2005). "The linear algebra of the Pell matrix". <i>Boletín de la Sociedad Matemática Mexicana, Tercera Serie</i>. <b>11</b> (2): 163–174. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2207722">2207722</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bolet%C3%ADn+de+la+Sociedad+Matem%C3%A1tica+Mexicana%2C+Tercera+Serie&rft.atitle=The+linear+algebra+of+the+Pell+matrix&rft.volume=11&rft.issue=2&rft.pages=163-174&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2207722%23id-name%3DMR&rft.au=Kilic%2C+Emrah&rft.au=Tasci%2C+Dursun&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnorr,_Wilbur1976" class="citation journal cs1"><a href="/wiki/Wilbur_Knorr" title="Wilbur Knorr">Knorr, Wilbur</a> (1976). "Archimedes and the measurement of the circle: A new interpretation". <i><a href="/wiki/Archive_for_History_of_Exact_Sciences" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a></i>. <b>15</b> (2): 115–140. <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%2FBF00348496">10.1007/BF00348496</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0497462">0497462</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:120954547">120954547</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Archimedes+and+the+measurement+of+the+circle%3A+A+new+interpretation&rft.volume=15&rft.issue=2&rft.pages=115-140&rft.date=1976&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0497462%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120954547%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00348496&rft.au=Knorr%2C+Wilbur&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnorr,_Wilbur1998" class="citation journal cs1"><a href="/wiki/Wilbur_Knorr" title="Wilbur Knorr">Knorr, Wilbur</a> (1998). "<span class="cs1-kern-left"></span>"Rational diameters" and the discovery of incommensurability". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>105</b> (5): 421–429. <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%2F3109803">10.2307/3109803</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/3109803">3109803</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=%22Rational+diameters%22+and+the+discovery+of+incommensurability&rft.volume=105&rft.issue=5&rft.pages=421-429&rft.date=1998&rft_id=info%3Adoi%2F10.2307%2F3109803&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3109803%23id-name%3DJSTOR&rft.au=Knorr%2C+Wilbur&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth,_Donald_E.1994" class="citation journal cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1994). "Leaper graphs". <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>. <b>78</b> (483): 274–297. <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/math.CO/9411240">math.CO/9411240</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994math.....11240K">1994math.....11240K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3620202">10.2307/3620202</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/3620202">3620202</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:16856513">16856513</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+Gazette&rft.atitle=Leaper+graphs&rft.volume=78&rft.issue=483&rft.pages=274-297&rft.date=1994&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3620202%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1994math.....11240K&rft_id=info%3Aarxiv%2Fmath.CO%2F9411240&rft_id=info%3Adoi%2F10.2307%2F3620202&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16856513%23id-name%3DS2CID&rft.au=Knuth%2C+Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin,_Artemas1875" class="citation journal cs1"><a href="/wiki/Artemas_Martin" title="Artemas Martin">Martin, Artemas</a> (1875). "Rational right angled triangles nearly isosceles". <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">The Analyst</a></i>. <b>3</b> (2): 47–50. <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%2F2635906">10.2307/2635906</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/2635906">2635906</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Analyst&rft.atitle=Rational+right+angled+triangles+nearly+isosceles&rft.volume=3&rft.issue=2&rft.pages=47-50&rft.date=1875&rft_id=info%3Adoi%2F10.2307%2F2635906&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2635906%23id-name%3DJSTOR&rft.au=Martin%2C+Artemas&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPethő,_A.1992" class="citation conference cs1">Pethő, A. (1992). "The Pell sequence contains only trivial perfect powers". <i>Sets, graphs, and numbers (Budapest, 1991)</i>. Colloq. Math. Soc. János Bolyai, 60, North-Holland. pp. 561–568. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1218218">1218218</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+Pell+sequence+contains+only+trivial+perfect+powers&rft.btitle=Sets%2C+graphs%2C+and+numbers+%28Budapest%2C+1991%29&rft.pages=561-568&rft.pub=Colloq.+Math.+Soc.+J%C3%A1nos+Bolyai%2C+60%2C+North-Holland&rft.date=1992&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1218218%23id-name%3DMR&rft.au=Peth%C5%91%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRidenhour,_J._R.1986" class="citation journal cs1">Ridenhour, J. R. (1986). "Ladder approximations of irrational numbers". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>59</b> (2): 95–105. <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%2F2690427">10.2307/2690427</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/2690427">2690427</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Ladder+approximations+of+irrational+numbers&rft.volume=59&rft.issue=2&rft.pages=95-105&rft.date=1986&rft_id=info%3Adoi%2F10.2307%2F2690427&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690427%23id-name%3DJSTOR&rft.au=Ridenhour%2C+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFalcón_SantanaDíaz-Barrero2006" class="citation journal cs1">Falcón Santana, Sergio; Díaz-Barrero, José Luis (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.35834%2F2006%2F1801033">"Some properties of sums involving Pell numbers"</a>. <i><a href="/w/index.php?title=Missouri_Journal_of_Mathematical_Sciences&action=edit&redlink=1" class="new" title="Missouri Journal of Mathematical Sciences (page does not exist)">Missouri Journal of Mathematical Sciences</a></i>. <b>18</b> (1). <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/10.35834%2F2006%2F1801033">10.35834/2006/1801033</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10553%2F72698">10553/72698</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Missouri+Journal+of+Mathematical+Sciences&rft.atitle=Some+properties+of+sums+involving+Pell+numbers&rft.volume=18&rft.issue=1&rft.date=2006&rft_id=info%3Ahdl%2F10553%2F72698&rft_id=info%3Adoi%2F10.35834%2F2006%2F1801033&rft.aulast=Falc%C3%B3n+Santana&rft.aufirst=Sergio&rft.au=D%C3%ADaz-Barrero%2C+Jos%C3%A9+Luis&rft_id=https%3A%2F%2Fdoi.org%2F10.35834%252F2006%252F1801033&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSellers,_James_A.2002" class="citation journal cs1">Sellers, James A. (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200705231252/https://www.emis.de/journals/JIS/VOL5/Sellers/sellers4.pdf">"Domino tilings and products of Fibonacci and Pell numbers"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Integer_Sequences" title="Journal of Integer Sequences">Journal of Integer Sequences</a></i>. <b>5</b>: 12. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JIntS...5...12S">2002JIntS...5...12S</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1919941">1919941</a>. Archived from <a rel="nofollow" class="external text" href="http://www.emis.de/journals/JIS/VOL5/Sellers/sellers4.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-07-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-01-28</span></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+Integer+Sequences&rft.atitle=Domino+tilings+and+products+of+Fibonacci+and+Pell+numbers&rft.volume=5&rft.pages=12&rft.date=2002&rft_id=info%3Abibcode%2F2002JIntS...5...12S&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1919941%23id-name%3DMR&rft.au=Sellers%2C+James+A.&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FJIS%2FVOL5%2FSellers%2Fsellers4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSesskin,_Sam1962" class="citation journal cs1">Sesskin, Sam (1962). "A "converse" to Fermat's last theorem?". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>35</b> (4): 215–217. <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%2F2688551">10.2307/2688551</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/2688551">2688551</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=A+%22converse%22+to+Fermat%27s+last+theorem%3F&rft.volume=35&rft.issue=4&rft.pages=215-217&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F2688551&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2688551%23id-name%3DJSTOR&rft.au=Sesskin%2C+Sam&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThibaut,_George1875" class="citation journal cs1"><a href="/wiki/George_Thibaut" title="George Thibaut">Thibaut, George</a> (1875). "On the Súlvasútras". <i><a href="/w/index.php?title=Journal_of_the_Royal_Asiatic_Society_of_Bengal&action=edit&redlink=1" class="new" title="Journal of the Royal Asiatic Society of Bengal (page does not exist)">Journal of the Royal Asiatic Society of Bengal</a></i>. <b>44</b>: 227–275.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Asiatic+Society+of+Bengal&rft.atitle=On+the+S%C3%BAlvas%C3%BAtras&rft.volume=44&rft.pages=227-275&rft.date=1875&rft.au=Thibaut%2C+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson,_D'Arcy_Wentworth1929" class="citation journal cs1"><a href="/wiki/D%27Arcy_Wentworth_Thompson" title="D'Arcy Wentworth Thompson">Thompson, D'Arcy Wentworth</a> (1929). "III.—Excess and defect: or the little more and the little less". <i><a href="/wiki/Mind_(journal)" title="Mind (journal)">Mind</a></i>. New Series. <b>38</b> (149): 43–55. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmind%2FXXXVIII.149.43">10.1093/mind/XXXVIII.149.43</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/2249223">2249223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mind&rft.atitle=III.%E2%80%94Excess+and+defect%3A+or+the+little+more+and+the+little+less&rft.volume=38&rft.issue=149&rft.pages=43-55&rft.date=1929&rft_id=info%3Adoi%2F10.1093%2Fmind%2FXXXVIII.149.43&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2249223%23id-name%3DJSTOR&rft.au=Thompson%2C+D%27Arcy+Wentworth&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVedova,_G._C.1951" class="citation journal cs1">Vedova, G. C. (1951). "Notes on Theon of Smyrna". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>58</b> (10): 675–683. <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%2F2307978">10.2307/2307978</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/2307978">2307978</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=Notes+on+Theon+of+Smyrna&rft.volume=58&rft.issue=10&rft.pages=675-683&rft.date=1951&rft_id=info%3Adoi%2F10.2307%2F2307978&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2307978%23id-name%3DJSTOR&rft.au=Vedova%2C+G.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pell_number&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Pell_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/PellNumber.html">"Pell 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=Pell+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPellNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APell+number" class="Z3988"></span></span></li> <li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A001333">sequence A001333 (Numerators of continued fraction convergents to sqrt(2))</a>—The numerators of the same sequence of approximations</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Prime_number_classes" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fermat_number" title="Fermat number">Fermat (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>n</i></sup></sup> + 1</span>)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>p</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple"><i>k</i>-tuples</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twin_prime" title="Twin prime">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a 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 href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a class="mw-selflink selflink">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="Metallic_means" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Metallic_ratios" title="Template:Metallic ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Metallic_ratios" title="Template talk:Metallic ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Metallic_ratios" title="Special:EditPage/Template:Metallic ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Metallic_means" style="font-size:114%;margin:0 4em"><a href="/wiki/Metallic_mean" title="Metallic mean">Metallic means</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"> <ul><li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot number</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Gold</a> <ul><li><a href="/wiki/Golden_angle" title="Golden angle">Angle</a></li> <li><a href="/wiki/Golden_ratio_base" title="Golden ratio base">Base</a></li> <li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler triangle</a></li> <li><a href="/wiki/Golden_rectangle" title="Golden rectangle">Rectangle</a></li> <li><a href="/wiki/Golden_rhombus" title="Golden rhombus">Rhombus</a></li> <li><a href="/wiki/Golden-section_search" title="Golden-section search">Section search</a></li> <li><a href="/wiki/Golden_spiral" title="Golden spiral">Spiral</a></li> <li><a href="/wiki/Golden_triangle_(mathematics)" title="Golden triangle (mathematics)">Triangle</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio</a></li></ul></li></ul> <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver</a> <ul><li><a class="mw-selflink selflink">Pell number</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio</a></li></ul></li> <li>Bronze</li> <li>Copper</li> <li>Nickel</li> <li>etc...</li></ul> </div></td><td class="noviewer navbox-image" rowspan="1" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Gold,_silver,_and_bronze_rectangles_vertical.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/50px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png" decoding="async" width="50" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/75px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/100px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png 2x" data-file-width="756" data-file-height="1074" /></a></span></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="Sequences_and_series" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Series_(mathematics)" title="Template:Series (mathematics)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Series_(mathematics)" title="Template talk:Series (mathematics)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Series_(mathematics)" title="Special:EditPage/Template:Series (mathematics)"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sequences_and_series" style="font-size:114%;margin:0 4em"><a href="/wiki/Sequence" title="Sequence">Sequences</a> and <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequences</a></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 scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</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/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li> <li><a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a></li> <li><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li> <li><a href="/wiki/Square_number" title="Square number">Square number</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Powers of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Powers of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></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/Complete_sequence" title="Complete sequence">Complete sequence</a></li> <li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li> <li><a href="/wiki/Figurate_number" title="Figurate number">Figurate number</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a></li> <li><a class="mw-selflink selflink">Pell number</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li> <li><a href="/wiki/Polygonal_number" title="Polygonal number">Polygonal number</a></li> <li><a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a> <ul><li><a href="/wiki/Triangular_array" title="Triangular array">array</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence"><img alt="Fibonacci spiral with square sizes up to 34." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/80px-Fibonacci_spiral_34.svg.png" decoding="async" width="80" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/120px-Fibonacci_spiral_34.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/160px-Fibonacci_spiral_34.svg.png 2x" data-file-width="915" data-file-height="579" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</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/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Monotonic function</a></li> <li><a href="/wiki/Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</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 scope="row" class="navbox-group" style="width:1%;text-align:left">Series</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/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Convergent_series" title="Convergent series">Convergent</a></li> <li><a href="/wiki/Divergent_series" title="Divergent series">Divergent</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</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/Absolute_convergence" title="Absolute convergence">Absolute</a></li> <li><a href="/wiki/Conditional_convergence" title="Conditional convergence">Conditional</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</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 scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</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/1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a></li> <li><a href="/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1/2 + 1/4 + 1/8 + 1/16 + ⋯</a></li> <li><a href="/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a></li> <li><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</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/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></li> <li><a href="/wiki/Grandi%27s_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">1 + 2 + 3 + 4 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF" title="1 − 2 + 4 − 8 + ⋯">1 − 2 + 4 − 8 + ⋯</a></li> <li><a href="/wiki/Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li> <li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_..." class="mw-redirect" title="1 − 1 + 2 − 6 + 24 − 120 + ...">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li> <li><a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</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/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Power_series" title="Power series">Power series</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a></li> <li><a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</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/Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li> <li><a href="/wiki/Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li> <li><a href="/wiki/Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li> <li><a href="/wiki/Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Mathematical_series" title="Category:Mathematical series">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Pell_number&oldid=1256792511">https://en.wikipedia.org/w/index.php?title=Pell_number&oldid=1256792511</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Integer_sequences" title="Category:Integer sequences">Integer sequences</a></li><li><a href="/wiki/Category:Recurrence_relations" title="Category:Recurrence relations">Recurrence relations</a></li><li><a href="/wiki/Category:Unsolved_problems_in_mathematics" title="Category:Unsolved problems in mathematics">Unsolved problems in mathematics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 11 November 2024, at 16:03<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Pell_number&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-49vgg","wgBackendResponseTime":159,"wgPageParseReport":{"limitreport":{"cputime":"0.655","walltime":"0.916","ppvisitednodes":{"value":5045,"limit":1000000},"postexpandincludesize":{"value":242802,"limit":2097152},"templateargumentsize":{"value":7806,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":97381,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 592.357 1 -total"," 25.43% 150.617 14 Template:Navbox"," 18.80% 111.386 1 Template:Prime_number_classes"," 15.73% 93.200 1 Template:Reflist"," 13.88% 82.205 1 Template:Short_description"," 13.15% 77.881 18 Template:Cite_journal"," 13.10% 77.608 1 Template:Citation"," 8.91% 52.755 2 Template:Pagetype"," 6.69% 39.617 1 Template:Classes_of_natural_numbers"," 5.80% 34.377 1 Template:Navbox_with_collapsible_groups"]},"scribunto":{"limitreport-timeusage":{"value":"0.300","limit":"10.000"},"limitreport-memusage":{"value":6394447,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-5dc468848-vgd46","timestamp":"20241123001442","ttl":85531,"transientcontent":true}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Pell number","url":"https:\/\/en.wikipedia.org\/wiki\/Pell_number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1386491","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1386491","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-01-02T19:47:42Z","dateModified":"2024-11-11T16:03:06Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/8c\/Silver_spiral_approximation.svg","headline":"denominator of close rational approximation to the square root of 2"}</script> </body> </html>

CINXE.COM

Pell number - Wikipedia