CINXE.COM

<!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>Gaussian integer - 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":"64d20535-8d5f-4980-bc14-a7d4ba5dbef4","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Gaussian_integer","wgTitle":"Gaussian integer","wgCurRevisionId":1254081435,"wgRevisionId":1254081435,"wgArticleId":48628,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Algebraic numbers","Cyclotomic fields","Lattice points","Quadratic irrational numbers","Integers","Complex numbers"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Gaussian_integer","wgRelevantArticleId":48628,"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":40000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q724975","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.math.styles":"ready","ext.cite.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","ext.scribunto.logs","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.quicksurveys.init","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/9/9e/Gaussian_integer_lattice.svg/1200px-Gaussian_integer_lattice.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="901"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Gaussian_integer_lattice.svg/800px-Gaussian_integer_lattice.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="601"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Gaussian_integer_lattice.svg/640px-Gaussian_integer_lattice.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="480"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Gaussian integer - 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/Gaussian_integer"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Gaussian_integer&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/Gaussian_integer"> <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-Gaussian_integer rootpage-Gaussian_integer 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" > Main menu </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">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</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">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</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">Recent changes</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">Upload file</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"> <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;"> </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"> Search </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" > </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" > Appearance </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="">Donate</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=Gaussian+integer" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</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=Gaussian+integer" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</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" > Personal tools </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">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Gaussian+integer" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Gaussian+integer" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</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">learn more</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">Contributions</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">Talk</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-Basic_definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_definitions"> <div class="vector-toc-text"> 1 Basic definitions </div> </a> <ul id="toc-Basic_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_division" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Euclidean_division"> <div class="vector-toc-text"> 2 Euclidean division </div> </a> <ul id="toc-Euclidean_division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principal_ideals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principal_ideals"> <div class="vector-toc-text"> 3 Principal ideals </div> </a> <ul id="toc-Principal_ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussian_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gaussian_primes"> <div class="vector-toc-text"> 4 Gaussian primes </div> </a> <ul id="toc-Gaussian_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unique_factorization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unique_factorization"> <div class="vector-toc-text"> 5 Unique factorization </div> </a> <ul id="toc-Unique_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussian_rationals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gaussian_rationals"> <div class="vector-toc-text"> 6 Gaussian rationals </div> </a> <ul id="toc-Gaussian_rationals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Greatest_common_divisor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Greatest_common_divisor"> <div class="vector-toc-text"> 7 Greatest common divisor </div> </a> <ul id="toc-Greatest_common_divisor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruences_and_residue_classes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Congruences_and_residue_classes"> <div class="vector-toc-text"> 8 Congruences and residue classes </div> </a> <button aria-controls="toc-Congruences_and_residue_classes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Congruences and residue classes subsection </button> <ul id="toc-Congruences_and_residue_classes-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 8.1 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Describing_residue_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Describing_residue_classes"> <div class="vector-toc-text"> 8.2 Describing residue classes </div> </a> <ul id="toc-Describing_residue_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Residue_class_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Residue_class_fields"> <div class="vector-toc-text"> 8.3 Residue class fields </div> </a> <ul id="toc-Residue_class_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Primitive_residue_class_group_and_Euler's_totient_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primitive_residue_class_group_and_Euler's_totient_function"> <div class="vector-toc-text"> 9 Primitive residue class group and Euler's totient function </div> </a> <ul id="toc-Primitive_residue_class_group_and_Euler's_totient_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_background" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_background"> <div class="vector-toc-text"> 10 Historical background </div> </a> <ul id="toc-Historical_background-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unsolved_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unsolved_problems"> <div class="vector-toc-text"> 11 Unsolved problems </div> </a> <ul id="toc-Unsolved_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </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"> 14 References </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"> 15 External links </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" > Toggle the table of contents </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">Gaussian integer</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 31 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-31" aria-hidden="true" > 31 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%D9%8A%D8%AD_%D8%BA%D8%A7%D9%88%D8%B3%D9%8A" title="عدد صحيح غاوسي – Arabic" lang="ar" hreflang="ar" data-title="عدد صحيح غاوسي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D0%B0%D0%B2%D1%8B_%D1%86%D1%8D%D0%BB%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D1%96" title="Гаусавы цэлыя лікі – Belarusian" lang="be" hreflang="be" data-title="Гаусавы цэлыя лікі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Enter_de_Gauss" title="Enter de Gauss – Catalan" lang="ca" hreflang="ca" data-title="Enter de Gauss" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D1%82%D1%83%D0%BB%D0%BB%D0%B8_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95%D1%81%D0%B5%D0%BC" title="Гаусс тулли хисепĕсем – Chuvash" lang="cv" hreflang="cv" data-title="Гаусс тулли хисепĕсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Gaussovo_cel%C3%A9_%C4%8D%C3%ADslo" title="Gaussovo celé číslo – Czech" lang="cs" hreflang="cs" data-title="Gaussovo celé číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gau%C3%9Fsche_Zahl" title="Gaußsche Zahl – German" lang="de" hreflang="de" data-title="Gaußsche Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Entero_gaussiano" title="Entero gaussiano – Spanish" lang="es" hreflang="es" data-title="Entero gaussiano" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ga%C5%ADsa_entjero" title="Gaŭsa entjero – Esperanto" lang="eo" hreflang="eo" data-title="Gaŭsa entjero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%DB%8C%D8%AD_%DA%AF%D8%A7%D9%88%D8%B3%DB%8C" title="عدد صحیح گاوسی – Persian" lang="fa" hreflang="fa" data-title="عدد صحیح گاوسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Entier_de_Gauss" title="Entier de Gauss – French" lang="fr" hreflang="fr" data-title="Entier de Gauss" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enteiro_de_Gauss" title="Enteiro de Gauss – Galician" lang="gl" hreflang="gl" data-title="Enteiro de Gauss" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4_%EC%A0%95%EC%88%98" title="가우스 정수 – Korean" lang="ko" hreflang="ko" data-title="가우스 정수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%BE%E0%A4%8A%E0%A4%B8%E0%A5%80_%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%BE%E0%A4%82%E0%A4%95" title="गाऊसी पूर्णांक – Hindi" lang="hi" hreflang="hi" data-title="गाऊसी पूर्णांक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Intero_di_Gauss" title="Intero di Gauss – Italian" lang="it" hreflang="it" data-title="Intero di Gauss" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%92_%D7%94%D7%A9%D7%9C%D7%9E%D7%99%D7%9D_%D7%A9%D7%9C_%D7%92%D7%90%D7%95%D7%A1" title="חוג השלמים של גאוס – Hebrew" lang="he" hreflang="he" data-title="חוג השלמים של גאוס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D1%81%D0%B0%D0%BD%D1%8B" title="Гаусс саны – Kazakh" lang="kk" hreflang="kk" data-title="Гаусс саны" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gauss-eg%C3%A9sz" title="Gauss-egész – Hungarian" lang="hu" hreflang="hu" data-title="Gauss-egész" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8B%E0%B4%B8%E0%B4%BF%E0%B4%AF%E0%B5%BB_%E0%B4%AA%E0%B5%82%E0%B5%BC%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="ഗോസിയൻ പൂർണ്ണസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="ഗോസിയൻ പൂർണ്ണസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geheel_getal_van_Gauss" title="Geheel getal van Gauss – Dutch" lang="nl" hreflang="nl" data-title="Geheel getal van Gauss" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%E6%95%B4%E6%95%B0" title="ガウス整数 – Japanese" lang="ja" hreflang="ja" data-title="ガウス整数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gaussisk_heltall" title="Gaussisk heltall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gaussisk heltall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_ca%C5%82kowite_Gaussa" title="Liczby całkowite Gaussa – Polish" lang="pl" hreflang="pl" data-title="Liczby całkowite Gaussa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Inteiro_de_Gauss" title="Inteiro de Gauss – Portuguese" lang="pt" hreflang="pt" data-title="Inteiro de Gauss" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81%D0%BE%D0%B2%D1%8B_%D1%86%D0%B5%D0%BB%D1%8B%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Гауссовы целые числа – Russian" lang="ru" hreflang="ru" data-title="Гауссовы целые числа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Gaussovo_pra%C5%A1tevilo" title="Gaussovo praštevilo – Slovenian" lang="sl" hreflang="sl" data-title="Gaussovo praštevilo" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gaussin_kokonaisluku" title="Gaussin kokonaisluku – Finnish" lang="fi" hreflang="fi" data-title="Gaussin kokonaisluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gaussiskt_heltal" title="Gaussiskt heltal – Swedish" lang="sv" hreflang="sv" data-title="Gaussiskt heltal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%9A%E0%AE%BF%E0%AE%AF_%E0%AE%AE%E0%AF%81%E0%AE%B4%E0%AF%81%E0%AE%B5%E0%AF%86%E0%AE%A3%E0%AF%8D" title="காசிய முழுவெண் – Tamil" lang="ta" hreflang="ta" data-title="காசிய முழுவெண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81%D0%BE%D0%B2%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Гауссові числа – Ukrainian" lang="uk" hreflang="uk" data-title="Гауссові числа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_nguy%C3%AAn_Gauss" title="Số nguyên Gauss – Vietnamese" lang="vi" hreflang="vi" data-title="Số nguyên Gauss" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%AB%98%E6%96%AF%E6%95%B4%E6%95%B8" title="高斯整數 – Chinese" lang="zh" hreflang="zh" data-title="高斯整數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q724975#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></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/Gaussian_integer" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Gaussian_integer" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</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" >English </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/Gaussian_integer">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Gaussian_integer&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Gaussian_integer&action=history" title="Past revisions of this page [h]" accesskey="h">View history</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" >Tools </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/Gaussian_integer">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Gaussian_integer&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Gaussian_integer&action=history">View history</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/Gaussian_integer" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Gaussian_integer" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</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">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Gaussian_integer&oldid=1254081435" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Gaussian_integer&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Gaussian_integer&id=1254081435&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</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%2FGaussian_integer">Get shortened URL</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%2FGaussian_integer">Download QR code</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=Gaussian_integer&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Gaussian_integer&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</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 class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Gaussian_integers" hreflang="en">Wikimedia Commons</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q724975" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</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">Complex number whose real and imaginary parts are both integers</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/Gaussian_integral" title="Gaussian integral">Gaussian integral</a>.</div> In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, a Gaussian integer is a <a href="/wiki/Complex_number" title="Complex number">complex number</a> whose real and imaginary parts are both <a href="/wiki/Integer" title="Integer">integers</a>. The Gaussian integers, with ordinary <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, form an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>, usually written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [i]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a617cf5867f951fefb72f3ab7278e0f6f1eedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.73ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} [i]}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [i].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03900897cdf515bbbc52879377653a871b9efc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.293ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i].}"><a href="#cite_note-Fraleigh_1976_286-1">[1]</a> Gaussian integers share many properties with integers: they form a <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a>, and have thus a <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> and a <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>; this implies <a href="/wiki/Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a> and many related properties. However, Gaussian integers do not have a <a href="/wiki/Total_order" title="Total order">total ordering</a> that respects arithmetic. Gaussian integers are <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a> and form the simplest ring of <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>. Gaussian integers are named after the German mathematician <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gaussian_integer_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Gaussian_integer_lattice.svg/217px-Gaussian_integer_lattice.svg.png" decoding="async" width="217" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Gaussian_integer_lattice.svg/326px-Gaussian_integer_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Gaussian_integer_lattice.svg/434px-Gaussian_integer_lattice.svg.png 2x" data-file-width="389" data-file-height="292" /></a><figcaption>Gaussian integers as <a href="/wiki/Lattice_point" class="mw-redirect" title="Lattice point">lattice points</a> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a></figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_definitions">Basic definitions</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=1" title="Edit section: Basic definitions">edit</a>]</div> The Gaussian integers are the set<a href="#cite_note-Fraleigh_1976_286-1">[1]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbf {Z} \},\qquad {\text{ where }}i^{2}=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbf {Z} \},\qquad {\text{ where }}i^{2}=-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e23ae09a25e0fde987eb1f99eb41e838ce6537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.057ex; height:3.176ex;" alt="{\displaystyle \mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbf {Z} \},\qquad {\text{ where }}i^{2}=-1.}"></dd></dl> In other words, a Gaussian integer is a <a href="/wiki/Complex_number" title="Complex number">complex number</a> such that its <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real</a> and <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary parts</a> are both <a href="/wiki/Integer" title="Integer">integers</a>. Since the Gaussian integers are closed under addition and multiplication, they form a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, which is a <a href="/wiki/Subring" title="Subring">subring</a> of the field of complex numbers. It is thus an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. When considered within the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, the Gaussian integers constitute the 2-dimensional <a href="/wiki/Integer_lattice" title="Integer lattice">integer lattice</a>. The conjugate of a Gaussian integer a + bi is the Gaussian integer a – bi. The <a href="/wiki/Field_norm" title="Field norm">norm</a> of a Gaussian integer is its product with its conjugate. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(a+bi)=(a+bi)(a-bi)=a^{2}+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(a+bi)=(a+bi)(a-bi)=a^{2}+b^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2a727f98a600cdef213b3215a50737f4de952a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.122ex; height:3.176ex;" alt="{\displaystyle N(a+bi)=(a+bi)(a-bi)=a^{2}+b^{2}.}"></dd></dl> The norm of a Gaussian integer is thus the square of its <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two <a href="/wiki/Square_number" title="Square number">squares</a>. Thus a norm <a href="/wiki/Sum_of_two_squares_theorem" title="Sum of two squares theorem">cannot be of the form 4k + 3, with k integer</a>. The norm is <a href="/wiki/Completely_multiplicative_function" title="Completely multiplicative function">multiplicative</a>, that is, one has<a href="#cite_note-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(zw)=N(z)N(w),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(zw)=N(z)N(w),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a5bf2b8d198337201ab9bcf9c36d858b187c66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.869ex; height:2.843ex;" alt="{\displaystyle N(zw)=N(z)N(w),}"></dd></dl> for every pair of Gaussian integers z, w. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers. The <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a> of the ring of Gaussian integers (that is the Gaussian integers whose <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, –1, i and –i.<a href="#cite_note-3">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Euclidean_division">Euclidean division</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=2" title="Edit section: Euclidean division">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gauss-euklid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Gauss-euklid.svg/250px-Gauss-euklid.svg.png" decoding="async" width="250" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Gauss-euklid.svg/375px-Gauss-euklid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Gauss-euklid.svg/500px-Gauss-euklid.svg.png 2x" data-file-width="531" data-file-height="432" /></a><figcaption>Visualization of maximal distance to some Gaussian integer</figcaption></figure> Gaussian integers have a <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> (division with remainder) similar to that of <a href="/wiki/Integer" title="Integer">integers</a> and <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>. This makes the Gaussian integers a <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a>, and implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> for computing <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisors</a>, <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a>, the <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal property</a>, <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a>, the <a href="/wiki/Unique_factorization_theorem" class="mw-redirect" title="Unique factorization theorem">unique factorization theorem</a>, and the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>, all of which can be proved using only Euclidean division. A Euclidean division algorithm takes, in the ring of Gaussian integers, a dividend a and divisor b ≠ 0, and produces a quotient q and remainder r such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)<N(b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>N</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)<N(b).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bfa05e12471603b88aec1d50027bd5ab58a761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.214ex; height:2.843ex;" alt="{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)<N(b).}"></dd></dl> In fact, one may make the remainder smaller: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)\leq {\frac {N(b)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>N</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)\leq {\frac {N(b)}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1a807c90c999ca3f9ac06cb27e5502eb450044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.051ex; height:5.676ex;" alt="{\displaystyle a=bq+r\quad {\text{and}}\quad N(r)\leq {\frac {N(b)}{2}}.}"></dd></dl> Even with this better inequality, the quotient and the remainder are not necessarily unique, but one may refine the choice to ensure uniqueness. To prove this, one may consider the <a href="/wiki/Complex_number" title="Complex number">complex number</a> quotient x + iy = <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>⁠a/b⁠. There are unique integers m and n such that –<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ < x – m ≤ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ and –<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ < y – n ≤ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, and thus N(x – m + i(y – n)) ≤ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. Taking q = m + in, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=bq+r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=bq+r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648db44ee70dc0e4933fc4d0b4930d0bc82e2f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.931ex; height:2.509ex;" alt="{\displaystyle a=bq+r,}"></dd></dl> with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=b{\bigl (}x-m+i(y-n){\bigr )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=b{\bigl (}x-m+i(y-n){\bigr )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f4fb6a031e780959385bf119b5876dcd6c9cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.975ex; height:3.176ex;" alt="{\displaystyle r=b{\bigl (}x-m+i(y-n){\bigr )},}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(r)\leq {\frac {N(b)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(r)\leq {\frac {N(b)}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e1230008572723dc77ffcb66b63a9cc6db09cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.373ex; height:5.676ex;" alt="{\displaystyle N(r)\leq {\frac {N(b)}{2}}.}"></dd></dl> The choice of x – m and y – n in a <a href="/wiki/Semi-open_interval" class="mw-redirect" title="Semi-open interval">semi-open interval</a> is required for uniqueness. This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number ξ to the closest Gaussian integer is at most <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠√2/2⁠.<a href="#cite_note-4">[4]</a> <div class="mw-heading mw-heading2"><h2 id="Principal_ideals">Principal ideals</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=3" title="Edit section: Principal ideals">edit</a>]</div> Since the ring G of Gaussian integers is a Euclidean domain, G is a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, which means that every <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of G is <a href="/wiki/Principal_ideal" title="Principal ideal">principal</a>. Explicitly, an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> I is a subset of a ring R such that every sum of elements of I and every product of an element of I by an element of R belong to I. An ideal is <a href="/wiki/Principal_ideal" title="Principal ideal">principal</a> if it consists of all multiples of a single element g, that is, it has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{gx\mid x\in G\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{gx\mid x\in G\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7d25693df349b3106625b0474359a9eb4ab917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.352ex; height:2.843ex;" alt="{\displaystyle \{gx\mid x\in G\}.}"></dd></dl> In this case, one says that the ideal is generated by g or that g is a generator of the ideal. Every ideal I in the ring of the Gaussian integers is principal, because, if one chooses in I a nonzero element g of minimal norm, for every element x of I, the remainder of Euclidean division of x by g belongs also to I and has a norm that is smaller than that of g; because of the choice of g, this norm is zero, and thus the remainder is also zero. That is, one has x = qg, where q is the quotient. For any g, the ideal generated by g is also generated by any associate of g, that is, g, gi, –g, –gi; no other element generates the same ideal. As all the generators of an ideal have the same norm, the norm of an ideal is the norm of any of its generators. In some circumstances, it is useful to choose, once for all, a generator for each ideal. There are two classical ways for doing that, both considering first the ideals of odd norm. If the g = a + bi has an odd norm a2 + b2, then one of a and b is odd, and the other is even. Thus g has exactly one associate with a real part a that is odd and positive. In his original paper, <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a> made another choice, by choosing the unique associate such that the remainder of its division by 2 + 2i is one. In fact, as N(2 + 2i) = 8, the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit. Multiplying g by the inverse of this unit, one finds an associate that has one as a remainder, when divided by 2 + 2i. If the norm of g is even, then either g = 2kh or g = 2kh(1 + i), where k is a positive integer, and N(h) is odd. Thus, one chooses the associate of g for getting a h which fits the choice of the associates for elements of odd norm. <div class="mw-heading mw-heading2"><h2 id="Gaussian_primes">Gaussian primes</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=4" title="Edit section: Gaussian primes">edit</a>]</div> As the Gaussian integers form a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a> they form also a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>. This implies that a Gaussian integer is <a href="/wiki/Irreducible_element" title="Irreducible element">irreducible</a> (that is, it is not the product of two <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">non-units</a>) if and only if it is <a href="/wiki/Prime_element" title="Prime element">prime</a> (that is, it generates a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>). The <a href="/wiki/Prime_element" title="Prime element">prime elements</a> of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes). A positive integer is a Gaussian prime if and only if it is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> that is <a href="/wiki/Congruence_class" class="mw-redirect" title="Congruence class">congruent to</a> 3 <a href="/wiki/Modulo_operator" class="mw-redirect" title="Modulo operator">modulo</a> 4 (that is, it may be written 4n + 3, with n a nonnegative integer) (sequence <a href="//oeis.org/A002145" class="extiw" title="oeis:A002145">A002145</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes. A Gaussian integer a + bi is a Gaussian prime if and only if either: <ul><li>one of a, b is zero and the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the other is a prime number of the form 4n + 3 (with n a nonnegative integer), or</li> <li>both are nonzero and a2 + b2 is a prime number (which will not be of the form 4n + 3).</li></ul> In other words, a Gaussian integer m is a Gaussian prime if and only if either its norm is a prime number, or m is the product of a unit (±1, ±i) and a prime number of the form 4n + 3. It follows that there are three cases for the factorization of a prime natural number p in the Gaussian integers: <ul><li>If p is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, p is said to be <a href="/wiki/Inert_prime" class="mw-redirect" title="Inert prime">inert</a> in the Gaussian integers.</li> <li>If p is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); p is said to be a <a href="/wiki/Decomposed_prime" class="mw-redirect" title="Decomposed prime">decomposed prime</a> in the Gaussian integers. For example, 5 = (2 + i)(2 − i) and 13 = (3 + 2i)(3 − 2i).</li> <li>If p = 2, we have 2 = (1 + i)(1 − i) = i(1 − i)2; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique <a href="/wiki/Ramification_(mathematics)#In_algebraic_number_theory" title="Ramification (mathematics)">ramified prime</a> in the Gaussian integers.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Unique_factorization">Unique factorization</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=5" title="Edit section: Unique factorization">edit</a>]</div> As for every <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>, every Gaussian integer may be factored as a product of a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with a corresponding change of the unit factor). If one chooses, once for all, a fixed Gaussian prime for each <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of associated primes, and if one takes only these selected primes in the factorization, then one obtains a prime factorization which is unique up to the order of the factors. With the <a href="#selected_associates">choices described above</a>, the resulting unique factorization has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(1+i)^{e_{0}}{p_{1}}^{e_{1}}\cdots {p_{k}}^{e_{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(1+i)^{e_{0}}{p_{1}}^{e_{1}}\cdots {p_{k}}^{e_{k}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac54202ebef1a4f49064fbc409600c290910b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.088ex; height:2.843ex;" alt="{\displaystyle u(1+i)^{e_{0}}{p_{1}}^{e_{1}}\cdots {p_{k}}^{e_{k}},}"></dd></dl> where u is a unit (that is, u ∈ {1, –1, i, –i}), e0 and k are nonnegative integers, e1, …, ek are positive integers, and p1, …, pk are distinct Gaussian primes such that, depending on the choice of selected associates, <ul><li>either pk = ak + ibk with a odd and positive, and b even,</li> <li>or the remainder of the Euclidean division of pk by 2 + 2i equals 1 (this is Gauss's original choice<a href="#cite_note-5">[5]</a>).</li></ul> An advantage of the second choice is that the selected associates behave well under products for Gaussian integers of odd norm. On the other hand, the selected associate for the real Gaussian primes are negative integers. For example, the factorization of 231 in the integers, and with the first choice of associates is 3 × 7 × 11, while it is (–1) × (–3) × (–7) × (–11) with the second choice. <div class="mw-heading mw-heading2"><h2 id="Gaussian_rationals">Gaussian rationals</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=6" title="Edit section: Gaussian rationals">edit</a>]</div> The <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a> is the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both <a href="/wiki/Rational_number" title="Rational number">rational</a>. The ring of Gaussian integers is the <a href="/wiki/Integral_closure" class="mw-redirect" title="Integral closure">integral closure</a> of the integers in the Gaussian rationals. This implies that Gaussian integers are <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a> and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+cx+d=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+cx+d=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbf1fe0b4e9a5688bce75380a8bebbc5de237e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.525ex; height:3.009ex;" alt="{\displaystyle x^{2}+cx+d=0,}"></dd></dl> with c and d integers. In fact a + bi is solution of the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-2ax+a^{2}+b^{2},}"> <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> <mi>a</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-2ax+a^{2}+b^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4378ba496bcf4f795b94d6ac4cd05f2795e73a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.61ex; height:3.009ex;" alt="{\displaystyle x^{2}-2ax+a^{2}+b^{2},}"></dd></dl> and this equation has integer coefficients if and only if a and b are both integers. <div class="mw-heading mw-heading2"><h2 id="Greatest_common_divisor">Greatest common divisor</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=7" title="Edit section: Greatest common divisor">edit</a>]</div> As for any <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>, a <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> (gcd) of two Gaussian integers a, b is a Gaussian integer d that is a common divisor of a and b, which has all common divisors of a and b as divisor. That is (where | denotes the <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a> relation), <ul><li>d | a and d | b, and</li> <li>c | a and c | b implies c | d.</li></ul> Thus, greatest is meant relatively to the divisibility relation, and not for an ordering of the ring (for integers, both meanings of greatest coincide). More technically, a greatest common divisor of a and b is a <a href="/wiki/Ideal_(ring_theory)#Ideal_generated_by_a_set" title="Ideal (ring theory)">generator</a> of the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by a and b (this characterization is valid for <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a>, but not, in general, for unique factorization domains). The greatest common divisor of two Gaussian integers is not unique, but is defined up to the multiplication by a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>. That is, given a greatest common divisor d of a and b, the greatest common divisors of a and b are d, –d, id, and –id. There are several ways for computing a greatest common divisor of two Gaussian integers a and b. When one knows the prime factorizations of a and b, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=i^{k}\prod _{m}{p_{m}}^{\nu _{m}},\quad b=i^{n}\prod _{m}{p_{m}}^{\mu _{m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=i^{k}\prod _{m}{p_{m}}^{\nu _{m}},\quad b=i^{n}\prod _{m}{p_{m}}^{\mu _{m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b8871b50dc93609ed9b4fe325e04969be6d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.455ex; height:5.509ex;" alt="{\displaystyle a=i^{k}\prod _{m}{p_{m}}^{\nu _{m}},\quad b=i^{n}\prod _{m}{p_{m}}^{\mu _{m}},}"></dd></dl> where the primes pm are pairwise non associated, and the exponents μm non-associated, a greatest common divisor is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{m}{p_{m}}^{\lambda _{m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{m}{p_{m}}^{\lambda _{m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af9e64f8fa6e92ac770d8dd793b68df75524f7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.374ex; height:5.509ex;" alt="{\displaystyle \prod _{m}{p_{m}}^{\lambda _{m}},}"></dd></dl> with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{m}=\min(\nu _{m},\mu _{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{m}=\min(\nu _{m},\mu _{m}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16009ca47cb4b98902f48636dabc44be36dc62e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.394ex; height:2.843ex;" alt="{\displaystyle \lambda _{m}=\min(\nu _{m},\mu _{m}).}"></dd></dl> Unfortunately, except in simple cases, the prime factorization is difficult to compute, and <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> leads to a much easier (and faster) computation. This algorithm consists of replacing of the input (a, b) by (b, r), where r is the remainder of the Euclidean division of a by b, and repeating this operation until getting a zero remainder, that is a pair (d, 0). This process terminates, because, at each step, the norm of the second Gaussian integer decreases. The resulting d is a greatest common divisor, because (at each step) b and r = a – bq have the same divisors as a and b, and thus the same greatest common divisor. This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated. Therefore, a third method is often preferred for hand-written computations. It consists in remarking that the norm N(d) of the greatest common divisor of a and b is a common divisor of N(a), N(b), and N(a + b). When the greatest common divisor D of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing D. For example, if a = 5 + 3i, and b = 2 – 8i, one has N(a) = 34, N(b) = 68, and N(a + b) = 74. As the greatest common divisor of the three norms is 2, the greatest common divisor of a and b has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessary associated to 1 + i, and as 1 + i divides a and b, then the greatest common divisor is 1 + i. If b is replaced by its conjugate b = 2 + 8i, then the greatest common divisor of the three norms is 34, the norm of a, thus one may guess that the greatest common divisor is a, that is, that a | b. In fact, one has 2 + 8i = (5 + 3i)(1 + i). <div class="mw-heading mw-heading2"><h2 id="Congruences_and_residue_classes">Congruences and residue classes</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=8" title="Edit section: Congruences and residue classes">edit</a>]</div> Given a Gaussian integer z0, called a modulus, two Gaussian integers z1,z2 are congruent modulo z0, if their difference is a multiple of z0, that is if there exists a Gaussian integer q such that z1 − z2 = qz0. In other words, two Gaussian integers are congruent modulo z0, if their difference belongs to the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by z0. This is denoted as z1 ≡ z2 (mod z0). The congruence modulo z0 is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> (also called a <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a>), which defines a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of the Gaussian integers into <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a>, called here <a href="/wiki/Congruence_class" class="mw-redirect" title="Congruence class">congruence classes</a> or residue classes. The set of the residue classes is usually denoted Z[i]/z0Z[i], or Z[i]/⟨z0⟩, or simply Z[i]/z0. The residue class of a Gaussian integer a is the set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}:=\left\{z\in \mathbf {Z} [i]\mid z\equiv a{\pmod {z_{0}}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>∣</mo> <mi>z</mi> <mo>≡</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}:=\left\{z\in \mathbf {Z} [i]\mid z\equiv a{\pmod {z_{0}}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff8f7224fcd09a0b310e70c5211a2283725c677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.132ex; height:2.843ex;" alt="{\displaystyle {\bar {a}}:=\left\{z\in \mathbf {Z} [i]\mid z\equiv a{\pmod {z_{0}}}\right\}}"></dd></dl> of all Gaussian integers that are congruent to a. It follows that a = b <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> a ≡ b (mod z0). Addition and multiplication are compatible with congruences. This means that a1 ≡ b1 (mod z0) and a2 ≡ b2 (mod z0) imply a1 + a2 ≡ b1 + b2 (mod z0) and a1a2 ≡ b1b2 (mod z0). This defines well-defined <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> (that is independent of the choice of representatives) on the residue classes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}+{\bar {b}}:={\overline {a+b}}\quad {\text{and}}\quad {\bar {a}}\cdot {\bar {b}}:={\overline {ab}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mi>b</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}+{\bar {b}}:={\overline {a+b}}\quad {\text{and}}\quad {\bar {a}}\cdot {\bar {b}}:={\overline {ab}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f74e762bceab7618ba65b513be6184ea2d117ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:33.359ex; height:3.176ex;" alt="{\displaystyle {\bar {a}}+{\bar {b}}:={\overline {a+b}}\quad {\text{and}}\quad {\bar {a}}\cdot {\bar {b}}:={\overline {ab}}.}"></dd></dl> With these operations, the residue classes form a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> of the Gaussian integers by the ideal generated by z0, which is also traditionally called the residue class ring modulo z0 (for more details, see <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a>). <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=9" title="Edit section: Examples">edit</a>]</div> <ul><li>There are exactly two residue classes for the modulus 1 + i, namely 0 = {0, ±2, ±4,…,±1 ± i, ±3 ± i,…} (all multiples of 1 + i), and 1 = {±1, ±3, ±5,…, ±i, ±2 ± i,…}, which form a checkerboard pattern in the complex plane. These two classes form thus a ring with two elements, which is, in fact, a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, the unique (up to an isomorphism) field with two elements, and may thus be identified with the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">integers modulo 2</a>. These two classes may be considered as a generalization of the partition of integers into even and odd integers. Thus one may speak of even and odd Gaussian integers (Gauss divided further even Gaussian integers into even, that is divisible by 2, and half-even).</li> <li>For the modulus 2 there are four residue classes, namely 0, 1, i, 1 + i. These form a ring with four elements, in which x = –x for every x. Thus this ring is not <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> with the ring of integers modulo 4, another ring with four elements. One has 1 + i2 = 0, and thus this ring is not the <a href="/wiki/Finite_field" title="Finite field">finite field</a> with four elements, nor the <a href="/wiki/Direct_product" title="Direct product">direct product</a> of two copies of the ring of integers modulo 2.</li> <li>For the modulus 2 + 2i = (i − 1)3 there are eight residue classes, namely 0, ±1, ±i, 1 ± i, 2, whereof four contain only even Gaussian integers and four contain only odd Gaussian integers.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Describing_residue_classes">Describing residue classes</h3>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=10" title="Edit section: Describing residue classes">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gauss-Restklassen-wiki.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Gauss-Restklassen-wiki.png/250px-Gauss-Restklassen-wiki.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Gauss-Restklassen-wiki.png/375px-Gauss-Restklassen-wiki.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/19/Gauss-Restklassen-wiki.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>All 13 residue classes with their minimal residues (blue dots) in the square Q00 (light green background) for the modulus z0 = 3 + 2i. One residue class with z = 2 − 4i ≡ −i (mod z0) is highlighted with yellow/orange dots.</figcaption></figure> Given a modulus z0, all elements of a residue class have the same remainder for the Euclidean division by z0, provided one uses the division with unique quotient and remainder, which is described <a href="#unique_remainder">above</a>. Thus enumerating the residue classes is equivalent with enumerating the possible remainders. This can be done geometrically in the following way. In the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, one may consider a <a href="/wiki/Square_grid" class="mw-redirect" title="Square grid">square grid</a>, whose squares are delimited by the two lines <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}V_{s}&=\left\{\left.z_{0}\left(s-{\tfrac {1}{2}}+ix\right)\right\vert x\in \mathbf {R} \right\}\quad {\text{and}}\\H_{t}&=\left\{\left.z_{0}\left(x+i\left(t-{\tfrac {1}{2}}\right)\right)\right\vert x\in \mathbf {R} \right\},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>i</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{s}&=\left\{\left.z_{0}\left(s-{\tfrac {1}{2}}+ix\right)\right\vert x\in \mathbf {R} \right\}\quad {\text{and}}\\H_{t}&=\left\{\left.z_{0}\left(x+i\left(t-{\tfrac {1}{2}}\right)\right)\right\vert x\in \mathbf {R} \right\},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73271fa220f64f2fe131304274db8df211169239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.196ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}V_{s}&=\left\{\left.z_{0}\left(s-{\tfrac {1}{2}}+ix\right)\right\vert x\in \mathbf {R} \right\}\quad {\text{and}}\\H_{t}&=\left\{\left.z_{0}\left(x+i\left(t-{\tfrac {1}{2}}\right)\right)\right\vert x\in \mathbf {R} \right\},\end{aligned}}}"></dd></dl> with s and t integers (blue lines in the figure). These divide the plane in <a href="/wiki/Semi-open_interval" class="mw-redirect" title="Semi-open interval">semi-open</a> squares (where m and n are integers) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{mn}=\left\{(s+it)z_{0}\left\vert s\in \left[m-{\tfrac {1}{2}},m+{\tfrac {1}{2}}\right),t\in \left[n-{\tfrac {1}{2}},n+{\tfrac {1}{2}}\right)\right.\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi>s</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{mn}=\left\{(s+it)z_{0}\left\vert s\in \left[m-{\tfrac {1}{2}},m+{\tfrac {1}{2}}\right),t\in \left[n-{\tfrac {1}{2}},n+{\tfrac {1}{2}}\right)\right.\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad979f96c1e4cf630c338899864ef9a861362d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:61.928ex; height:3.509ex;" alt="{\displaystyle Q_{mn}=\left\{(s+it)z_{0}\left\vert s\in \left[m-{\tfrac {1}{2}},m+{\tfrac {1}{2}}\right),t\in \left[n-{\tfrac {1}{2}},n+{\tfrac {1}{2}}\right)\right.\right\}.}"></dd></dl> The semi-open intervals that occur in the definition of Qmn have been chosen in order that every complex number belong to exactly one square; that is, the squares Qmn form a <a href="/wiki/Partition_(set_theory)" class="mw-redirect" title="Partition (set theory)">partition</a> of the complex plane. One has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{mn}=(m+in)z_{0}+Q_{00}=\left\{(m+in)z_{0}+z\mid z\in Q_{00}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>i</mi> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>i</mi> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>z</mi> <mo>∣</mo> <mi>z</mi> <mo>∈</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{mn}=(m+in)z_{0}+Q_{00}=\left\{(m+in)z_{0}+z\mid z\in Q_{00}\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c837b7636248d825945918a98e6528c9caeb969b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.165ex; height:2.843ex;" alt="{\displaystyle Q_{mn}=(m+in)z_{0}+Q_{00}=\left\{(m+in)z_{0}+z\mid z\in Q_{00}\right\}.}"></dd></dl> This implies that every Gaussian integer is congruent modulo z0 to a unique Gaussian integer in Q00 (the green square in the figure), which its remainder for the division by z0. In other words, every residue class contains exactly one element in Q00. The Gaussian integers in Q00 (or in its <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>) are sometimes called minimal residues because their norm are not greater than the norm of any other Gaussian integer in the same residue class (Gauss called them absolutely smallest residues). From this one can deduce by geometrical considerations, that the number of residue classes modulo a Gaussian integer z0 = a + bi equals its norm N(z0) = a2 + b2 (see below for a proof; similarly, for integers, the number of residue classes modulo n is its absolute value |n|). <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style="">Proof The relation Qmn = (m + in)z0 + Q00 means that all Qmn are obtained from Q00 by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translating</a> it by a Gaussian integer. This implies that all Qmn have the same area N = N(z0), and contain the same number ng of Gaussian integers. Generally, the number of grid points (here the Gaussian integers) in an arbitrary square with the area A is A + Θ(√A) (see <a href="/wiki/Big_theta" class="mw-redirect" title="Big theta">Big theta</a> for the notation). If one considers a big square consisting of k × k squares Qmn, then it contains k2N + O(k√N) grid points. It follows k2ng = k2N + Θ(k√N), and thus ng = N + Θ(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠√N/k⁠), after a division by k2. Taking the limit when k tends to the infinity gives ng = N = N(z0). </div> <div class="mw-heading mw-heading3"><h3 id="Residue_class_fields">Residue class fields</h3>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=11" title="Edit section: Residue class fields">edit</a>]</div> The residue class ring modulo a Gaussian integer z0 is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"> is a Gaussian prime. If z0 is a decomposed prime or the ramified prime 1 + i (that is, if its norm N(z0) is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, N(z0)). It is thus <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the field of the integers modulo N(z0). If, on the other hand, z0 is an inert prime (that is, N(z0) = p2 is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has p2 elements, and it is an <a href="/wiki/Field_extension" title="Field extension">extension</a> of degree 2 (unique, up to an isomorphism) of the <a href="/wiki/Prime_field" class="mw-redirect" title="Prime field">prime field</a> with p elements (the integers modulo p). <div class="mw-heading mw-heading2"><h2 id="Primitive_residue_class_group_and_Euler's_totient_function">Primitive residue class group and Euler's totient function</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=12" title="Edit section: Primitive residue class group and Euler's totient function">edit</a>]</div> Many theorems (and their proofs) for moduli of integers can be directly transferred to moduli of Gaussian integers, if one replaces the absolute value of the modulus by the norm. This holds especially for the primitive residue class group (also called <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo n</a>) and <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a>. The primitive residue class group of a modulus z is defined as the subset of its residue classes, which contains all residue classes a that are coprime to z, i.e. (a,z) = 1. Obviously, this system builds a <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a>. The number of its elements shall be denoted by ϕ(z) (analogously to Euler's totient function φ(n) for integers n). For Gaussian primes it immediately follows that ϕ(p) = |p|2 − 1 and for arbitrary composite Gaussian integers <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=i^{k}\prod _{m}{p_{m}}^{\nu _{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=i^{k}\prod _{m}{p_{m}}^{\nu _{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038a7c8e3432ef17c86c0de20a6da431bade1b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.046ex; height:5.509ex;" alt="{\displaystyle z=i^{k}\prod _{m}{p_{m}}^{\nu _{m}}}"></dd></dl> <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's product formula</a> can be derived as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (z)=\prod _{m\,(\nu _{m}>0)}{\bigl |}{p_{m}}^{\nu _{m}}{\bigr |}^{2}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)=|z|^{2}\prod _{p_{m}|z}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msup> <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> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (z)=\prod _{m\,(\nu _{m}>0)}{\bigl |}{p_{m}}^{\nu _{m}}{\bigr |}^{2}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)=|z|^{2}\prod _{p_{m}|z}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7f71a2bd6a3b6aa56ad822ffd4533297a11004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:60.688ex; height:7.176ex;" alt="{\displaystyle \phi (z)=\prod _{m\,(\nu _{m}>0)}{\bigl |}{p_{m}}^{\nu _{m}}{\bigr |}^{2}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)=|z|^{2}\prod _{p_{m}|z}\left(1-{\frac {1}{|p_{m}|{}^{2}}}\right)}"></dd></dl> where the product is to build over all prime divisors pm of z (with νm > 0). Also the important <a href="/wiki/Euler%27s_theorem" title="Euler's theorem">theorem of Euler</a> can be directly transferred: <dl><dd>For all a with (a,z) = 1, it holds that aϕ(z) ≡ 1 (mod z).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=13" title="Edit section: Historical background">edit</a>]</div> The ring of Gaussian integers was introduced by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> in his second monograph on <a href="/wiki/Quartic_reciprocity" title="Quartic reciprocity">quartic reciprocity</a> (1832).<a href="#cite_note-6">[6]</a> The theorem of <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a> (which he had first succeeded in proving in 1796) relates the solvability of the congruence x2 ≡ q (mod p) to that of x2 ≡ p (mod q). Similarly, cubic reciprocity relates the solvability of x3 ≡ q (mod p) to that of x3 ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation between x4 ≡ q (mod p) and x4 ≡ p (mod q). Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers). In a footnote he notes that the <a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a> are the natural domain for stating and proving results on <a href="/wiki/Cubic_reciprocity" title="Cubic reciprocity">cubic reciprocity</a> and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws. This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory. <div class="mw-heading mw-heading2"><h2 id="Unsolved_problems">Unsolved problems</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=14" title="Edit section: Unsolved problems">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gauss-primes-768x768.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Gauss-primes-768x768.png/170px-Gauss-primes-768x768.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Gauss-primes-768x768.png/255px-Gauss-primes-768x768.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Gauss-primes-768x768.png/340px-Gauss-primes-768x768.png 2x" data-file-width="768" data-file-height="768" /></a><figcaption>The distribution of the small Gaussian primes in the complex plane</figcaption></figure> Most of the unsolved problems are related to distribution of Gaussian primes in the plane. <ul><li><a href="/wiki/Gauss%27s_circle_problem" class="mw-redirect" title="Gauss's circle problem">Gauss's circle problem</a> does not deal with the Gaussian integers per se, but instead asks for the number of <a href="/wiki/Lattice_point" class="mw-redirect" title="Lattice point">lattice points</a> inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.</li></ul> There are also conjectures and unsolved problems about the Gaussian primes. Two of them are: <ul><li>The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form 1 + ki?<a href="#cite_note-7">[7]</a></li> <li>Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of a uniformly bounded length? This is known as the <a href="/wiki/Gaussian_moat" title="Gaussian moat">Gaussian moat</a> problem; it was posed in 1962 by <a href="/wiki/Basil_Gordon" title="Basil Gordon">Basil Gordon</a> and remains unsolved.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=15" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Algebraic_integer" title="Algebraic integer">Algebraic integer</a></li> <li><a href="/wiki/Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a href="/wiki/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Hurwitz_quaternion" title="Hurwitz quaternion">Hurwitz quaternion</a></li> <li><a href="/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares" class="mw-redirect" title="Proofs of Fermat's theorem on sums of two squares">Proofs of Fermat's theorem on sums of two squares</a></li> <li><a href="/wiki/Proofs_of_quadratic_reciprocity" title="Proofs of quadratic reciprocity">Proofs of quadratic reciprocity</a></li> <li><a href="/wiki/Quadratic_integer" title="Quadratic integer">Quadratic integer</a></li> <li><a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">Splitting of prime ideals in Galois extensions</a> describes the structure of prime ideals in the Gaussian integers</li> <li><a href="/wiki/Table_of_Gaussian_integer_factorizations" title="Table of Gaussian integer factorizations">Table of Gaussian integer factorizations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=16" title="Edit section: Notes">edit</a>]</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-Fraleigh_1976_286-1">^ <a href="#cite_ref-Fraleigh_1976_286_1-0">a</a> <a href="#cite_ref-Fraleigh_1976_286_1-1">b</a> <a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 286) </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 289) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 288) </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 287) </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFGauss1831">Gauss (1831</a>, p. 546) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFKleiner1998">Kleiner (1998)</a> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F) </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <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="CITEREFGethnerWagonWick1998" class="citation journal cs1">Gethner, Ellen; <a href="/wiki/Stan_Wagon" title="Stan Wagon">Wagon, Stan</a>; Wick, Brian (1998). "A stroll through the Gaussian primes". <a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 105 (4): 327–337. <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%2F2589708">10.2307/2589708</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/2589708">2589708</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=1614871">1614871</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0946.11002">0946.11002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=A+stroll+through+the+Gaussian+primes&rft.volume=105&rft.issue=4&rft.pages=327-337&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0946.11002%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1614871%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589708%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2589708&rft.aulast=Gethner&rft.aufirst=Ellen&rft.au=Wagon%2C+Stan&rft.au=Wick%2C+Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004). Unsolved problems in number theory (3rd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 55–57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20860-2" title="Special:BookSources/978-0-387-20860-2"><bdi>978-0-387-20860-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1058.11001">1058.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+problems+in+number+theory&rft.pages=55-57&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11001%23id-name%3DZbl&rft.isbn=978-0-387-20860-2&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=17" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1831" class="citation cs2">Gauss, C. F. (1831), <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=285">"Theoria residuorum biquadraticorum. Commentatio secunda."</a>, Comm. Soc. Reg. Sci. Göttingen, 7: 89–148</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comm.+Soc.+Reg.+Sci.+G%C3%B6ttingen&rft.atitle=Theoria+residuorum+biquadraticorum.+Commentatio+secunda.&rft.volume=7&rft.pages=89-148&rft.date=1831&rft.aulast=Gauss&rft.aufirst=C.+F.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015073697180%26view%3D1up%26seq%3D285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988">; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93–148. A German translation of this paper is available online in ″H. Maser (ed.): <a rel="nofollow" class="external text" href="https://archive.org/details/carlfriedrichga00gausgoog">Carl Friedrich Gauss’ Arithmetische Untersuchungen über höhere Arithmetik.</a> Springer, Berlin 1889, pp. 534″.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraleigh1976" class="citation cs2">Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-01984-1" title="Special:BookSources/0-201-01984-1"><bdi>0-201-01984-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+In+Abstract+Algebra&rft.place=Reading&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1976&rft.isbn=0-201-01984-1&rft.aulast=Fraleigh&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1998" class="citation journal cs1">Kleiner, Israel (1998). <a rel="nofollow" class="external text" href="https://ems.press/journals/em/articles/664">"From Numbers to Rings: The Early History of Ring Theory"</a>. <a href="/wiki/Elem._Math." class="mw-redirect" title="Elem. Math.">Elem. Math.</a> 53 (1): 18–35. <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%2Fs000170050029">10.1007/s000170050029</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0908.16001">0908.16001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Elem.+Math.&rft.atitle=From+Numbers+to+Rings%3A+The+Early+History+of+Ring+Theory&rft.volume=53&rft.issue=1&rft.pages=18-35&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0908.16001%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1007%2Fs000170050029&rft.aulast=Kleiner&rft.aufirst=Israel&rft_id=https%3A%2F%2Fems.press%2Fjournals%2Fem%2Farticles%2F664&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation book cs1"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996). The New Book of Prime Number Records (3rd ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94457-5" title="Special:BookSources/0-387-94457-5"><bdi>0-387-94457-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0856.11001">0856.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.edition=3rd&rft.pub=Springer&rft.date=1996&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0856.11001%23id-name%3DZbl&rft.isbn=0-387-94457-5&rft.aulast=Ribenboim&rft.aufirst=Paulo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenry_G._Baker1993" class="citation journal cs1">Henry G. Baker (1993). "Complex Gaussian Integers for "Gaussian Graphics"". ACM SIGPLAN Notices. 28 (11): 22–27. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F165564.165571">10.1145/165564.165571</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:8083226">8083226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGPLAN+Notices&rft.atitle=Complex+Gaussian+Integers+for+%22Gaussian+Graphics%22&rft.volume=28&rft.issue=11&rft.pages=22-27&rft.date=1993&rft_id=info%3Adoi%2F10.1145%2F165564.165571&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8083226%23id-name%3DS2CID&rft.au=Henry+G.+Baker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGaussian+integer" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Gaussian_integer&action=edit&section=18" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120306225505/http://www.imocompendium.com/index.php?options=mbb%7Ctekstkut&page=0&art=extensions_ddj%7Cf&ttn=Dushan%20D%3Bjukic1%7C%20Arithmetic%20in%20Quadratic%20Fields%7CN%2FA&knj=&p=3nbbw45001">IMO Compendium</a> text on quadratic extensions and Gaussian Integers in problem solving</li> <li>Keith Conrad, <a rel="nofollow" class="external text" href="https://kconrad.math.uconn.edu/blurbs/ugradnumthy/Zinotes.pdf">The Gaussian Integers</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .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="Algebraic_numbers" 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:Algebraic_numbers" title="Template:Algebraic numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_numbers" title="Template talk:Algebraic numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_numbers" title="Special:EditPage/Template:Algebraic numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Algebraic_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic 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"> <ul><li><a href="/wiki/Algebraic_integer" title="Algebraic integer">Algebraic integer</a></li> <li><a href="/wiki/Chebyshev_nodes" title="Chebyshev nodes">Chebyshev nodes</a></li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible number</a></li> <li><a href="/wiki/Look-and-say_sequence" title="Look-and-say sequence">Conway's constant</a></li> <li><a href="/wiki/Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a class="mw-selflink selflink">Gaussian integer</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio (φ)</a></li> <li><a href="/wiki/Perron_number" title="Perron number">Perron number</a></li> <li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li> <li><a href="/wiki/Plastic_ratio" title="Plastic ratio">Plastic ratio (ρ)</a></li> <li><a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational number</a></li> <li><a href="/wiki/Root_of_unity" title="Root of unity">Root of unity</a></li> <li><a href="/wiki/Salem_number" title="Salem number">Salem number</a></li> <li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver ratio (δS)</a></li> <li><a href="/wiki/Square_root_of_2" title="Square root of 2">Square root of 2</a></li> <li><a href="/wiki/Square_root_of_3" title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Square_root_of_6" title="Square root of 6">Square root of 6</a></li> <li><a href="/wiki/Square_root_of_7" title="Square root of 7">Square root of 7</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (ψ)</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (ς)</a></li> <li><a href="/wiki/Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</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="Prime_number_classes" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fermat_number" title="Fermat number">Fermat (22n + 1)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (2p − 1)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (22p−1 − 1)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff (2p + 1)/3</a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (k·2n + 1)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (n! ± 1)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (pn# ± 1)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (pn# + 1)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (4n + 1)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (2m·3n + 1)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (x4 + y4)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (2m ± 2n ± 1)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (n·2n + 1)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (n·2n − 1)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (x3 − y3)/(x − y)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (xy + yx)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (3·2n − 1)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams ((b−1)·bn − 1)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (⌊A3n⌋)</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 (10n − 1)/9</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">k-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 (p, p + 2)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (p, p + 2 or p + 4, p + 6)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (p, p + 2, p + 6, p + 8)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (p, p + 4)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (p, p + 6)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (consecutive p − n, p, p + n)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)</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 (p, 2p + 1)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)</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 class="mw-selflink-fragment" href="#Gaussian_primes">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Gaussian_integer&oldid=1254081435">https://en.wikipedia.org/w/index.php?title=Gaussian_integer&oldid=1254081435</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:Algebraic_numbers" title="Category:Algebraic numbers">Algebraic numbers</a></li><li><a href="/wiki/Category:Cyclotomic_fields" title="Category:Cyclotomic fields">Cyclotomic fields</a></li><li><a href="/wiki/Category:Lattice_points" title="Category:Lattice points">Lattice points</a></li><li><a href="/wiki/Category:Quadratic_irrational_numbers" title="Category:Quadratic irrational numbers">Quadratic irrational numbers</a></li><li><a href="/wiki/Category:Integers" title="Category:Integers">Integers</a></li><li><a href="/wiki/Category:Complex_numbers" title="Category:Complex numbers">Complex numbers</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_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches 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 29 October 2024, at 10:03 (UTC).</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=Gaussian_integer&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-688fc9465-gbnnw","wgBackendResponseTime":173,"wgPageParseReport":{"limitreport":{"cputime":"0.788","walltime":"1.003","ppvisitednodes":{"value":15600,"limit":1000000},"postexpandincludesize":{"value":131208,"limit":2097152},"templateargumentsize":{"value":32702,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":45630,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 750.915 1 -total"," 45.47% 341.408 295 Template:Math"," 21.06% 158.179 3 Template:Navbox"," 18.78% 141.026 1 Template:Reflist"," 13.19% 99.036 1 Template:Algebraic_numbers"," 11.49% 86.260 3 Template:Cite_journal"," 8.91% 66.927 1 Template:Short_description"," 7.36% 55.294 1 Template:Prime_number_classes"," 7.36% 55.237 298 Template:Main_other"," 5.78% 43.433 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.368","limit":"10.000"},"limitreport-memusage":{"value":6515704,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFFraleigh1976\"] = 1,\n [\"CITEREFGauss1831\"] = 1,\n [\"CITEREFGethnerWagonWick1998\"] = 1,\n [\"CITEREFGuy2004\"] = 1,\n [\"CITEREFHenry_G._Baker1993\"] = 1,\n [\"CITEREFKleiner1998\"] = 1,\n [\"CITEREFRibenboim1996\"] = 1,\n [\"congruences\"] = 1,\n [\"gcd\"] = 1,\n [\"selected_associates\"] = 1,\n [\"unique_remainder\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 7,\n [\"=\"] = 24,\n [\"Abs\"] = 2,\n [\"Algebraic numbers\"] = 1,\n [\"Anchor\"] = 4,\n [\"Angbr\"] = 1,\n [\"Citation\"] = 2,\n [\"Cite book\"] = 2,\n [\"Cite journal\"] = 3,\n [\"Distinguish\"] = 1,\n [\"Harvtxt\"] = 6,\n [\"Math\"] = 265,\n [\"Math proof\"] = 1,\n [\"Mvar\"] = 9,\n [\"Nowrap\"] = 2,\n [\"OEIS\"] = 1,\n [\"Overline\"] = 16,\n [\"Prime number classes\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfrac\"] = 8,\n [\"Short description\"] = 1,\n [\"Sqrt\"] = 5,\n [\"Void\"] = 3,\n}\narticle_whitelist = table#1 {\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-8pvr2","timestamp":"20241125141948","ttl":34816,"transientcontent":true}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Gaussian integer","url":"https:\/\/en.wikipedia.org\/wiki\/Gaussian_integer","sameAs":"http:\/\/www.wikidata.org\/entity\/Q724975","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q724975","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":"2002-04-10T21:11:22Z","dateModified":"2024-10-29T10:03:49Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/9e\/Gaussian_integer_lattice.svg","headline":"complex number whose real and imaginary parts are both integers"}</script> </body> </html>