Number theory | Definition, Topics, & History

<!doctype html> <html lang="en" class="topic-desktop ui-ie7 ui-ie"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb#"> <meta charset="utf-8"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <link rel="dns-prefetch" href="https://cdn.britannica.com/mendel-resources/3-130"> <link rel="preconnect" href="https://cdn.britannica.com/mendel-resources/3-130"> <link rel="preload" as="script" href="https://www.googletagservices.com/tag/js/gpt.js" /> <link rel="icon" href="/favicon.png" /> <meta name="description" content="Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory." /> <meta name="keywords" content="number theory, encyclopedia, encyclopeadia, britannica, article" /> <link rel="canonical" href="https://www.britannica.com/science/number-theory" /> <title>Number theory | Definition, Topics, & History | Britannica</title> <script type="text/javascript" data-type="Init Mendel"> window.$UI = {}; window.Constants = {"LICENSE_URL": "/bps/license","DEFAULT_TEST_VERSION": "A","DEFAULT_STATE": "XX","QUIZ_URL": "/quiz","SPOTLIGHT_BROWSE_URL": "/stories/spotlight","CONTENT_TYPE_TEXT": "text/plain;charset=UTF-8","TOPIC_FACTS_DATA_URL": "/facts","QUIZ_BROWSE_IMAGE_QUIZZES": "images","TOPIC_MEDIA_PATH": "/images-videos","USER_PROFILE_URL": "/user","DEBUG_URL": "/debug","ONE_GOOD_FACT_URL": "/one-good-fact","ERROR_404_URL": "/error404","PROCON_CITED_IN_THE_NEWS_URL": "/procon/ProCon-in-the-News","PROCON_URL": "/procon","TOPIC_PAGE_CONTENT_AJAX_URL": "/topic-content/page","INFINITE_SCROLL_PREFIX_URL": "/scroll","TOPIC_TOP_QUESTION_BROWSE_URL": "/questions","CC_USD": "USD","domain": "britannica.com","SURVEY_URL": "/survey","CATEGORY_BROWSE_URL": "/browse","STORY_BROWSE_URL": "/stories","COUNTRY_US": "US","OPEN_MEDIA_OVERLAY_PARAMETER": "/media","NEWSLETTER_SUBSCRIPTION_URL": "/newsletter-subscription","MAINTENANCE_ERROR_URL": "https://maintenance.eb.com","IMARS_EDITOR_ID": "12365882","PROFILE_EB_EDITOR_URL": "/editor","WEB_INF_RESOURCES_PATH": "WEB-INF/resources","AI_ABOUT_PAGE_URL": "/about-britannica-ai","TOPIC_ADDITIONAL_INFO_PATH": "/additional-info","SUDOKU_GAME_URL": "/games/sudoku","CC_INR": "INR","ARTICLE_PRINT_URL": "/print/article","FIRST_EDITION_URL": "/subscriber/firstedition","WW1_PORTAL_URL": "/discover/World-War-I","MENDEL_COOKIE": "__mendel","DEMYSTIFIED_BROWSE_URL": "/stories/demystified","LIST_BROWSE_URL": "/list/browse","PROFILE_EXPERT_URL": "/contributor","ASSEMBLY_IMAGE_URL": "/image/assembly","DAY_IN_HISTORY_URL": "/on-this-day","DEFAULT_CURRENCY": "USD","CONTENT_TYPE_XML": "text/xml;charset=UTF-8","PORTAL_FINANCE_BROWSE_URL_PREFIX": "/money/browse","MONEY_IMARS_CATEGORY": "13000","AJAX_PREFIX_URL": "/ajax","TOPIC_BROWSE_URL": "/topic-browse","MARKETING_CONTENT": "/marketing-content","ENV_RUNTIME": "runtime","GALLERY_URL": "/gallery","topicUrlClassesList": "topic|animal|art|biography|event|place|plant|science|sports|technology|procon","CONTENT_TYPE_HTML": "text/html;charset=UTF-8","ENV_LOCAL": "override","MEDIA_OVERLAY_URL": "/media-overlay","CHATBOT_PAGE_URL": "/chatbot","NEWSLETTER_PAGE_URL": "/newsletters","ENV_DEV": "development","MEDIA_URL": "/media","TOPIC_TOP_QUESTION_URL": "/question","PORTAL_FINANCE_URL_PREFIX": "/money","PODCASTS_URL": "/podcasts","STAND_ALONE_VIDEO_URL": "/video","MORE_ON_THIS_DAY_URL": "/more-on-this-day","TOPIC_QUOTES_URL": "/quotes","SEARCH_PAGE_URL": "/search","PROCON_CLASS": "PROCON","KUSTOM_MENDEL_APPLICATION_ID": "1","TOPIC_CONTENT_AJAX_URL": "/topic-content/topic","ENV_BRANCH": "branch","ERROR_URL": "/error","MAIN_VERSION": "mainVersion","DEFAULT_S3_REGION": "US_EAST_1","TOPIC_COLLECTION_URL": "/summary","LOGINBOX_URL": "/auth/loginbox","ONE_GOOD_FACT_BROWSE_URL": "/one-good-fact/all-good-facts","QUIZ_BROWSE_URL": "/quiz/browse","BIO_BROWSE_URL": "/browse/biographies","LIST_URL": "/list","TIGHTROPE_QUIZ_URL": "/quiz/tightrope","ALPHA_BROWSE_URL": "/sitemap","CONTENT_TYPE_JSON": "application/json","DICTIONARY_URL": "/dictionary","COBRAND_IMAGE_URL": "/image/cobrand","PROCON_IN_THE_NEWS_URL": "/procon/pro-and-con-issues-in-the-news","PROCON_BROWSE_URL": "/procon","QUIZ_BROWSE_VOCAB_QUIZZES": "vocabulary-quizzes","SUBMISSION_URL": "/submission","EB_LOG_OUT": "/auth2/logout","ENV_PRODUCTION": "production","EXPLORE_PORTAL_URL": "/explore","TOPIC_AJAX_URL": "/ajax/topic","TOPIC_SUMMARY_BROWSE_URL": "/summaries","WTFACT_BROWSE_URL": "/stories/wtfact","VIDEO_CHANNEL_URL": "/videos","GALLERY_BROWSE_URL": "/gallery/browse","CACHE_URL": "/cache","PROCON_ABOUT_URL": "/procon/About-ProCon","COMPANION_BROWSE_URL": "/stories/companion","MEDIA_FOLDER": "/eb-media","SHOW_ALL_CONTRIBUTORS": "/additional-info#contributors","BRITANNICA_EDITORS_ID": "4419","ENV_CACHE_DISABLED": "mendelCache","CALCULATORS_BROWSE_URL": "/calculators","STORY_URL": "/story","DEFAULT_COUNTRY": "US","NAVBAR_URL": "/ajax/navbar","EB_LOGIN_URL": "/auth/eb-login","NEW_ARTICLES_URL": "/new-articles",}; window.CDN = "https://cdn.britannica.com"; window.CAM_LOGIN_URL = "https://cam.britannica.com"; window.CAM_SIGN_UP_URL = "https://cam.britannica.com/registration" window.Mendel = { "config" : { "domain": "britannica.com", "page": "Topic", "videoPlayerId": "UyMCoK2v", "sharedUrl": "https://www.britannica.com/science/number-theory", "amuselabsUrl": "https://cdn3.amuselabs.com", "resourcesPrefixUrl": "https://cdn.britannica.com/mendel-resources/3-130/[url]?v=3.130.14", "date": 20241128, "userInfo": { "type": "ANONYMOUS" ,"currency": "AUUS" ,"country": "SG" ,"state": "XX" ,"timezone": "Asia/Singapore" ,"bcomId": "751141081076087470" ,"hasAds": true ,"testVersion": "C" ,"adsTestVersion": "C" ,"consumerId": "" ,"instId": "" ,"consumerUserName": "" ,"instUserName": "" ,"cognito": null }, "tvs":{ "r":[25,25,25,25], "a": [25,25,25,25]}, "isLoggedInAsUser": false, "isPhone": false, "isDesktop": true, "logoutUrl": "/auth2/logout", "selfServiceUrl": "https://myaccount.britannica.com", "cdnUrl": "https://cdn.britannica.com", "chatbotApi": "https://www.britannica.com/chat-api", "fetchOffset": 800, "mendelCookieName": "__mendel", "mendelCookie": {"surveyShown":false,"visitedTopicId":422325,"currentDate":20241128}, "autocompleteToSearchPage": false,"topicUrl": "https://www.britannica.com/science/number-theory" ,"freeTopicReason": "PERMANENT_FREE_TOPIC" ,"topicId": 422325 ,"template": "DESKTOP" ,"type": "CORE" ,"hasToc": true ,"chatbotApi": "https://www.britannica.com/chat-api" ,"showPreview": false ,"infiniteScrollList": [{"p":4,"t":422325},{"p":1,"t":920687},{"p":1,"t":164341},{"p":1,"t":422375},{"p":1,"t":1372353},{"p":1,"t":476309},{"p":3,"t":34730},{"p":1,"t":204685},{"p":3,"t":682032},{"p":1,"t":205791}] ,"sequence": 1 ,"topics": {} }, "GA": {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} }; </script> <meta property="fb:app_id" content="1887621861548296"/ <meta name="twitter:card" content="summary_large_image" /> <meta name="twitter:site" content="@britannica" /> <meta name="twitter:image" content="https://cdn.britannica.com/mendel-resources/3-130/images/social/eb-thistle-social-image.jpg?v=3.130.14" /> <meta name="twitter:description" content="Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory."/> <meta property="og:type" content="ARTICLE"/> <meta property="og:title" content="Number theory | Definition, Topics, & History | Britannica"/> <meta property="og:description" content="Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory."/> <meta property="og:site_name" content="Encyclopedia Britannica" /> <meta property="og:url" content="https://www.britannica.com/science/number-theory"/> <meta property="og:image" content="https://cdn.britannica.com/mendel-resources/3-130/images/social/eb-thistle-social-image.jpg?v=3.130.14" /> <meta property="og:image:type" content="" /> <script type="text/javascript" data-type="init opengraph"> Mendel.openGraph = {"type":"ARTICLE","title":"Number theory | Definition, Topics, & History","description":"Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory.","imageUrl":"https://cdn.britannica.com/mendel-resources/3-130/images/social/eb-thistle-social-image.jpg?v=3.130.14","pageUrl":"https://www.britannica.com/science/number-theory"}</script> <link rel="preconnect" href="https://fonts.googleapis.com/"> <link rel="dns-prefetch" href="https://fonts.googleapis.com/" > <link rel="stylesheet" href="https://fonts.googleapis.com/icon?family=Material+Icons"> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/vendor-bundle.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/mendel-css.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.css?v=3.130.14" rel="stylesheet" /> <script type="text/javascript"> if (self !== top) { top.location = self.location; } // if ('scrollRestoration' in history) { history.scrollRestoration = 'manual'; } </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/at.js?v=3.130.14" async ></script> <script> dataLayer = []; </script> <script type="text/javascript">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= '//www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-5W6NC8'); </script> <script type="application/ld+json"> {"headline":"Number theory | Definition, Topics, & History","author":[{"name":"William Dunham","url":"https://www.britannica.com/contributor/William-Dunham/5198","@type":"Person"}],"keywords":"number theory","wordcount":3820,"url":"https://www.britannica.com/science/number-theory","datePublished":"1999-07-26T00:00:00Z","description":"Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory.","publisher":{"name":"Encyclopedia Britannica","@type":"Organization","logo":{"url":"https://corporate.britannica.com/wp-content/themes/eb-corporate/_img/logo.png","@type":"ImageObject"}},"@context":"https://schema.org","@type":"article"} </script></head> <body data-leg="C" class="new-topic topic-desktop first-page-true user-ANONYMOUS user-ads md-desktop leg-cb-ie">  <script>Mendel.config.adProvider='ay';</script> <script async src="https://securepubads.g.doubleclick.net/tag/js/gpt.js"></script> <script> window.googletag = window.googletag || {cmd: []}; googletag.cmd.push(function() { googletag.defineSlot('/15510053/CMP_1x1', [1, 1], 'div-gpt-ad-1709766812090-0').addService(googletag.pubads()); googletag.pubads().enableSingleRequest(); googletag.enableServices(); }); </script> <script async defer src="https://launchpad-wrapper.privacymanager.io/0ccc6fe8-1870-4ad8-b47b-6d029ac116fc/launchpad-liveramp.js"></script> <script async src="https://JRyhoywLYXNLYMAhs.ay.delivery/manager/JRyhoywLYXNLYMAhs" type="text/javascript" referrerpolicy="no-referrer-when-downgrade" ></script><div class="ie-warning d-flex align-items-center align-self-center justify-content-center site-alert bg-orange"> <div> You are using an <strong>outdated</strong> browser. Please <a class="text-white text-underscore" href="https://browsehappy.com/">upgrade your browser</a> to improve your experience and security. </div> </div> <script id="json-navbar-info" type="application/json"> {"topSectionLinks":[{"title":"Ask the Chatbot","url":"/chatbot","navbarId":"CHATBOT"},{"title":"Games & Quizzes","url":"/quiz/browse","navbarId":"QUIZZES"},{"title":"ProCon","url":"/procon","navbarId":"PROCON"},{"title":"History & Society","url":"/History-Society","navbarId":"HISTORY"},{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"},{"title":"Biographies","url":"/Biographies","navbarId":"BIOS"},{"title":"Animals & Nature","url":"/Animals-Nature","navbarId":"ANIMALS"},{"title":"Geography & Travel","url":"/Geography-Travel","navbarId":"GEOGRAPHY"},{"title":"Arts & Culture","url":"/Arts-Culture","navbarId":"ART"},{"title":"Money","url":"/money","navbarId":"MONEY"},{"title":"Videos","url":"/videos","navbarId":"VIDEOS"}],"selectedSuperCategory":{"id":6,"title":"Science & Tech","url":"Science-Tech","description":"Explore science and technology; astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","keywords":"astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","classId":"SCIENCE","sortOrder":2},"selectedNavbarLink":{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"}} </script> <script id="json-hamburger-menu" type="application/json"> {"britannicaMenu1":[{"title":"Home","url":"/"},{"title":"ProCon","url":"/procon"},{"title":"History & Society","url":"/History-Society"},{"title":"Science & Tech","url":"/Science-Tech"},{"title":"Biographies","url":"/Biographies"},{"title":"Animals & Nature","url":"/Animals-Nature"},{"title":"Geography & Travel","url":"/Geography-Travel"},{"title":"Arts & Culture","url":"/Arts-Culture"},{"title":"Money","url":"/money"}],"britannicaMenu2":[{"title":"Games & Quizzes","url":"/quiz/browse"},{"title":"Videos","url":"/videos"},{"title":"On This Day","url":"/on-this-day"},{"title":"One Good Fact","url":"/one-good-fact"},{"title":"Dictionary","url":"/dictionary"},{"title":"New Articles","url":"/new-articles"}],"browseByCategory":[{"title":{"id":5,"title":"History & Society","url":"/History-Society"},"links":[{"title":"Lifestyles & Social Issues","url":"/browse/Lifestyles-Social-Issues"},{"title":"Philosophy & Religion","url":"/browse/Philosophy-Religion"},{"title":"Politics, Law & Government","url":"/browse/Politics-Law-Government"},{"title":"World History","url":"/browse/World-History"}]},{"title":{"id":6,"title":"Science & Tech","url":"/Science-Tech"},"links":[{"title":"Health & Medicine","url":"/browse/Health-Medicine"},{"title":"Science","url":"/browse/Science"},{"title":"Technology","url":"/browse/Technology"}]},{"title":{"id":3,"title":"Biographies","url":"/Biographies"},"links":[{"title":"Browse Biographies","url":"/browse/biographies"}]},{"title":{"id":1,"title":"Animals & Nature","url":"/Animals-Nature"},"links":[{"title":"Birds, Reptiles & Other Vertebrates","url":"/browse/Birds-Reptiles-Vertebrates"},{"title":"Bugs, Mollusks & Other Invertebrates","url":"/browse/Bugs-Mollusks-Invertebrates"},{"title":"Environment","url":"/browse/Environment"},{"title":"Fossils & Geologic Time","url":"/browse/Fossil-Geologic-Time"},{"title":"Mammals","url":"/browse/Mammals"},{"title":"Plants","url":"/browse/Plants"}]},{"title":{"id":4,"title":"Geography & Travel","url":"/Geography-Travel"},"links":[{"title":"Geography & Travel","url":"/browse/Geography-Travel"}]},{"title":{"id":2,"title":"Arts & Culture","url":"/Arts-Culture"},"links":[{"title":"Entertainment & Pop Culture","url":"/browse/Entertainment-Pop-Culture"},{"title":"Literature","url":"/browse/Literature"},{"title":"Sports & Recreation","url":"/browse/Sports-Recreation"},{"title":"Visual Arts","url":"/browse/Visual-Arts"}]}],"browseByFeature":[{"title":"Companions","url":"/stories/companion"},{"title":"Demystified","url":"/stories/demystified"},{"title":"Image Galleries","url":"/gallery/browse"},{"title":"Lists","url":"/list/browse"},{"title":"Podcasts","url":"/podcasts"},{"title":"Spotlight","url":"/stories/spotlight"},{"title":"Summaries","url":"/summary"},{"title":"The Forum","url":"/stories/the-forum"},{"title":"Top Questions","url":"/question"},{"title":"#WTFact","url":"/stories/wtfact"}],"moreFromBritannica":[{"title":"Britannica Kids","url":"https://kids.britannica.com/","newTab":true}],"menuType":"DEFAULT"} </script> <header id="header" class="bg-navy-dark"> <div class="global-nav-top-bar"> <div class="grid gx-0 h-100 justify-content-between align-items-center container-lg mx-auto p-0 position-relative"> <div class="d-flex align-items-center"> <button class="d-flex align-items-center justify-self-start js-toggle js-toggle-hamburger btn btn-link link-white btn-sm rounded-0 p-10"> <div class="hamburger-tooltip"> <em class="material-icons d-inline-block font-24" id="nav-toggle" data-icon="menu"></em> </div> <em class="material-icons d-inline-block font-24 global-nav-search-icon" id="nav-search-icon" data-icon="search" ></em> </button> <a href="/" class="d-flex align-items-center justify-content-center ml-10"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-logo global-nav-logo-left" /> </a> <div class="global-nav-top-search-bar global-nav-top-search-container global-nav-search-container" id="global-nav-top-search-bar"> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> </div> </div> <a href="/" class="d-flex align-items-center justify-content-center"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-center global-nav-logo non-homepage-logo" /> </a> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small global-nav-center search global-nav-center-search-container"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> <div class="col-35 col-sm-auto text-right order-3 mr-lg-15 align-items-center d-flex justify-content-end"> <div class="d-none d-md-inline-block"> <SPAN class="marketing-HEADER_SUBSCRIPTION_DESKTOP2 marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_DESKTOP2"><a href="https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-sm btn-outline-white-orange py-5 mr-10" target="_blank" rel="noopener"> Subscribe </a></SPAN></div> <div class="d-inline-block d-md-none mr-5 mr-sm-10"> <SPAN class="marketing-HEADER_SUBSCRIPTION_MOBILE marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_MOBILE"><a href="https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-xs btn-orange-dark p-5" target="_blank" rel="noopener"> Subscribe </a></SPAN></div> <button class="js-toggle-user-dropdown js-toggle btn btn-sm btn-link link-white rounded-0 px-md-15 pl-5 pr-5"> <span class="d-none d-md-inline-block mr-5">Login</span> <em class="material-icons d-inline-block d-md-none font-16 font-sm-20" data-icon="account_circle"></em> <div class="d-none dropdown-menu-subscription-link">https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS</div> <em class="material-icons inactive-icon d-inline-block font-18" data-icon="keyboard_arrow_down"></em> <em class="material-icons active-icon d-inline-block font-18" data-icon="keyboard_arrow_up"></em> </button> </div> </div> </div> <div class="d-none hamburger-menu-subscription-link"><DIV class="marketing-HAMBURGER_MENU_CTA marketing-content" data-marketing-id="HAMBURGER_MENU_CTA"><a href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=black-friday-2024" class="subscribe-link btn btn-sm btn-orange py-5" target="_blank"> Subscribe <span class="d-none d-md-inline"> Now</span> </a></DIV></div> <div id="global-nav-react"> <div class="d-none"> <ul> <li><a href="/">Home</a></li> <li><a href="/procon">ProCon</a></li> <li><a href="/History-Society">History & Society</a></li> <li><a href="/Science-Tech">Science & Tech</a></li> <li><a href="/Biographies">Biographies</a></li> <li><a href="/Animals-Nature">Animals & Nature</a></li> <li><a href="/Geography-Travel">Geography & Travel</a></li> <li><a href="/Arts-Culture">Arts & Culture</a></li> <li><a href="/money">Money</a></li> </ul> <ul> <li><a href="/quiz/browse">Games & Quizzes</a></li> <li><a href="/videos">Videos</a></li> <li><a href="/on-this-day">On This Day</a></li> <li><a href="/one-good-fact">One Good Fact</a></li> <li><a href="/dictionary">Dictionary</a></li> <li><a href="/new-articles">New Articles</a></li> </ul> <a href="/History-Society">History & Society</a> <ul> <li><a href="/browse/Lifestyles-Social-Issues">Lifestyles & Social Issues</a></li> <li><a href="/browse/Philosophy-Religion">Philosophy & Religion</a></li> <li><a href="/browse/Politics-Law-Government">Politics, Law & Government</a></li> <li><a href="/browse/World-History">World History</a></li> </ul> <a href="/Science-Tech">Science & Tech</a> <ul> <li><a href="/browse/Health-Medicine">Health & Medicine</a></li> <li><a href="/browse/Science">Science</a></li> <li><a href="/browse/Technology">Technology</a></li> </ul> <a href="/Biographies">Biographies</a> <ul> <li><a href="/browse/biographies">Browse Biographies</a></li> </ul> <a href="/Animals-Nature">Animals & Nature</a> <ul> <li><a href="/browse/Birds-Reptiles-Vertebrates">Birds, Reptiles & Other Vertebrates</a></li> <li><a href="/browse/Bugs-Mollusks-Invertebrates">Bugs, Mollusks & Other Invertebrates</a></li> <li><a href="/browse/Environment">Environment</a></li> <li><a href="/browse/Fossil-Geologic-Time">Fossils & Geologic Time</a></li> <li><a href="/browse/Mammals">Mammals</a></li> <li><a href="/browse/Plants">Plants</a></li> </ul> <a href="/Geography-Travel">Geography & Travel</a> <ul> <li><a href="/browse/Geography-Travel">Geography & Travel</a></li> </ul> <a href="/Arts-Culture">Arts & Culture</a> <ul> <li><a href="/browse/Entertainment-Pop-Culture">Entertainment & Pop Culture</a></li> <li><a href="/browse/Literature">Literature</a></li> <li><a href="/browse/Sports-Recreation">Sports & Recreation</a></li> <li><a href="/browse/Visual-Arts">Visual Arts</a></li> </ul> <ul> <li><a href="/stories/companion">Companions</a></li> <li><a href="/stories/demystified">Demystified</a></li> <li><a href="/gallery/browse">Image Galleries</a></li> <li><a href="/list/browse">Lists</a></li> <li><a href="/podcasts">Podcasts</a></li> <li><a href="/stories/spotlight">Spotlight</a></li> <li><a href="/summary">Summaries</a></li> <li><a href="/stories/the-forum">The Forum</a></li> <li><a href="/question">Top Questions</a></li> <li><a href="/stories/wtfact">#WTFact</a></li> </ul> <ul> <li><a href="https://kids.britannica.com/">Britannica Kids</a></li> </ul> </div> </div> </header> <div class="bg-navy-dark"> <div class="container-lg p-0 d-flex justify-content-center global-nav-categories-bar overflow-hidden"> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 global-nav-slider category-snap-slider"> <div class="slider-container js-slider-container overflow-hidden d-flex font-14 overflow-hidden text-nowrap mx-5"> <a class="nav-bar-category mx-5 category-link-CHATBOT " href="/chatbot">Ask the Chatbot</a> <a class="nav-bar-category mx-5 category-link-QUIZZES " href="/quiz/browse">Games & Quizzes</a> <a class="nav-bar-category mx-5 category-link-PROCON " href="/procon">ProCon</a> <a class="nav-bar-category mx-5 category-link-HISTORY " href="/History-Society">History & Society</a> <a class="nav-bar-category mx-5 category-link-SCIENCE selected selected" href="/Science-Tech">Science & Tech</a> <a class="nav-bar-category mx-5 category-link-BIOS " href="/Biographies">Biographies</a> <a class="nav-bar-category mx-5 category-link-ANIMALS " href="/Animals-Nature">Animals & Nature</a> <a class="nav-bar-category mx-5 category-link-GEOGRAPHY " href="/Geography-Travel">Geography & Travel</a> <a class="nav-bar-category mx-5 category-link-ART " href="/Arts-Culture">Arts & Culture</a> <a class="nav-bar-category mx-5 category-link-MONEY " href="/money">Money</a> <a class="nav-bar-category mx-5 category-link-VIDEOS " href="/videos">Videos</a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> </div> <main> <div class="md-page-wrapper"> <div id="content" class="md-content"> <div class="md-article-container template-desktop infinite-pagination"> <div class="infinite-scroll-container article last"> <article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="422325"> <div class="grid gx-0"> <div class="col-auto"> <div class="topic-left-rail md-article-drawer position-relative d-flex border-right-sm border-left-sm open"> <div class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/number-theory">number theory</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <ul class="list-unstyled my-0" data-level="h1"><li data-target="#ref1"><div class="pl-25"><a class="link-gray-900 w-100" href="/science/number-theory">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref233895"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/number-theory#ref233895">From prehistory through Classical Greece</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref233896"><a class="w-100 link-gray-900" href="/science/number-theory#ref233896">Pythagoras</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref233897"><a class="w-100 link-gray-900" href="/science/number-theory/Euclid">Euclid</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref233898"><a class="w-100 link-gray-900" href="/science/number-theory/Euclid#ref233898">Diophantus</a></li></ul></div></li><li data-target="#ref233899"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/number-theory/Euclid#ref233899">Number theory in the East</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref233900"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/number-theory/Euclid#ref233900">Modern number theory</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref233901"><a class="w-100 link-gray-900" href="/science/number-theory/Pierre-de-Fermat">Pierre de Fermat</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref233902"><a class="w-100 link-gray-900" href="/science/number-theory/Pierre-de-Fermat#ref233902">Number theory in the 18th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref233903"><a class="w-100 link-gray-900" href="/science/number-theory/Pierre-de-Fermat#ref233903">Number theory in the 19th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref233904"><a class="w-100 link-gray-900" href="/science/number-theory/Pierre-de-Fermat#ref233904"><em>Disquisitiones Arithmeticae</em></a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref233905"><a class="w-100 link-gray-900" href="/science/number-theory/Pierre-de-Fermat#ref233905">From classical to analytic number theory</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref233906"><a class="w-100 link-gray-900" href="/science/number-theory/Prime-number-theorem">Prime number theorem</a></li></ul></li></ul></div></li><li data-target="#ref233907"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/number-theory/Prime-number-theorem#ref233907">Number theory in the 20th century</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref233908"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/number-theory/Prime-number-theorem#ref233908">Unsolved problems</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li></ul> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/number-theory/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/number-theory">Quick Facts & Related Topics</a> </div> <div class="mb-30 tlr-student-links"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> For Students </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/summary/number-theory"> <img loading="lazy" src="https://cdn.britannica.com/mendel-resources/3-130/images/shared/default3.png?v=3.130.14" class="default " height="200" width="200"/> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/summary/number-theory" >number theory summary</a> </div> </div> </div> <div class="mb-30 tlr-related-quizzes"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Quizzes </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/Numbers-and-mathematics"> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg?w=200&h=200&c=crop" alt="Equations written on blackboard" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/Numbers-and-mathematics" >Numbers and Mathematics</a> </div> </div> </div> <div class="mb-30 tlr-discover"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Discover </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/why-is-christmas-in-december"> <img loading="lazy" src="https://cdn.britannica.com/24/190024-131-988B5D32/Nativity-Scene-Adoration-of-the-Magi-Church.jpg?w=200&h=200&c=crop" alt="Nativity Scene, Adoration of the Magi, Church of the Birth of the Virgin Mary, Montenegro" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/why-is-christmas-in-december" >Why Is Christmas in December?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/was-martin-luther-king-jr-a-republican-or-a-democrat"> <img loading="lazy" src="https://cdn.britannica.com/57/203257-131-48B2BCA1/Martin-Luther-King-Jr-marchers-speech-I-August-28th-1963.jpg?w=200&h=200&c=crop" alt="Dr. Martin Luther King Jr. addresses marchers during his "I Have a Dream" speech at the Lincoln Memorial in Washington. August 28th 1963" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/was-martin-luther-king-jr-a-republican-or-a-democrat" >Was Martin Luther King, Jr., a Republican or a Democrat?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/whats-the-difference-between-buffalo-and-bison"> <img loading="lazy" src="https://cdn.britannica.com/04/187404-131-2839AF60/bison-American-buffalo-plains-prairie.jpg?w=200&h=200&c=crop" alt="American bison (Bison bison) also known as buffalo or plains buffalo on the prairie, western U.S." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/whats-the-difference-between-buffalo-and-bison" >What’s the Difference Between Bison and Buffalo?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/10-best-hockey-players-of-all-time"> <img loading="lazy" src="https://cdn.britannica.com/13/134913-131-D9520745/Ice-Sledge-Hockey-Canada-Cup-USA-ice-2010.jpg?w=200&h=200&c=crop" alt="Ice Sledge Hockey, Hockey Canada Cup, USA (left) vs Canada, 2009. UBC Thunderbird Arena, Vancouver, BC, competition site for Olympic ice hockey and Paralympic ice sledge hockey. Vancouver 2010 Olympic and Paralympic Winter Games, Vancouver Olympics" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/10-best-hockey-players-of-all-time" >10 Best Hockey Players of All Time</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/secret-service-code-names-of-11-us-presidents"> <img loading="lazy" src="https://cdn.britannica.com/45/189145-131-45FF672E/Secret-Service-Agent-Earpiece.jpg?w=200&h=200&c=crop" alt="Secret Service Agent Listens To Earpiece" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/secret-service-code-names-of-11-us-presidents" >Secret Service Code Names of 11 U.S. Presidents</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/whats-the-difference-between-venomous-and-poisonous"> <img loading="lazy" src="https://cdn.britannica.com/57/194857-131-F5FF4C32/meadow-adder-snake-viper-Ursini-tongue.jpg?w=200&h=200&c=crop" alt="Female meadow adder snake (Vipera ursinii) Also called meadow viper or Ursini's viper. Reptile venomous poisonous tongue" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/whats-the-difference-between-venomous-and-poisonous" >What’s the Difference Between Venomous and Poisonous?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/10-famous-artworks-by-leonardo-da-vinci"> <img loading="lazy" src="https://cdn.britannica.com/06/200006-131-ABB681CF/Leonardo-da-Vinci-Italian-Renaissance-Florence-Engraving-1500.jpg?w=200&h=200&c=crop" alt="Leonardo da Vinci (1452-1519), Italian Renaissance painter from Florence. Engraving by Cosomo Colombini (d. 1812) after a Leonardo self portrait. Ca. 1500." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/10-famous-artworks-by-leonardo-da-vinci" >10 Famous Artworks by Leonardo da Vinci</a> </div> </div> </div> </div> </div> </div> <button class="drawerToggle btn position-sticky border btn-xs btn-white btn-circle rounded-sm d-none d-lg-flex " type="button" aria-label="Toggle Drawer"> <em class="material-icons font-18 text-blue" data-icon="keyboard_arrow_left"></em> </button> </div> </div> <div class="col"> <div class="h-100 ml-0 pr-lg-0 "> <div class="h-100 grid gx-0 gx-lg-20"> <div class="h-100 col-sm"> <div class="h-100 infinite-pagination-container d-flex flex-column position-relative"> <div class="position-absolute top-0 h-100 w-100"> <div class="toc-sticky-header d-none d-lg-none bg-gray-50 px-10 px-sm-30 position-sticky w-100 "> <div class="toc-sticky-header-inner-container align-items-center d-flex mx-auto h-100 w-100"> <button class="d-flex d-lg-none btn btn-sm btn-white text-blue border-2 border-gray-100 gtm-mobile-toc-header-button js-sections-button d-lg-none p-10"> <em class="material-icons my-n5 md-icon" data-icon="toc"></em> Contents </button> <div class="header-ai-ask-button-placeholder"></div> <div class="header-ai-summarize-button-placeholder"></div> </div> </div> </div> <div class="grey-box w-100 grey-box-top"> <div class="grey-box-content mx-auto w-100"> <script type="application/ld+json"> { "@context" : "https://schema.org", "@type" : "BreadcrumbList", "itemListElement" : [ { "@type" : "ListItem", "position" : 1, "item" : { "@id" : "https://www.britannica.com/browse/Science", "name": "Science" } } , { "@type" : "ListItem", "position" : 2, "item" : { "@id" : "https://www.britannica.com/browse/Mathematics", "name": "Mathematics" } } ] } </script> <nav class="breadcrumb mt-20"> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Science">Science</a> </span> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Mathematics">Mathematics</a> </span> </nav> <div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="false"> <script class="page-description-json" type="application/json"> { "url": "/science/number-theory", "shareUrl": "https://www.britannica.com/science/number-theory", "browserTitle": "Number theory | Definition, Topics, & History", "firstTopicPage": true, "topicId":422325 } </script> <div class="reading-channel"> <div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>number theory</h1></div> </div> <div class="topic-identifier font-16 font-md-20">mathematics</div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn border-2 btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400"> Ask the Chatbot a Question </button> </div> <div class="d-none d-sm-block md-topic-tools qa-action-buttons mb-15" data-topic-id="422325"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <button class="js-print-modal-button js-modal gtm-topic-tool btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=422325] .md-print-modal" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </button> <div class="md-print-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10">print</em> Print </div> <div class="mt-20 mb-10"> Please select which sections you would like to print: </div> <form action="/print/article/422325" method="post" target="_blank" rel="noopener"> <div class="print-box-items"> <ul class="list-unstyled"> <li><label><input class="mr-10" type="checkbox" name="sequence[]" value="0">Table Of Contents</label></li> </ul> </div> <input type="submit" class="btn btn-blue md-disabled" value="Print" /> </form> </div> </div> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=422325] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. </div> <div class="label mt-20 mb-10">Select Citation Style</div> <select class="js-citation-format-select form-select"> <option selected value="mla">MLA</option> <option value="apa">APA</option> <option value="chicago">Chicago Manual of Style</option> </select> <div class="citation font-serif border rounded p-15 mt-20" data-authors="William Dunham" data-title="number theory" data-published-date="17 Jul. 2024" data-url="https://www.britannica.com/science/number-theory" > <div class="citation-text"></div> </div> <button class="js-copy-citation-button mt-20 btn btn-xs btn-outline-blue border shadow-sm pr-10" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="file_copy"></em> <span class="js-citation-status-text">Copy Citation</span> </button> </div> </div> </div> <div> <button class="js-share-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=422325] .md-share-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="share"></em> Share </button> <div class="md-share-modal size-lg d-none qa-share-modal"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10" data-icon="share"></em> Share </div> <div class="label my-20">Share to social media</div> <div class="md-social-toolbar-circle d-flex align-items-start inverted" data-value="share" title="number theory" data-url="https://www.britannica.com/science/number-theory" > <a class="social-icon facebook justify-content-center d-flex align-items-center align-self-center" data-provider="facebook" href="https://www.facebook.com/BRITANNICA/" target="_blank" rel="noopener"><span>Facebook</span></a> <a class="social-icon x justify-content-center d-flex align-items-center align-self-center" data-provider="x" href="https://x.com/britannica" target="_blank" rel="noopener"><span>X</span></a> </div> <div class="label pt-20 mt-20 mb-5 border-top">URL</div> <a class="font-serif text-truncate d-inline-block" href="https://www.britannica.com/science/number-theory">https://www.britannica.com/science/number-theory</a> </div> </div> </div> <div> <button class="js-feedback-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal=".md-feedback-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="message"></em> Feedback </button> </div> <div> <button class="qa-external-website-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=422325] .md-websites-modal"> <em class="material-icons md-icon ml-n10 mr-5" data-icon="link"></em> External Websites </button> </div> </div> <div class="md-feedback-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom"> Feedback </div> <form method="post" action="/submission/feedback/422325"> <div class="my-20"> Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). </div> <div class="type-menu"> <label for="feedback-type" class="label mb-10">Feedback Type</label> <select id="feedback-type" class="form-select mb-30" name="feedbackTypeId" required> <option value="" selected="selected">Select a type (Required)</option> <option value="1">Factual Correction</option> <option value="2">Spelling/Grammar Correction</option> <option value="3">Link Correction</option> <option value="4">Additional Information</option> <option value="5">Other</option> </select> </div> <label for="feedback" class="label mb-10">Your Feedback</label> <textarea id="feedback" class="form-control mb-30" name="feedback" maxlength="3000" rows="7" required></textarea> <button class="btn btn-blue" type="submit">Submit Feedback</button> </form> <div class="success-messaging d-none mt-30"> <div class="title">Thank you for your feedback</div> <p>Our editors will review what you’ve submitted and determine whether to revise the article.</p> </div> </div> </div> <div class="md-websites-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom font-weight-bold"> External Websites </div> <div class="pb-20"> <ul class="list-unstyled mt-20 lh-lg"> <li><a class="external" href="https://www.academia.edu/45067505/Introduction_to_Number_Theory_Step_by_Step" target="_blank" rel="noopener ">Academia - Introduction to Number Theory Step by Step</a></li> <li><a class="external" href="https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/6%3A_Additional_Topics/6.2%3A_Introduction_to_Number_Theory" target="_blank" rel="noopener ">Mathematics LibreTexts - Introduction to Number Theory</a></li> <li><a class="external" href="https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1240&context=tme" target="_blank" rel="noopener ">University of Montana - ScholarWorks - Number theory and the Queen of Mathematics</a></li> <li><a class="external" href="https://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_1.pdf" target="_blank" rel="noopener ">Florida State University - Department of Mathematics - Number Theory</a></li> <li><a class="external" href="https://www.eolss.net/sample-chapters/C02/E6-01-02-05.pdf" target="_blank" rel="noopener ">UNESCO-EOLSS - Number theory and applications</a></li> </ul> </div> </div> </div> </div> </div> <div class="toc-header-marker"></div> <button class="ai-ask-button btn border-2 js-header-ai-ask-button d-none btn-sm btn-outline-red-400 border-red-400 mr-0 mr-lg-10 ml-5 ml-sm-10 ml-lg-0 p-10"> Ask the Chatbot a Question </button> <div class="caption alternate-titles">Also known as: higher arithmetic</div> <div class="md-byline module-spacing "> <div class="font-serif font-12"> <span class="written-by text-gray-700"> Written by </span> <div class="editor-popover popover p-0"> <a class="d-block p-20 gtm-byline font-12 byline-contributor" href="/contributor/William-Dunham/5198" > <div class="editor-title font-16 font-weight-bold">William Dunham</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Professor of Mathematics, Muhlenberg College, Allentown, Pennsylvania. Author of <i>Euler the Master of Us All </i>and others.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> William Dunham</span></div> <div class="font-serif font-12 text-gray-700"> <span class="qa-fact-checked-by">Fact-checked by</span> <div class="editor-popover popover p-0"> <a class="d-block p-20 font-12" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <a class="byline-edit-history" href="https://www.britannica.com/science/number-theory/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Key People: </dt> <dd><a href="/biography/Carl-Friedrich-Gauss" topicid="227204">Carl Friedrich Gauss</a></dd> <dd><a href="/biography/Pierre-de-Fermat" topicid="204668">Pierre de Fermat</a></dd> <dd><a href="/biography/Diophantus" topicid="164347">Diophantus</a></dd> <dd><a href="/biography/Paul-Erdos" topicid="191138">Paul Erdős</a></dd> <dd><a href="/biography/Leonhard-Euler" topicid="195201">Leonhard Euler</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> </div> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/Riemann-hypothesis" topicid="503221">Riemann hypothesis</a></dd> <dd><a href="/science/twin-prime-conjecture" topicid="1556762">twin prime conjecture</a></dd> <dd><a href="/science/prime-number-theorem" topicid="476362">prime number theorem</a></dd> <dd><a href="/science/perfect-number" topicid="451491">perfect number</a></dd> <dd><a href="/science/Bertrands-postulate" topicid="62873">Bertrand’s postulate</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/number-theory"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref508677"></span>number theory</strong>, branch of <a href="https://www.britannica.com/science/mathematics" class="md-crosslink " data-show-preview="true">mathematics</a> concerned with properties of the positive <span id="ref508678"></span><a href="https://www.britannica.com/science/integer" class="md-crosslink " data-show-preview="true">integers</a> (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural of mathematical pursuits.</p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">Number theory has always fascinated <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="amateurs" href="https://www.britannica.com/dictionary/amateurs" data-type="EB">amateurs</a> as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background.</p><span class="marker MOD2 mod-inline"></span><span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">Until the mid-20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. The advent of digital <span id="ref796034"></span><a href="https://www.britannica.com/technology/computer" class="md-crosslink " data-show-preview="true">computers</a> and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining <a href="https://www.britannica.com/science/prime-number-theorem" class="md-crosslink " data-show-preview="true">primes</a>, testing conjectures, and solving numerical problems once considered out of reach.</p><span class="marker MOD3 mod-inline"></span><span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, <a href="https://www.britannica.com/science/algebraic-number" class="md-crosslink autoxref " data-show-preview="true">algebraic number</a> theory, <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="analytic" href="https://www.merriam-webster.com/dictionary/analytic" data-type="MW">analytic</a> number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers.</p><span class="marker MOD4 mod-inline"></span></section> <span class="marker h2"></span><section data-level="1" id="ref233895"> <h2 class="h1">From prehistory through Classical Greece</h2> <span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">The ability to count dates back to prehistoric times. This is evident from archaeological <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="artifacts" href="https://www.merriam-webster.com/dictionary/artifacts" data-type="MW">artifacts</a>, such as a 10,000-year-old bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something. Very near the dawn of civilization, people had grasped the idea of “multiplicity” and thereby had taken the first steps toward a study of numbers.</p><a class="link-module shadow-sm d-block qa-quiz-module" href="/quiz/Numbers-and-mathematics" data-link-module-iframe-link=""> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg" alt="Equations written on blackboard" class="rounded-sm mr-15" width="70" /> <div class="line-clamp clamp-5"> <div class="module-title bg-green">Britannica Quiz</div> <div class="font-weight-semi-bold mt-5">Numbers and Mathematics</div> </div> </a><span class="marker MOD5 mod-inline"></span> <span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">It is certain that an understanding of numbers existed in ancient Mesopotamia, <a href="https://www.britannica.com/place/Egypt" class="md-crosslink autoxref " data-show-preview="true">Egypt</a>, China, and India, for tablets, papyri, and temple carvings from these early <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="cultures" href="https://www.merriam-webster.com/dictionary/cultures" data-type="MW">cultures</a> have survived. A <span id="ref796455"></span><a href="https://www.britannica.com/place/Babylon-ancient-city-Mesopotamia-Asia" class="md-crosslink " data-show-preview="true">Babylonian</a> tablet known as <span id="ref796456"></span>Plimpton 322 (c. 1700 <span class="text-smallcaps">bce</span>) is a case in point. In modern notation, it displays number triples <em>x</em>, <em>y</em>, and <em>z</em> with the property that <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = <em>z</em><sup>2</sup>. One such triple is 2,291, 2,700, and 3,541, where 2,291<sup>2</sup> + 2,700<sup>2</sup> = 3,541<sup>2</sup>. This certainly reveals a degree of number theoretic sophistication in ancient Babylon.</p><span class="marker MOD6 mod-inline"></span> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">Despite such isolated results, a general theory of numbers was nonexistent. For this—as with so much of theoretical mathematics—one must look to the Classical <span id="ref796457"></span><a href="https://www.britannica.com/place/ancient-Greece" class="md-crosslink " data-show-preview="true">Greeks</a>, whose groundbreaking achievements displayed an odd fusion of the mystical tendencies of the <span id="ref508698"></span><a href="https://www.britannica.com/science/Pythagoreanism" class="md-crosslink " data-show-preview="true">Pythagoreans</a> and the severe logic of <a href="https://www.britannica.com/biography/Euclid-Greek-mathematician" class="md-crosslink " data-show-preview="true">Euclid</a>’s <em>Elements</em> (c. 300 <span class="text-smallcaps">bce</span>).</p><div class="module-spacing"> <DIV class="marketing-INLINE_SUBSCRIPTION marketing-content" data-marketing-id="INLINE_SUBSCRIPTION"><style> .student-promo-banner-wrapper { container-type: inline-size; margin-bottom: 15px; } @container (min-width: 475px) { .student-promo-banner { flex-direction: row; } .student-promo-banner-img-wrapper { margin-bottom: 0; margin-right: 10px; justify-content: flex-start; } .student-promo-banner-text-wrapper { text-align: left; margin-bottom: 0px; margin-left: 10px; } .student-promo-banner-button-wrapper { margin-right: 0; } }</style> <div class="student-promo-banner-wrapper"> <div class="student-promo-banner d-flex flex-column align-items-center bg-blue rounded p-20"> <div class="student-promo-banner-img-wrapper mb-20 mr-0 d-flex justify-content-center"> <img class="rounded" style="max-width: 100px; min-width: 80px" src="https://cdn.britannica.com/marketing/BlueThistle.webp" /> </div> <div class="student-promo-banner-text-wrapper ml-0 mb-10 text-center text-white"> <div class="h2 mb-10">Get Unlimited Access</div> <div class="h4 font-weight-semi-bold">Try Britannica Premium for free and discover more.</div> </div> <div class="student-promo-banner-button-wrapper d-flex justify-content-center align-items-center ml-auto mr-auto"> <a class="btn btn-m btn-orange" href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=black-friday-2024">Subscribe</a> </div> </div> </div> </DIV></div><span class="marker MOD7 mod-inline"></span> <section data-level="2" id="ref233896"> <h2 class="h2"><span id="ref796458"></span><a href="https://www.britannica.com/biography/Pythagoras" class="md-crosslink " data-show-preview="true">Pythagoras</a></h2> <span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">According to tradition, <a href="https://www.britannica.com/biography/Pythagoras" class="md-crosslink " data-show-preview="true">Pythagoras</a> (c. 580–500 <span class="text-smallcaps">bce</span>) worked in southern Italy amid devoted followers. His philosophy enshrined number as the unifying <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="concept" href="https://www.britannica.com/dictionary/concept" data-type="EB">concept</a> necessary for understanding everything from planetary motion to musical harmony. Given this viewpoint, it is not surprising that the Pythagoreans attributed quasi-rational properties to certain numbers.</p><span class="marker MOD8 mod-inline"></span> <span class="marker PREMOD9 mod-inline"></span><p class="topic-paragraph">For instance, they attached significance to <span id="ref508696"></span><a href="https://www.britannica.com/science/perfect-number" class="md-crosslink " data-show-preview="true">perfect numbers</a>—i.e., those that equal the sum of their proper divisors. Examples are 6 (whose proper divisors 1, 2, and 3 sum to 6) and 28 (1 + 2 + 4 + 7 + 14). The Greek philosopher <span id="ref796459"></span><a href="https://www.britannica.com/biography/Nicomachus-of-Gerasa" class="md-crosslink " data-show-preview="true">Nicomachus of Gerasa</a> (flourished c. 100 <span class="text-smallcaps">ce</span>), writing centuries after Pythagoras but clearly in his philosophical debt, stated that perfect numbers represented “virtues, wealth, moderation, propriety, and beauty.” (Some modern writers label such nonsense numerical theology.)</p><div class="one-good-fact-module"> </div><span class="marker MOD9 mod-inline"></span> <span class="marker PREMOD10 mod-inline"></span><p class="topic-paragraph">In a similar vein, the Greeks called a pair of integers <span id="ref796460"></span><a href="https://www.britannica.com/science/amicable-numbers" class="md-crosslink " data-show-preview="true">amicable</a> (“friendly”) if each was the sum of the proper divisors of the other. They knew only a single <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="amicable" href="https://www.merriam-webster.com/dictionary/amicable" data-type="MW">amicable</a> pair: 220 and 284. One can easily check that the sum of the proper divisors of 284 is 1 + 2 + 4 + 71 + 142 = 220 and the sum of the proper divisors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. For those prone to number mysticism, such a phenomenon must have seemed like magic.</p><span class="marker MOD10 mod-inline"></span> </section> </section><span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article></div> </div></div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":4,"pageId":422325,"pageLength":659,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/libs/jquery-3.5.0.min.js?v=3.130.14"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.js?v=3.130.14' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

CINXE.COM

Number theory | Definition, Topics, & History | Britannica