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-disabled skin-theme-clientpref-day vector-toc-available" lang="it" dir="ltr"> <head> <meta charset="UTF-8"> <title>Teorema dell'infinità dei numeri primi - 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-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )itwikimwclientpreferences=([^;]+)/);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":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","gennaio","febbraio","marzo","aprile","maggio","giugno","luglio","agosto","settembre","ottobre","novembre","dicembre"],"wgRequestId":"55a452b8-a7f0-445f-a9de-5641c7408b4b","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Teorema_dell'infinità_dei_numeri_primi","wgTitle":"Teorema dell'infinità dei numeri primi","wgCurRevisionId":137505791,"wgRevisionId":137505791,"wgArticleId":53586,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["P2812 letta da Wikidata","Teoremi sui numeri primi"],"wgPageViewLanguage":"it","wgPageContentLanguage":"it","wgPageContentModel":"wikitext","wgRelevantPageName":"Teorema_dell'infinità_dei_numeri_primi","wgRelevantArticleId":53586,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia", "wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"it","pageLanguageDir":"ltr","pageVariantFallbacks":"it"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":8000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1506253","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.coloriDarkMode-default":"ready", "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.MainPageWikiList","ext.gadget.stru-commonsupload","ext.gadget.HiddenCat","ext.gadget.ReferenceTooltips","ext.gadget.TitoloErrato","ext.gadget.NewSection","ext.gadget.RichiediRevisioneBozza","ext.urlShortener.toolbar","ext.centralauth.centralautologin", "mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=it&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=it&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=it&modules=ext.gadget.coloriDarkMode-default&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=it&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <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 name="viewport" content="width=1120"> <meta property="og:title" content="Teorema dell'infinità dei numeri primi - 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="//it.m.wikipedia.org/wiki/Teorema_dell%27infinit%C3%A0_dei_numeri_primi"> <link rel="alternate" type="application/x-wiki" title="Modifica" href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&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 (it)"> <link rel="EditURI" type="application/rsd+xml" href="//it.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://it.wikipedia.org/wiki/Teorema_dell%27infinit%C3%A0_dei_numeri_primi"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it"> <link rel="alternate" type="application/atom+xml" title="Feed Atom di Wikipedia" href="/w/index.php?title=Speciale:UltimeModifiche&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-Teorema_dell_infinità_dei_numeri_primi rootpage-Teorema_dell_infinità_dei_numeri_primi skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Vai al contenuto</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="Sito"> <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="Menu principale" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menu principale</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu principale</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">nascondi</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigazione </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Pagina_principale" title="Visita la pagina principale [z]" accesskey="z"><span>Pagina principale</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Speciale:UltimeModifiche" title="Elenco delle ultime modifiche del sito [r]" accesskey="r"><span>Ultime modifiche</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Speciale:PaginaCasuale" title="Mostra una pagina a caso [x]" accesskey="x"><span>Una voce a caso</span></a></li><li id="n-nearby-pages-title" class="mw-list-item"><a href="/wiki/Speciale:NelleVicinanze"><span>Nelle vicinanze</span></a></li><li id="n-vetrina" class="mw-list-item"><a href="/wiki/Wikipedia:Vetrina"><span>Vetrina</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Aiuto:Aiuto" title="Pagine di aiuto"><span>Aiuto</span></a></li><li id="n-Sportello-informazioni" class="mw-list-item"><a href="/wiki/Aiuto:Sportello_informazioni"><span>Sportello informazioni</span></a></li> </ul> </div> </div> <div id="p-Comunità" class="vector-menu mw-portlet mw-portlet-Comunità" > <div class="vector-menu-heading"> Comunità </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Portale:Comunit%C3%A0" title="Descrizione del progetto, cosa puoi fare, dove trovare le cose"><span>Portale Comunità</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:Bar"><span>Bar</span></a></li><li id="n-wikipediano" class="mw-list-item"><a href="/wiki/Wikipedia:Wikipediano"><span>Il Wikipediano</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Wikipedia:Contatti"><span>Contatti</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Pagina_principale" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="L'enciclopedia libera" src="/static/images/mobile/copyright/wikipedia-tagline-it.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Speciale:Ricerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca in Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Ricerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca in Wikipedia" aria-label="Cerca in Wikipedia" autocapitalize="sentences" title="Cerca in Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Speciale:Ricerca"> </div> <button class="cdx-button cdx-search-input__end-button">Ricerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Strumenti personali"> <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="Aspetto"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Modifica la dimensione, la larghezza e il colore del testo" > <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="Aspetto" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aspetto</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_it.wikipedia.org&uselang=it" class=""><span>Fai una donazione</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speciale:CreaUtenza&returnto=Teorema+dell%27infinit%C3%A0+dei+numeri+primi" title="Si consiglia di registrarsi e di effettuare l'accesso, anche se non è obbligatorio" class=""><span>registrati</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speciale:Entra&returnto=Teorema+dell%27infinit%C3%A0+dei+numeri+primi" title="Si consiglia di effettuare l'accesso, anche se non è obbligatorio [o]" accesskey="o" class=""><span>entra</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Altre opzioni" > <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="Strumenti personali" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Strumenti personali</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu utente" > <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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_it.wikipedia.org&uselang=it"><span>Fai una donazione</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciale:CreaUtenza&returnto=Teorema+dell%27infinit%C3%A0+dei+numeri+primi" title="Si consiglia di registrarsi e di effettuare l'accesso, anche se non è obbligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>registrati</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciale:Entra&returnto=Teorema+dell%27infinit%C3%A0+dei+numeri+primi" title="Si consiglia di effettuare l'accesso, anche se non è obbligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>entra</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pagine per utenti anonimi <a href="/wiki/Aiuto:Benvenuto" aria-label="Ulteriori informazioni sulla contribuzione"><span>ulteriori informazioni</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speciale:MieiContributi" title="Un elenco delle modifiche fatte da questo indirizzo IP [y]" accesskey="y"><span>contributi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speciale:MieDiscussioni" title="Discussioni sulle modifiche fatte da questo indirizzo IP [n]" accesskey="n"><span>discussioni</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sito"> <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="Indice" 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">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</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">Inizio</div> </a> </li> <li id="toc-Dimostrazioni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dimostrazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Dimostrazioni</span> </div> </a> <button aria-controls="toc-Dimostrazioni-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Dimostrazioni</span> </button> <ul id="toc-Dimostrazioni-sublist" class="vector-toc-list"> <li id="toc-Dimostrazione_di_Euclide" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimostrazione_di_Euclide"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Dimostrazione di Euclide</span> </div> </a> <ul id="toc-Dimostrazione_di_Euclide-sublist" class="vector-toc-list"> <li id="toc-Corollario" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Corollario"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Corollario</span> </div> </a> <ul id="toc-Corollario-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dimostrazione_di_Eulero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimostrazione_di_Eulero"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Dimostrazione di Eulero</span> </div> </a> <ul id="toc-Dimostrazione_di_Eulero-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimostrazione_topologica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimostrazione_topologica"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Dimostrazione topologica</span> </div> </a> <ul id="toc-Dimostrazione_topologica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Teorema dell'infinità dei numeri primi</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 29 lingue" > <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-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A5%D9%82%D9%84%D9%8A%D8%AF%D8%B3" title="مبرهنة إقليدس - arabo" lang="ar" hreflang="ar" data-title="مبرهنة إقليدس" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_d%27Euclides" title="Teorema d'Euclides - catalano" lang="ca" hreflang="ca" data-title="Teorema d'Euclides" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3" title="تیۆرمی ئیقلیدس - curdo centrale" lang="ckb" hreflang="ckb" data-title="تیۆرمی ئیقلیدس" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidova_v%C4%9Bta_o_prvo%C4%8D%C3%ADslech" title="Eukleidova věta o prvočíslech - ceco" lang="cs" hreflang="cs" data-title="Eukleidova věta o prvočíslech" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="voce in vetrina"><a href="https://de.wikipedia.org/wiki/Satz_des_Euklid" title="Satz des Euklid - tedesco" lang="de" hreflang="de" data-title="Satz des Euklid" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7" title="Θεώρημα του Ευκλείδη - greco" lang="el" hreflang="el" data-title="Θεώρημα του Ευκλείδη" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclid%27s_theorem" title="Euclid's theorem - inglese" lang="en" hreflang="en" data-title="Euclid's theorem" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Euclides" title="Teorema de Euclides - spagnolo" lang="es" hreflang="es" data-title="Teorema de Euclides" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Euklidesen_teorema" title="Euklidesen teorema - basco" lang="eu" hreflang="eu" data-title="Euklidesen teorema" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3" title="قضیه اقلیدس - persiano" lang="fa" hreflang="fa" data-title="قضیه اقلیدس" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_d%27Euclide_sur_les_nombres_premiers" title="Théorème d'Euclide sur les nombres premiers - francese" lang="fr" hreflang="fr" data-title="Théorème d'Euclide sur les nombres premiers" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Euclides" title="Teorema de Euclides - galiziano" lang="gl" hreflang="gl" data-title="Teorema de Euclides" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%99%D7%95%D7%9E%D7%9D_%D7%A9%D7%9C_%D7%90%D7%99%D7%A0%D7%A1%D7%95%D7%A3_%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99%D7%99%D7%9D" title="קיומם של אינסוף מספרים ראשוניים - ebraico" lang="he" hreflang="he" data-title="קיומם של אינסוף מספרים ראשוניים" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%BE%D5%AF%D5%AC%D5%AB%D5%A4%D5%A5%D5%BD%D5%AB_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Էվկլիդեսի թեորեմ - armeno" lang="hy" hreflang="hy" data-title="Էվկլիդեսի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B4%A0%E6%95%B0%E3%81%8C%E7%84%A1%E6%95%B0%E3%81%AB%E5%AD%98%E5%9C%A8%E3%81%99%E3%82%8B%E3%81%93%E3%81%A8%E3%81%AE%E8%A8%BC%E6%98%8E" title="素数が無数に存在することの証明 - giapponese" lang="ja" hreflang="ja" data-title="素数が無数に存在することの証明" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C%EC%9D%98_%EC%A0%95%EB%A6%AC" title="유클리드의 정리 - coreano" lang="ko" hreflang="ko" data-title="유클리드의 정리" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Theor%C3%A8me_vum_Euklid" title="Theorème vum Euklid - lussemburghese" lang="lb" hreflang="lb" data-title="Theorème vum Euklid" data-language-autonym="Lëtzebuergesch" data-language-local-name="lussemburghese" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Euklido_teorema" title="Euklido teorema - lituano" lang="lt" hreflang="lt" data-title="Euklido teorema" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Евклидова теорема - macedone" lang="mk" hreflang="mk" data-title="Евклидова теорема" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC" title="Евклидийн теорем - mongolo" lang="mn" hreflang="mn" data-title="Евклидийн теорем" data-language-autonym="Монгол" data-language-local-name="mongolo" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Euclides" title="Stelling van Euclides - olandese" lang="nl" hreflang="nl" data-title="Stelling van Euclides" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Euclides" title="Teorema de Euclides - portoghese" lang="pt" hreflang="pt" data-title="Teorema de Euclides" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B0" title="Теорема Евклида - russo" lang="ru" hreflang="ru" data-title="Теорема Евклида" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Еуклидова теорема - serbo" lang="sr" hreflang="sr" data-title="Еуклидова теорема" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Euklides_sats" title="Euklides sats - svedese" lang="sv" hreflang="sv" data-title="Euklides sats" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid_teoremi" title="Öklid teoremi - turco" lang="tr" hreflang="tr" data-title="Öklid teoremi" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D0%B0" title="Теорема Евкліда - ucraino" lang="uk" hreflang="uk" data-title="Теорема Евкліда" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Euclid" title="Định lý Euclid - vietnamita" lang="vi" hreflang="vi" data-title="Định lý Euclid" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%AE%9A%E7%90%86" title="欧几里得定理 - cinese" lang="zh" hreflang="zh" data-title="欧几里得定理" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1506253#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <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/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Teorema_dell%27infinit%C3%A0_dei_numeri_primi" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t"><span>Discussione</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambia versione linguistica" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">italiano</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visite"> <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/Teorema_dell%27infinit%C3%A0_dei_numeri_primi"><span>Leggi</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <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="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Strumenti</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div class="vector-menu-heading"> Azioni </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/Teorema_dell%27infinit%C3%A0_dei_numeri_primi"><span>Leggi</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=history"><span>Cronologia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&oldid=137505791" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&id=137505791&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FTeorema_dell%2527infinit%25C3%25A0_dei_numeri_primi"><span>Ottieni URL breve</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FTeorema_dell%2527infinit%25C3%25A0_dei_numeri_primi"><span>Scarica codice QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Teorema+dell%27infinit%C3%A0+dei+numeri+primi"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1506253" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <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">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</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">Da Wikipedia, l'enciclopedia libera.</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="it" dir="ltr"><p>Il <b>teorema dell'infinità dei numeri primi</b> afferma che, per quanto grande si scelga un <a href="/wiki/Numero_naturale" title="Numero naturale">numero naturale</a> <i>n</i>, esiste sempre un <a href="/wiki/Numero_primo" title="Numero primo">numero primo</a> maggiore di <i>n</i>. </p><p>È stato dimostrato per la prima volta da <a href="/wiki/Euclide" title="Euclide">Euclide</a> nei suoi <i><a href="/wiki/Elementi_(Euclide)" title="Elementi (Euclide)">Elementi</a></i> (libro IX, proposizione 20), ma ne sono state trovate circa altre cinquanta dimostrazioni, che usano una gran varietà di tecniche diverse: ad esempio <a href="/wiki/Eulero" title="Eulero">Eulero</a> lo ricavò dalla <a href="/wiki/Serie_divergente" title="Serie divergente">divergenza</a> della <a href="/wiki/Serie_armonica" title="Serie armonica">serie armonica</a> e dalla possibilità di scrivere ogni numero come prodotto di numeri primi; <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a> usò i <a href="/wiki/Numeri_di_Fermat" class="mw-redirect" title="Numeri di Fermat">numeri di Fermat</a>, mentre <a href="/wiki/Harry_Furstenberg" class="mw-redirect" title="Harry Furstenberg">Harry Furstenberg</a> ideò una dimostrazione che sfrutta i metodi della <a href="/wiki/Topologia" title="Topologia">topologia</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Alcune di queste dimostrazioni (quella di Euclide, quella di Goldbach e un'altra che usa i <a href="/wiki/Numero_di_Mersenne" class="mw-redirect" title="Numero di Mersenne">numeri di Mersenne</a>) si basano su una strategia simile, ovvero dimostrare che esiste una successione infinita di numeri che sono a due a due <a href="/wiki/Coprimo" class="mw-redirect" title="Coprimo">coprimi</a>, da cui segue necessariamente l'infinità dei numeri primi. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Dimostrazioni">Dimostrazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=1" title="Modifica la sezione Dimostrazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=1" title="Edit section's source code: Dimostrazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione_di_Euclide">Dimostrazione di Euclide</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=2" title="Modifica la sezione Dimostrazione di Euclide" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=2" title="Edit section's source code: Dimostrazione di Euclide"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La dimostrazione, molto semplice in termini moderni, è esposta negli <a href="/wiki/Elementi_(Euclide)" title="Elementi (Euclide)">Elementi</a> di <a href="/wiki/Euclide" title="Euclide">Euclide</a> e può a buon diritto essere considerata la prima dimostrazione di un teorema di <a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">teoria dei numeri</a>. </p><p>La dimostrazione avviene per assurdo, con il seguente ragionamento: </p><p>Si supponga che i numeri primi non siano infiniti ma solo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P=\{2,3,\dots ,p_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle P=\{2,3,\dots ,p_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cdcd20df88e53d41c845f37c66052da8a5a15b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.094ex; height:2.843ex;" alt="{\textstyle P=\{2,3,\dots ,p_{n}\}}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> sarebbe allora il più grande dei numeri primi. Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49947ab1bd3083f203a150b28037828fc17afdba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\textstyle a\in \mathbb {N} }"></span> il prodotto degli <i>n</i> numeri primi, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8028f23cfdcb108712e2bc53369305574afe820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}"></span> non è divisibile per 2, perché lo è <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\textstyle a}"></span> e quindi ha resto 1. Non è divisibile per 3, per lo stesso motivo. In generale, detto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> l'i-esimo numero primo, la divisione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {a+1}{p_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {a+1}{p_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ab76aab95f151ca831a8fe6e5457778e77d061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.806ex; height:4.009ex;" alt="{\textstyle {\frac {a+1}{p_{i}}}}"></span> ha sempre resto 1: assumendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35204c48caf0ef2a3b92614a7e42fce2747cbafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:2.843ex;" alt="{\textstyle (a+1)}"></span> come dividendo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> come divisore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle q\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle q\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa658a4a7ede995062e212a7107e9755537c8a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\textstyle q\in \mathbb {N} }"></span> come quoziente della divisione, è sufficiente dimostrare che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a+1)-(q\cdot p_{i})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a+1)-(q\cdot p_{i})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0130d89ade8049eb916edaee8934af4fd15cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.67ex; height:2.843ex;" alt="{\textstyle (a+1)-(q\cdot p_{i})=1}"></span>, cioè che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(q\cdot p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(q\cdot p_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ac7b70c7831981c0723ad238274387bf115ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.855ex; height:2.843ex;" alt="{\displaystyle a=(q\cdot p_{i})}"></span> assunto come ipotesi. </p><p>A questo punto, per il <a href="/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica">teorema fondamentale dell'aritmetica</a>, sono possibili due casi: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8028f23cfdcb108712e2bc53369305574afe820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}"></span> è primo, e ovviamente essendo maggiore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> quest'ultimo non è il più grande dei numeri primi;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8028f23cfdcb108712e2bc53369305574afe820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}"></span>, non essendo primo, è il prodotto di numeri primi che non possono figurare tra gli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ipotizzati (in quanto, come abbiamo appena mostrato, nessun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> divide <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35204c48caf0ef2a3b92614a7e42fce2747cbafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:2.843ex;" alt="{\textstyle (a+1)}"></span>) e che devono quindi essere maggiori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span>; anche in questo caso segue che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> non è il più grande dei numeri primi.</li></ol> <p>In entrambi i casi si perviene alla conclusione che non può non esistere un numero primo più grande di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span>, dunque i numeri primi sono infiniti. </p><p>In realtà solo pochi dei numeri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8028f23cfdcb108712e2bc53369305574afe820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}"></span> (detti <a href="/wiki/Numero_di_Euclide" title="Numero di Euclide">numeri di Euclide</a>) così trovati sono primi, perché il divario tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> cresce circa come il <a href="/wiki/Fattoriale" title="Fattoriale">fattoriale</a>, e quindi c'è sempre più possibilità che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8028f23cfdcb108712e2bc53369305574afe820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}"></span> abbia un divisore tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea43e3ce4af2ace1740c27fabee3f6e91eda8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.169ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a+1}}}"></span>. </p><p>Una conseguenza immediata di questa dimostrazione è la seguente <a href="/wiki/Disuguaglianza_(matematica)" class="mw-redirect" title="Disuguaglianza (matematica)">disuguaglianza</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n+1}<p_{1}p_{2}\cdots p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n+1}<p_{1}p_{2}\cdots p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17406c2f9ae09b4a045d5c0c71842db4f59ac047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:18.009ex; height:2.176ex;" alt="{\displaystyle p_{n+1}<p_{1}p_{2}\cdots p_{n}}"></span></dd></dl> <p>La <a href="/wiki/Disuguaglianza_di_Bonse" title="Disuguaglianza di Bonse">disuguaglianza di Bonse</a> e le sue generalizzazioni forniscono risultati più forti. </p> <div class="mw-heading mw-heading4"><h4 id="Corollario">Corollario</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=3" title="Modifica la sezione Corollario" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=3" title="Edit section's source code: Corollario"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Primo_fattoriale" title="Primo fattoriale">Primo fattoriale</a></b>.</span></div> </div> <p>Un interessante corollario, che è evidente rigirando la dimostrazione, è che si può sempre costruire un intervallo, lungo a piacere, di numeri consecutivi che non siano numeri primi. Infatti se vogliamo avere un intervallo di 99 numeri consecutivi senza primi, è possibile costruirlo prendendo, ad esempio, il fattoriale di 100, ossia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 100!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>100</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 100!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7714815cf4e0282c097639053f8c604b627cfceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\textstyle 100!}"></span> ≈ <span class="nowrap"><span data-sort-value="7157933262153999999♠"></span>9,33262154<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>157</sup></span>. Questo numero, enorme, è divisibile per tutti i numeri tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"></span>, per costruzione. Se poniamo che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> sia uno di questi numeri tra 2 e 100, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 100!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>100</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 100!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7714815cf4e0282c097639053f8c604b627cfceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\textstyle 100!}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100!+k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo>!</mo> <mo>+</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100!+k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b365b6fe685236f78f6f98d53442aff57ea6e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.186ex; height:2.343ex;" alt="{\displaystyle 100!+k}"></span> sono entrambi divisibili per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. Abbiamo quindi 99 numeri consecutivi senza primi, da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100!+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo>!</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100!+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f04000847177056de59c511d29443a04557e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.137ex; height:2.343ex;" alt="{\displaystyle 100!+2}"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100!+100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo>!</mo> <mo>+</mo> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100!+100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e863423cdac350d26ca01eefa5730d86e252ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.462ex; height:2.343ex;" alt="{\displaystyle 100!+100}"></span>. Si noti che, data la lunghezza dell'intervallo, gli estremi dell'intervallo costruito in questo modo non sono i minimi possibili. </p> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione_di_Eulero">Dimostrazione di Eulero</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=4" title="Modifica la sezione Dimostrazione di Eulero" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=4" title="Edit section's source code: Dimostrazione di Eulero"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La dimostrazione di Eulero parte dal fatto che la <a href="/wiki/Serie_armonica" title="Serie armonica">serie armonica</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=1+{\frac {1}{2}}+{\frac {1}{3}}+\dots +{\frac {1}{n}}+\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=1+{\frac {1}{2}}+{\frac {1}{3}}+\dots +{\frac {1}{n}}+\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb112415d01f59dfc326e0c131aa0b44e7f1b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.637ex; height:5.176ex;" alt="{\displaystyle S=1+{\frac {1}{2}}+{\frac {1}{3}}+\dots +{\frac {1}{n}}+\dots }"></span></dd></dl> <p>è divergente. </p><p><a href="/wiki/Eulero" title="Eulero">Eulero</a> osserva che la serie armonica può vedersi come il prodotto di queste <a href="/wiki/Serie_geometriche" class="mw-redirect" title="Serie geometriche">serie geometriche</a>, una per ogni numero primo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots +{\frac {1}{2^{n}}}+\dots =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots +{\frac {1}{2^{n}}}+\dots =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee24f85a3a70823ecc137f85e85a4dda29abf38d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.703ex; height:5.343ex;" alt="{\displaystyle S_{2}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots +{\frac {1}{2^{n}}}+\dots =2}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3}=1+{\frac {1}{3}}+{\frac {1}{9}}+{\frac {1}{27}}+\dots +{\frac {1}{3^{n}}}+\dots ={\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>27</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}=1+{\frac {1}{3}}+{\frac {1}{9}}+{\frac {1}{27}}+\dots +{\frac {1}{3^{n}}}+\dots ={\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276cfd319f48d00546d39dbe0e6e6231064c910c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.701ex; height:5.343ex;" alt="{\displaystyle S_{3}=1+{\frac {1}{3}}+{\frac {1}{9}}+{\frac {1}{27}}+\dots +{\frac {1}{3^{n}}}+\dots ={\frac {3}{2}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{5}=1+{\frac {1}{5}}+{\frac {1}{25}}+{\frac {1}{125}}+\dots +{\frac {1}{5^{n}}}+\dots ={\frac {5}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>25</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>125</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{5}=1+{\frac {1}{5}}+{\frac {1}{25}}+{\frac {1}{125}}+\dots +{\frac {1}{5^{n}}}+\dots ={\frac {5}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5315176505c11b4c087bfe99482f6683d7c0ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.026ex; height:5.343ex;" alt="{\displaystyle S_{5}=1+{\frac {1}{5}}+{\frac {1}{25}}+{\frac {1}{125}}+\dots +{\frac {1}{5^{n}}}+\dots ={\frac {5}{4}}}"></span> </p><p>... </p><p>Le serie si calcolano facilmente ricordando che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}={\frac {1}{1-{\frac {1}{n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}={\frac {1}{1-{\frac {1}{n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757743901f0ee937b786e6c3a31db575a540788b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.403ex; height:6.509ex;" alt="{\displaystyle S_{n}={\frac {1}{1-{\frac {1}{n}}}}}"></span> (vedi <a href="/wiki/Serie_geometrica" title="Serie geometrica">serie geometrica</a>). </p><p>Infatti la serie armonica è la somma dei reciproci di tutti i numeri naturali e ogni numero naturale può rappresentarsi come il prodotto dei suoi fattori primi. Ci si convince allora facilmente che ogni elemento della serie armonica corrisponde a un possibile prodotto di elementi presi uno ad uno dalle serie suddette. Per esempio per l'elemento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b853f046397cffb4795c16aac2694ec88cb67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle 1/15}"></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{15}}=1\times {\frac {1}{3}}\times {\frac {1}{5}}\times 1\times 1\times \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{15}}=1\times {\frac {1}{3}}\times {\frac {1}{5}}\times 1\times 1\times \dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfc889536a65d7a326d7861890ed58cda82398c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.669ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{15}}=1\times {\frac {1}{3}}\times {\frac {1}{5}}\times 1\times 1\times \dots }"></span> </p><p>D'altra parte le serie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2},S_{3},S_{5},S_{7},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2},S_{3},S_{5},S_{7},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5da7f9f91a9d56ff1975082fb80eae5e8bbb429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.776ex; height:2.509ex;" alt="{\displaystyle S_{2},S_{3},S_{5},S_{7},\ldots }"></span> sono tutte finite. Ma allora se i numeri primi fossero finiti, il loro prodotto sarebbe anch'esso finito, mentre sappiamo che la serie armonica diverge. </p><p>Ne segue che i numeri primi devono essere infiniti. </p> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione_topologica">Dimostrazione topologica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=5" title="Modifica la sezione Dimostrazione topologica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=5" title="Edit section's source code: Dimostrazione topologica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per ogni coppia di interi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"></span> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28dc4d9d3d37d7552c7c3b21d641a6559810f25f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.138ex; height:2.509ex;" alt="{\displaystyle a>0,}"></span> si ponga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mathbb {Z} +b:=\{an+b:n\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mi>b</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mi>n</mi> <mo>+</mo> <mi>b</mi> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbb {Z} +b:=\{an+b:n\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cfac79c1abe03752b50118f6d0d495f84186e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.874ex; height:2.843ex;" alt="{\displaystyle a\mathbb {Z} +b:=\{an+b:n\in \mathbb {Z} \}}"></span>. La famiglia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a\mathbb {Z} +b:a,b\in \mathbb {Z} ,a>0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a\mathbb {Z} +b:a,b\in \mathbb {Z} ,a>0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa09a597980d8fdffc478be3736e0894116a37a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.057ex; height:2.843ex;" alt="{\displaystyle \{a\mathbb {Z} +b:a,b\in \mathbb {Z} ,a>0\}}"></span> è una base di una topologia di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, detta <a href="/wiki/Topologia_degli_interi_equispaziati" title="Topologia degli interi equispaziati">topologia degli interi equispaziati</a>: la prova dell'infinitudine dei numeri primi si cela dietro le sue proprietà topologiche. Gli aperti di tale topologia godono di tre proprietà: </p> <ol><li>ogni aperto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mathbb {Z} +b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbb {Z} +b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52ba170ebc94cd997edc17220101560b110296a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.618ex; height:2.343ex;" alt="{\displaystyle a\mathbb {Z} +b}"></span> è chiuso;</li> <li>ogni aperto non vuoto ha infiniti elementi;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \backslash \{-1,1\}=\bigcup _{p\in \mathbb {P} }p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mrow> </munder> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \backslash \{-1,1\}=\bigcup _{p\in \mathbb {P} }p\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d37d403e983b77f2c572faa1640c6896770423d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.337ex; height:5.843ex;" alt="{\displaystyle \mathbb {Z} \backslash \{-1,1\}=\bigcup _{p\in \mathbb {P} }p\mathbb {Z} }"></span>, ove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span> è l'insieme dei numeri primi positivi.</li></ol> <p>La 2 è immediata e la 3 discende dal teorema fondamentale dell'aritmetica. Per quanto riguarda la 1 è sufficiente notare che </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mathbb {Z} +b=\mathbb {Z} \backslash \bigcup _{m=b+1}^{a+b-1}(a\mathbb {Z} +m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mi class="MJX-variant" mathvariant="normal">∖</mi> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbb {Z} +b=\mathbb {Z} \backslash \bigcup _{m=b+1}^{a+b-1}(a\mathbb {Z} +m).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb85fb515939c22651b28d3e70cfa8b08caa8534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.461ex; height:7.509ex;" alt="{\displaystyle a\mathbb {Z} +b=\mathbb {Z} \backslash \bigcup _{m=b+1}^{a+b-1}(a\mathbb {Z} +m).}"></span></dd></dl> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span> fosse finito, in virtù della 3 e della 1, avremmo che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-1,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-1,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ffb8c7a09dad8456eee3669ee9a7e462fe3c34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{-1,1\}}"></span> è aperto, ma ciò è in contraddizione con la 2. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=6" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=6" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <cite class="citation pubblicazione" style="font-style:normal"> Harry Furstenberg, <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1955-05_62_5/page/353"><span style="font-style:italic;">On the infinitude of primes</span></a>, in <span style="font-style:italic;"><a href="/wiki/American_Mathematical_Monthly" title="American Mathematical Monthly">Amer. Math. Monthly</a></span>, vol. 62, n. 5, 1955, p. 353, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F2307043">10.2307/2307043</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=7" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal"> Martin Aigner e Günter M. Ziegler, <span style="font-style:italic;"><a href="/wiki/Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a></span>, traduzione di Silvia Quarterni, Milano, Springer, 2006, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-470-0435-7" title="Speciale:RicercaISBN/88-470-0435-7">88-470-0435-7</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=8" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=8" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Crivello_di_Eratostene" title="Crivello di Eratostene">Crivello di Eratostene</a></li> <li><a href="/wiki/Euclide" title="Euclide">Euclide</a></li> <li><a href="/wiki/Eulero" title="Eulero">Eulero</a></li> <li><a href="/wiki/Numero_primo" title="Numero primo">Numero primo</a></li> <li><a href="/wiki/Piccolo_teorema_di_Fermat" title="Piccolo teorema di Fermat">Piccolo teorema di Fermat</a></li> <li><a href="/wiki/Serie_geometrica" title="Serie geometrica">Serie geometrica</a></li> <li><a href="/wiki/Dimostrazione_della_divergenza_della_serie_dei_reciproci_dei_primi" title="Dimostrazione della divergenza della serie dei reciproci dei primi">Dimostrazione della divergenza della serie dei reciproci dei primi</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&veaction=edit&section=9" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&action=edit&section=9" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/EuclidsTheorems.html"><span style="font-style:italic;">Euclid's Theorems</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q1506253#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Algebra"><tbody><tr><th colspan="3" style="background:#ffc0cb;"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Algebra" title="Template:Algebra"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Algebra&action=edit&redlink=1" class="new" title="Discussioni template:Algebra (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Algebra&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Algebra" title="Algebra">Algebra</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Numero" title="Numero">Numeri</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Numero_naturale" title="Numero naturale">Naturali</a><b> ·</b> <a href="/wiki/Numero_intero" title="Numero intero">Interi</a><b> ·</b> <a href="/wiki/Numero_razionale" title="Numero razionale">Razionali</a><b> ·</b> <a href="/wiki/Numero_irrazionale" title="Numero irrazionale">Irrazionali</a><b> ·</b> <a href="/wiki/Numero_algebrico" title="Numero algebrico">Algebrici</a><b> ·</b> <a href="/wiki/Numero_trascendente" title="Numero trascendente">Trascendenti</a><b> ·</b> <a href="/wiki/Numero_reale" title="Numero reale">Reali</a><b> ·</b> <a href="/wiki/Numero_complesso" title="Numero complesso">Complessi</a><b> ·</b> <a href="/wiki/Numero_ipercomplesso" title="Numero ipercomplesso">Numero ipercomplesso</a><b> ·</b> <a href="/wiki/Numero_p-adico" title="Numero p-adico">Numero p-adico</a><b> ·</b> <a href="/wiki/Numero_duale" title="Numero duale">Duali</a><b> ·</b> <a href="/wiki/Numero_complesso_iperbolico" title="Numero complesso iperbolico">Complessi iperbolici</a></td><td rowspan="10" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/58px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="58" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/87px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/116px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><figcaption></figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Principi fondamentali</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Principio_d%27induzione" title="Principio d'induzione">Principio d'induzione</a><b> ·</b> <a href="/wiki/Principio_del_buon_ordinamento" title="Principio del buon ordinamento">Principio del buon ordinamento</a><b> ·</b> <a href="/wiki/Relazione_di_equivalenza" title="Relazione di equivalenza">Relazione di equivalenza</a><b> ·</b> <a href="/wiki/Relazione_d%27ordine" title="Relazione d'ordine">Relazione d'ordine</a><b> ·</b> <a href="/wiki/Associativit%C3%A0_della_potenza" title="Associatività della potenza">Associatività della potenza</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Algebra_elementare" title="Algebra elementare">Algebra elementare</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Equazione" title="Equazione">Equazione</a><b> ·</b> <a href="/wiki/Disequazione" title="Disequazione">Disequazione</a><b> ·</b> <a href="/wiki/Polinomio" title="Polinomio">Polinomio</a><b> ·</b> <a href="/wiki/Triangolo_di_Tartaglia" title="Triangolo di Tartaglia">Triangolo di Tartaglia</a><b> ·</b> <a href="/wiki/Teorema_binomiale" title="Teorema binomiale">Teorema binomiale</a><b> ·</b> <a href="/wiki/Teorema_del_resto" title="Teorema del resto">Teorema del resto</a><b> ·</b> <a href="/wiki/Lemma_di_Gauss_(polinomi)" title="Lemma di Gauss (polinomi)">Lemma di Gauss</a><b> ·</b> <a href="/wiki/Teorema_delle_radici_razionali" title="Teorema delle radici razionali">Teorema delle radici razionali</a><b> ·</b> <a href="/wiki/Regola_di_Ruffini" title="Regola di Ruffini">Regola di Ruffini</a><b> ·</b> <a href="/wiki/Criterio_di_Eisenstein" title="Criterio di Eisenstein">Criterio di Eisenstein</a><b> ·</b> <a href="/wiki/Criterio_di_Cartesio" title="Criterio di Cartesio">Criterio di Cartesio</a><b> ·</b> <a href="/wiki/Disequazione_con_il_valore_assoluto" title="Disequazione con il valore assoluto">Disequazione con il valore assoluto</a><b> ·</b> <a href="/wiki/Segno_(matematica)" title="Segno (matematica)">Segno</a><b> ·</b> <a href="/wiki/Metodo_di_Gauss-Seidel" title="Metodo di Gauss-Seidel">Metodo di Gauss-Seidel</a><b> ·</b> <a href="/wiki/Polinomio_simmetrico" title="Polinomio simmetrico">Polinomio simmetrico</a><b> ·</b> <a href="/wiki/Funzione_simmetrica" title="Funzione simmetrica">Funzione simmetrica</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Elementi di <a href="/wiki/Calcolo_combinatorio" title="Calcolo combinatorio">Calcolo combinatorio</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a><b> ·</b> <a href="/wiki/Permutazione" title="Permutazione">Permutazione</a><b> ·</b> <a href="/wiki/Disposizione" title="Disposizione">Disposizione</a><b> ·</b> <a href="/wiki/Combinazione" title="Combinazione">Combinazione</a><b> ·</b> <a href="/wiki/Dismutazione_(matematica)" title="Dismutazione (matematica)">Dismutazione</a><b> ·</b> <a href="/wiki/Principio_di_inclusione-esclusione" title="Principio di inclusione-esclusione">Principio di inclusione-esclusione</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Concetti fondamentali di <a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">Teoria dei numeri</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Primi</th><td colspan="1"><a href="/wiki/Numero_primo" title="Numero primo">Numero primo</a><b> ·</b> <a class="mw-selflink selflink">Teorema dell'infinità dei numeri primi</a><b> ·</b> <a href="/wiki/Crivello_di_Eratostene" title="Crivello di Eratostene">Crivello di Eratostene</a><b> ·</b> <a href="/wiki/Crivello_di_Atkin" title="Crivello di Atkin">Crivello di Atkin</a><b> ·</b> <a href="/wiki/Test_di_primalit%C3%A0" title="Test di primalità">Test di primalità</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica">Teorema fondamentale dell'aritmetica</a></td></tr><tr><th class="subnavbox_group">Divisori</th><td colspan="1"><a href="/wiki/Interi_coprimi" title="Interi coprimi">Interi coprimi</a><b> ·</b> <a href="/wiki/Identit%C3%A0_di_B%C3%A9zout" title="Identità di Bézout">Identità di Bézout</a><b> ·</b> <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">MCD</a><b> ·</b> <a href="/wiki/Minimo_comune_multiplo" title="Minimo comune multiplo">mcm</a><b> ·</b> <a href="/wiki/Algoritmo_di_Euclide" title="Algoritmo di Euclide">Algoritmo di Euclide</a><b> ·</b> <a href="/wiki/Algoritmo_esteso_di_Euclide" title="Algoritmo esteso di Euclide">Algoritmo esteso di Euclide</a><b> ·</b> <a href="/wiki/Criteri_di_divisibilit%C3%A0" title="Criteri di divisibilità">Criteri di divisibilità</a><b> ·</b> <a href="/wiki/Divisore" title="Divisore">Divisore</a></td></tr><tr><th class="subnavbox_group"><a href="/wiki/Aritmetica_modulare" title="Aritmetica modulare">Aritmetica modulare</a></th><td colspan="1"><a href="/wiki/Teorema_cinese_del_resto" title="Teorema cinese del resto">Teorema cinese del resto</a><b> ·</b> <a href="/wiki/Piccolo_teorema_di_Fermat" title="Piccolo teorema di Fermat">Piccolo teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Eulero_(aritmetica_modulare)" title="Teorema di Eulero (aritmetica modulare)">Teorema di Eulero</a><b> ·</b> <a href="/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero">Funzione φ di Eulero</a><b> ·</b> <a href="/wiki/Teorema_di_Wilson" title="Teorema di Wilson">Teorema di Wilson</a><b> ·</b> <a href="/wiki/Reciprocit%C3%A0_quadratica" title="Reciprocità quadratica">Reciprocità quadratica</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_dei_gruppi" title="Teoria dei gruppi">Teoria dei gruppi</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Gruppi</th><td colspan="1"><a href="/wiki/Gruppo_(matematica)" title="Gruppo (matematica)">Gruppo</a> (<a href="/wiki/Gruppo_finito" title="Gruppo finito">finito</a><b> ·</b> <a href="/wiki/Gruppo_ciclico" title="Gruppo ciclico">ciclico</a><b> ·</b> <a href="/wiki/Gruppo_abeliano" title="Gruppo abeliano">abeliano</a>)<b> ·</b> <a href="/wiki/Gruppo_primario" title="Gruppo primario">Gruppo primario</a><b> ·</b> <a href="/wiki/Gruppo_quoziente" title="Gruppo quoziente">Gruppo quoziente</a><b> ·</b> <a href="/wiki/Gruppo_nilpotente" title="Gruppo nilpotente">Gruppo nilpotente</a><b> ·</b> <a href="/wiki/Gruppo_risolubile" title="Gruppo risolubile">Gruppo risolubile</a><b> ·</b> <a href="/wiki/Gruppo_simmetrico" title="Gruppo simmetrico">Gruppo simmetrico</a><b> ·</b> <a href="/wiki/Gruppo_diedrale" title="Gruppo diedrale">Gruppo diedrale</a><b> ·</b> <a href="/wiki/Gruppo_semplice" title="Gruppo semplice">Gruppo semplice</a><b> ·</b> <a href="/wiki/Gruppo_sporadico" title="Gruppo sporadico">Gruppo sporadico</a><b> ·</b> <a href="/wiki/Gruppo_mostro" title="Gruppo mostro">Gruppo mostro</a><b> ·</b> <a href="/wiki/Gruppo_di_Klein" title="Gruppo di Klein">Gruppo di Klein</a><b> ·</b> <a href="/wiki/Gruppo_dei_quaternioni" title="Gruppo dei quaternioni">Gruppo dei quaternioni</a><b> ·</b> <a href="/wiki/Gruppo_generale_lineare" title="Gruppo generale lineare">Gruppo generale lineare</a><b> ·</b> <a href="/wiki/Gruppo_ortogonale" title="Gruppo ortogonale">Gruppo ortogonale</a><b> ·</b> <a href="/wiki/Gruppo_unitario" title="Gruppo unitario">Gruppo unitario</a><b> ·</b> <a href="/wiki/Gruppo_unitario_speciale" title="Gruppo unitario speciale">Gruppo unitario speciale</a><b> ·</b> <a href="/wiki/Gruppo_residualmente_finito" title="Gruppo residualmente finito">Gruppo residualmente finito</a><b> ·</b> <a href="/wiki/Gruppo_spaziale" title="Gruppo spaziale">Gruppo spaziale</a><b> ·</b> <a href="/wiki/Gruppo_profinito" title="Gruppo profinito">Gruppo profinito</a><b> ·</b> <a href="/wiki/Out(Fn)" title="Out(Fn)">Out(F<sub>n</sub>)</a><b> ·</b> <a href="/wiki/Parola_(teoria_dei_gruppi)" title="Parola (teoria dei gruppi)">Parola</a><b> ·</b> <a href="/wiki/Prodotto_diretto" title="Prodotto diretto">Prodotto diretto</a><b> ·</b> <a href="/wiki/Prodotto_semidiretto" title="Prodotto semidiretto">Prodotto semidiretto</a><b> ·</b> <a href="/wiki/Prodotto_intrecciato" title="Prodotto intrecciato">Prodotto intrecciato</a></td></tr><tr><th class="subnavbox_group">Teoremi</th><td colspan="1"><a href="/wiki/Alternativa_di_Tits" title="Alternativa di Tits">Alternativa di Tits</a><b> ·</b> <a href="/wiki/Teorema_di_isomorfismo" title="Teorema di isomorfismo">Teorema di isomorfismo</a><b> ·</b> <a href="/wiki/Teorema_di_Lagrange_(teoria_dei_gruppi)" title="Teorema di Lagrange (teoria dei gruppi)">Teorema di Lagrange</a><b> ·</b> <a href="/wiki/Teorema_di_Cauchy_(teoria_dei_gruppi)" title="Teorema di Cauchy (teoria dei gruppi)">Teorema di Cauchy</a><b> ·</b> <a href="/wiki/Teoremi_di_Sylow" title="Teoremi di Sylow">Teoremi di Sylow</a><b> ·</b> <a href="/wiki/Teorema_di_Cayley" title="Teorema di Cayley">Teorema di Cayley</a><b> ·</b> <a href="/wiki/Gruppo_abeliano#Classificazione" title="Gruppo abeliano">Teorema di struttura dei gruppi abeliani finiti</a><b> ·</b> <a href="/wiki/Lemma_della_farfalla" title="Lemma della farfalla">Lemma della farfalla</a><b> ·</b> <a href="/wiki/Lemma_del_ping-pong" title="Lemma del ping-pong">Lemma del ping-pong</a><b> ·</b> <a href="/wiki/Classificazione_dei_gruppi_semplici_finiti" title="Classificazione dei gruppi semplici finiti">Classificazione dei gruppi semplici finiti</a></td></tr><tr><th class="subnavbox_group">Sottoinsiemi</th><td colspan="1"><a href="/wiki/Sottogruppo" title="Sottogruppo">Sottogruppo</a><b> ·</b> <a href="/wiki/Sottogruppo_normale" title="Sottogruppo normale">Sottogruppo normale</a><b> ·</b> <a href="/wiki/Sottogruppo_caratteristico" title="Sottogruppo caratteristico">Sottogruppo caratteristico</a><b> ·</b> <a href="/wiki/Sottogruppo_di_Frattini" title="Sottogruppo di Frattini">Sottogruppo di Frattini</a><b> ·</b> <a href="/wiki/Sottogruppo_di_torsione" title="Sottogruppo di torsione">Sottogruppo di torsione</a><b> ·</b> <a href="/wiki/Classe_laterale" title="Classe laterale">Classe laterale</a><b> ·</b> <a href="/wiki/Classe_di_coniugio" title="Classe di coniugio">Classe di coniugio</a><b> ·</b> <a href="/wiki/Serie_di_composizione" title="Serie di composizione">Serie di composizione</a></td></tr><tr><td colspan="2" class="navbox_center"><a href="/wiki/Omomorfismo_di_gruppi" title="Omomorfismo di gruppi">Omomorfismo</a><b> ·</b> <a href="/wiki/Isomorfismo_tra_gruppi" title="Isomorfismo tra gruppi">Isomorfismo</a><b> ·</b> <a href="/wiki/Automorfismo_interno" title="Automorfismo interno">Automorfismo interno</a><b> ·</b> <a href="/wiki/Automorfismo_esterno" title="Automorfismo esterno">Automorfismo esterno</a><b> ·</b> <a href="/wiki/Permutazione" title="Permutazione">Permutazione</a><b> ·</b> <a href="/wiki/Presentazione_di_un_gruppo" title="Presentazione di un gruppo">Presentazione di un gruppo</a><b> ·</b> <a href="/wiki/Azione_di_gruppo" title="Azione di gruppo">Azione di gruppo</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_degli_anelli" title="Teoria degli anelli">Teoria degli anelli</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Anello_(algebra)" title="Anello (algebra)">Anello</a> (<a href="/wiki/Anello_artiniano" title="Anello artiniano">artiniano</a><b> ·</b> <a href="/wiki/Anello_noetheriano" title="Anello noetheriano">noetheriano</a><b> ·</b> <a href="/wiki/Anello_locale" title="Anello locale">locale</a>)<b> ·</b> <a href="/wiki/Caratteristica_(algebra)" title="Caratteristica (algebra)">Caratteristica</a><b> ·</b> <a href="/wiki/Ideale_(matematica)" title="Ideale (matematica)">Ideale</a> (<a href="/wiki/Ideale_primo" title="Ideale primo">primo</a><b> ·</b> <a href="/wiki/Ideale_massimale" title="Ideale massimale">massimale</a>)<b> ·</b> <a href="/wiki/Dominio_d%27integrit%C3%A0" title="Dominio d'integrità">Dominio</a> (<a href="/wiki/Dominio_a_fattorizzazione_unica" title="Dominio a fattorizzazione unica">a fattorizzazione unica</a><b> ·</b> <a href="/wiki/Dominio_ad_ideali_principali" title="Dominio ad ideali principali">a ideali principali</a><b> ·</b> <a href="/wiki/Dominio_euclideo" title="Dominio euclideo">euclideo</a>)<b> ·</b> <a href="/wiki/Matrice" title="Matrice">Matrice</a><b> ·</b> <a href="/wiki/Anello_semplice" title="Anello semplice">Anello semplice</a><b> ·</b> <a href="/wiki/Anello_degli_endomorfismi" title="Anello degli endomorfismi">Anello degli endomorfismi</a><b> ·</b> <a href="/wiki/Teorema_di_Artin-Wedderburn" title="Teorema di Artin-Wedderburn">Teorema di Artin-Wedderburn</a><b> ·</b> <a href="/wiki/Modulo_(algebra)" title="Modulo (algebra)">Modulo</a><b> ·</b> <a href="/wiki/Dominio_di_Dedekind" title="Dominio di Dedekind">Dominio di Dedekind</a><b> ·</b> <a href="/wiki/Estensione_di_anelli" title="Estensione di anelli">Estensione di anelli</a><b> ·</b> <a href="/wiki/Teorema_della_base_di_Hilbert" title="Teorema della base di Hilbert">Teorema della base di Hilbert</a><b> ·</b> <a href="/wiki/Anello_di_Gorenstein" title="Anello di Gorenstein">Anello di Gorenstein</a><b> ·</b> <a href="/wiki/Base_di_Gr%C3%B6bner" title="Base di Gröbner">Base di Gröbner</a><b> ·</b> <a href="/wiki/Prodotto_tensoriale" title="Prodotto tensoriale">Prodotto tensoriale</a><b> ·</b> <a href="/wiki/Primo_associato" title="Primo associato">Primo associato</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_dei_campi_(matematica)" class="mw-redirect" title="Teoria dei campi (matematica)">Teoria dei campi</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><table class="subnavbox"><tbody><tr><td colspan="2" class="navbox_center"><a href="/wiki/Campo_(matematica)" title="Campo (matematica)">Campo</a><b> ·</b> <a href="/wiki/Polinomio_irriducibile" title="Polinomio irriducibile">Polinomio irriducibile</a><b> ·</b> <a href="/wiki/Polinomio_ciclotomico" title="Polinomio ciclotomico">Polinomio ciclotomico</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_dell%27algebra" title="Teorema fondamentale dell'algebra">Teorema fondamentale dell'algebra</a><b> ·</b> <a href="/wiki/Campo_finito" title="Campo finito">Campo finito</a><b> ·</b> <a href="/wiki/Automorfismo" title="Automorfismo">Automorfismo</a><b> ·</b> <a href="/wiki/Endomorfismo_di_Frobenius" title="Endomorfismo di Frobenius">Endomorfismo di Frobenius</a></td></tr><tr><th class="subnavbox_group">Estensioni</th><td colspan="1"><a href="/wiki/Campo_di_spezzamento" title="Campo di spezzamento">Campo di spezzamento</a><b> ·</b> <a href="/wiki/Estensione_di_campi" title="Estensione di campi">Estensione di campi</a><b> ·</b> <a href="/wiki/Estensione_algebrica" title="Estensione algebrica">Estensione algebrica</a><b> ·</b> <a href="/wiki/Estensione_separabile" title="Estensione separabile">Estensione separabile</a><b> ·</b> <a href="/wiki/Chiusura_algebrica" title="Chiusura algebrica">Chiusura algebrica</a><b> ·</b> <a href="/wiki/Campo_di_numeri" title="Campo di numeri">Campo di numeri</a><b> ·</b> <a href="/wiki/Estensione_normale" title="Estensione normale">Estensione normale</a><b> ·</b> <a href="/wiki/Estensione_di_Galois" title="Estensione di Galois">Estensione di Galois</a><b> ·</b> <a href="/wiki/Estensione_abeliana" title="Estensione abeliana">Estensione abeliana</a><b> ·</b> <a href="/wiki/Estensione_ciclotomica" title="Estensione ciclotomica">Estensione ciclotomica</a><b> ·</b> <a href="/wiki/Teoria_di_Kummer" title="Teoria di Kummer">Teoria di Kummer</a></td></tr><tr><th class="subnavbox_group">Teoria di Galois</th><td colspan="1"><a href="/wiki/Gruppo_di_Galois" title="Gruppo di Galois">Gruppo di Galois</a><b> ·</b> <a href="/wiki/Teoria_di_Galois" title="Teoria di Galois">Teoria di Galois</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_della_teoria_di_Galois" title="Teorema fondamentale della teoria di Galois">Teorema fondamentale della teoria di Galois</a><b> ·</b> <a href="/wiki/Teorema_di_Abel-Ruffini" title="Teorema di Abel-Ruffini">Teorema di Abel-Ruffini</a><b> ·</b> <a href="/wiki/Costruzioni_con_riga_e_compasso" title="Costruzioni con riga e compasso">Costruzioni con riga e compasso</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Altre <a href="/wiki/Struttura_algebrica" title="Struttura algebrica">strutture algebriche</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Magma_(matematica)" title="Magma (matematica)">Magma</a><b> ·</b> <a href="/wiki/Semigruppo" title="Semigruppo">Semigruppo</a><b> ·</b> <a href="/wiki/Corpo_(matematica)" title="Corpo (matematica)">Corpo</a><b> ·</b> <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a><b> ·</b> <a href="/wiki/Algebra_su_campo" title="Algebra su campo">Algebra su campo</a><b> ·</b> <a href="/wiki/Algebra_di_Lie" title="Algebra di Lie">Algebra di Lie</a><b> ·</b> <a href="/wiki/Algebra_differenziale" title="Algebra differenziale">Algebra differenziale</a><b> ·</b> <a href="/wiki/Algebra_di_Clifford" title="Algebra di Clifford">Algebra di Clifford</a><b> ·</b> <a href="/wiki/Gruppo_topologico" title="Gruppo topologico">Gruppo topologico</a><b> ·</b> <a href="/wiki/Gruppo_ordinato" title="Gruppo ordinato">Gruppo ordinato</a><b> ·</b> <a href="/wiki/Quasi-anello" title="Quasi-anello">Quasi-anello</a><b> ·</b> <a href="/wiki/Algebra_di_Boole" title="Algebra di Boole">Algebra di Boole</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">argomenti</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teoria_delle_categorie" title="Teoria delle categorie">Teoria delle categorie</a><b> ·</b> <a href="/wiki/Algebra_lineare" title="Algebra lineare">Algebra lineare</a><b> ·</b> <a href="/wiki/Algebra_commutativa" title="Algebra commutativa">Algebra commutativa</a><b> ·</b> <a href="/wiki/Algebra_omologica" title="Algebra omologica">Algebra omologica</a><b> ·</b> <a href="/wiki/Algebra_astratta" title="Algebra astratta">Algebra astratta</a><b> ·</b> <a href="/wiki/Algebra_computazionale" class="mw-redirect" title="Algebra computazionale">Algebra computazionale</a><b> ·</b> <a href="/wiki/Algebra_differenziale" title="Algebra differenziale">Algebra differenziale</a><b> ·</b> <a href="/wiki/Algebra_universale" title="Algebra universale">Algebra universale</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Teoria_dei_numeri"><tbody><tr><th colspan="3" style="background:#ffc0cb;"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Teoria_dei_numeri" title="Template:Teoria dei numeri"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Teoria_dei_numeri&action=edit&redlink=1" class="new" title="Discussioni template:Teoria dei numeri (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Teoria_dei_numeri&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">Teoria dei numeri</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Numero" title="Numero">Numeri</a> più usati</th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Numero_naturale" title="Numero naturale">Naturali</a><b> ·</b> <a href="/wiki/Numero_intero" title="Numero intero">Interi</a><b> ·</b> <a href="/wiki/Numeri_pari_e_dispari" title="Numeri pari e dispari">Pari e dispari</a></td><td rowspan="10" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/58px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="58" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/87px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/116px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><figcaption></figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Principi generali</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Principio_d%27induzione" title="Principio d'induzione">Principio d'induzione</a><b> ·</b> <a href="/wiki/Principio_del_buon_ordinamento" title="Principio del buon ordinamento">Principio del buon ordinamento</a><b> ·</b> <a href="/wiki/Relazione_di_equivalenza" title="Relazione di equivalenza">Relazione di equivalenza</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Successione_di_interi" title="Successione di interi">Successioni di interi</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a><b> ·</b> <a href="/wiki/Successione_di_Fibonacci" title="Successione di Fibonacci">Successione di Fibonacci</a><b> ·</b> <a href="/wiki/Numero_di_Catalan" title="Numero di Catalan">Numero di Catalan</a><b> ·</b> <a href="/wiki/Numero_di_Perrin" title="Numero di Perrin">Numero di Perrin</a><b> ·</b> <a href="/wiki/Numero_di_Eulero_(teoria_dei_numeri)" class="mw-redirect" title="Numero di Eulero (teoria dei numeri)">Numero di Eulero</a><b> ·</b> <a href="/wiki/Successione_di_Mian-Chowla" title="Successione di Mian-Chowla">Successione di Mian-Chowla</a><b> ·</b> <a href="/wiki/Successione_di_Thue-Morse" title="Successione di Thue-Morse">Successione di Thue-Morse</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Caratteristiche dei numeri primi</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Numero_primo" title="Numero primo">Numero primo</a><b> ·</b> <a href="/wiki/Lemma_di_Euclide" title="Lemma di Euclide">Lemma di Euclide</a><b> ·</b> <a class="mw-selflink selflink">Teorema dell'infinità dei numeri primi</a><b> ·</b> <a href="/wiki/Crivello_di_Eratostene" title="Crivello di Eratostene">Crivello di Eratostene</a><b> ·</b> <a href="/wiki/Test_di_primalit%C3%A0" title="Test di primalità">Test di primalità</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica">Teorema fondamentale dell'aritmetica</a><b> ·</b> <a href="/wiki/Interi_coprimi" title="Interi coprimi">Interi coprimi</a><b> ·</b> <a href="/wiki/Identit%C3%A0_di_B%C3%A9zout" title="Identità di Bézout">Identità di Bézout</a><b> ·</b> <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">MCD</a><b> ·</b> <a href="/wiki/Minimo_comune_multiplo" title="Minimo comune multiplo">mcm</a><b> ·</b> <a href="/wiki/Algoritmo_di_Euclide" title="Algoritmo di Euclide">Algoritmo di Euclide</a><b> ·</b> <a href="/wiki/Algoritmo_esteso_di_Euclide" title="Algoritmo esteso di Euclide">Algoritmo esteso di Euclide</a><b> ·</b> <a href="/wiki/Teorema_dei_numeri_primi" title="Teorema dei numeri primi">Teorema dei numeri primi</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Funzione_aritmetica" title="Funzione aritmetica">Funzioni aritmetiche</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Funzione_moltiplicativa" title="Funzione moltiplicativa">Funzione moltiplicativa</a><b> ·</b> <a href="/wiki/Funzione_additiva" title="Funzione additiva">Funzione additiva</a><b> ·</b> <a href="/wiki/Convoluzione_di_Dirichlet" title="Convoluzione di Dirichlet">Convoluzione di Dirichlet</a><b> ·</b> <a href="/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero">Funzione φ di Eulero</a><b> ·</b> <a href="/wiki/Funzione_di_M%C3%B6bius" title="Funzione di Möbius">Funzione di Möbius</a><b> ·</b> <a href="/wiki/Funzione_tau_sui_positivi" title="Funzione tau sui positivi">Funzione tau sui positivi</a><b> ·</b> <a href="/wiki/Funzione_sigma" title="Funzione sigma">Funzione sigma</a><b> ·</b> <a href="/wiki/Funzione_di_Liouville" title="Funzione di Liouville">Funzione di Liouville</a><b> ·</b> <a href="/wiki/Funzione_di_Mertens" title="Funzione di Mertens">Funzione di Mertens</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Aritmetica_modulare" title="Aritmetica modulare">Aritmetica modulare</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teorema_cinese_del_resto" title="Teorema cinese del resto">Teorema cinese del resto</a><b> ·</b> <a href="/wiki/Piccolo_teorema_di_Fermat" title="Piccolo teorema di Fermat">Piccolo teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Eulero_(aritmetica_modulare)" title="Teorema di Eulero (aritmetica modulare)">Teorema di Eulero</a><b> ·</b> <a href="/wiki/Criteri_di_divisibilit%C3%A0" title="Criteri di divisibilità">Criteri di divisibilità</a><b> ·</b> <a href="/wiki/Teorema_di_Fermat_sulle_somme_di_due_quadrati" title="Teorema di Fermat sulle somme di due quadrati">Teorema di Fermat sulle somme di due quadrati</a><b> ·</b> <a href="/wiki/Teorema_di_Wilson" title="Teorema di Wilson">Teorema di Wilson</a><b> ·</b> <a href="/wiki/Reciprocit%C3%A0_quadratica" title="Reciprocità quadratica">Legge di reciprocità quadratica</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Congettura" title="Congettura">Congetture</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Congettura_di_Goldbach" title="Congettura di Goldbach">Congettura di Goldbach</a><b> ·</b> <a href="/wiki/Congettura_di_Polignac" title="Congettura di Polignac">Congettura di Polignac</a><b> ·</b> <a href="/wiki/Congettura_abc" title="Congettura abc">Congettura abc</a><b> ·</b> <a href="/wiki/Congettura_dei_numeri_primi_gemelli" title="Congettura dei numeri primi gemelli">Congettura dei numeri primi gemelli</a><b> ·</b> <a href="/wiki/Congettura_di_Legendre" title="Congettura di Legendre">Congettura di Legendre</a><b> ·</b> <a href="/wiki/Nuova_congettura_di_Mersenne" title="Nuova congettura di Mersenne">Nuova congettura di Mersenne</a><b> ·</b> <a href="/wiki/Congettura_di_Collatz" title="Congettura di Collatz">Congettura di Collatz</a><b> ·</b> <a href="/wiki/Ipotesi_di_Riemann" title="Ipotesi di Riemann">Ipotesi di Riemann</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Altro</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Problema_di_Waring" title="Problema di Waring">Problema di Waring</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Principali teorici</th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci">Fibonacci</a><b> ·</b> <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a><b> ·</b> <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a><b> ·</b> <a href="/wiki/Eulero" title="Eulero">Eulero</a><b> ·</b> <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a><b> ·</b> <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a><b> ·</b> <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Dirichlet</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Discipline connesse</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teoria_algebrica_dei_numeri" title="Teoria algebrica dei numeri">Teoria algebrica dei numeri</a><b> ·</b> <a href="/wiki/Teoria_analitica_dei_numeri" title="Teoria analitica dei numeri">Teoria analitica dei numeri</a><b> ·</b> <a href="/wiki/Crittografia" title="Crittografia">Crittografia</a><b> ·</b> <a href="/wiki/Teoria_computazionale_dei_numeri" title="Teoria computazionale dei numeri">Teoria computazionale dei numeri</a></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Teorema_dell%27infinità_dei_numeri_primi&oldid=137505791">https://it.wikipedia.org/w/index.php?title=Teorema_dell%27infinità_dei_numeri_primi&oldid=137505791</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categoria</a>: <ul><li><a href="/wiki/Categoria:Teoremi_sui_numeri_primi" title="Categoria:Teoremi sui numeri primi">Teoremi sui numeri primi</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria nascosta: <ul><li><a href="/wiki/Categoria:P2812_letta_da_Wikidata" title="Categoria:P2812 letta da Wikidata">P2812 letta da 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"> Questa pagina è stata modificata per l'ultima volta il 23 gen 2024 alle 21:41.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; possono applicarsi condizioni ulteriori. Vedi le <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/it">condizioni d'uso</a> per i dettagli.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/it">Informativa sulla privacy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Sala_stampa/Wikipedia">Informazioni su Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Avvertenze_generali">Avvertenze</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Codice di condotta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Sviluppatori</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/it.wikipedia.org">Statistiche</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Dichiarazione sui cookie</a></li> <li id="footer-places-mobileview"><a href="//it.m.wikipedia.org/w/index.php?title=Teorema_dell%27infinit%C3%A0_dei_numeri_primi&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Versione mobile</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-5b4c46c98-k7h72","wgBackendResponseTime":142,"wgPageParseReport":{"limitreport":{"cputime":"0.292","walltime":"0.622","ppvisitednodes":{"value":2240,"limit":1000000},"postexpandincludesize":{"value":68262,"limit":2097152},"templateargumentsize":{"value":424,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":10440,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 326.178 1 -total"," 35.97% 117.333 1 Template:Collegamenti_esterni"," 18.45% 60.169 2 Template:Navbox"," 17.65% 57.582 1 Template:Algebra"," 16.66% 54.336 1 Template:Vedi_anche"," 8.88% 28.964 1 Template:M"," 7.79% 25.415 1 Template:Portale"," 4.93% 16.089 1 Template:Cita_pubblicazione"," 4.17% 13.593 3 Template:Navbox_subgroup"," 3.57% 11.633 1 Template:Icona_argomento"]},"scribunto":{"limitreport-timeusage":{"value":"0.155","limit":"10.000"},"limitreport-memusage":{"value":6662477,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-864bbfd546-b82x2","timestamp":"20241130090330","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Teorema dell'infinit\u00e0 dei numeri primi","url":"https:\/\/it.wikipedia.org\/wiki\/Teorema_dell%27infinit%C3%A0_dei_numeri_primi","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1506253","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1506253","author":{"@type":"Organization","name":"Contributori ai progetti Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-10-22T11:06:12Z","dateModified":"2024-01-23T20:41:22Z"}</script> </body> </html>