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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-Dimostrazione_del_teorema" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dimostrazione_del_teorema"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Dimostrazione del teorema</span> </div> </a> <button aria-controls="toc-Dimostrazione_del_teorema-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 Dimostrazione del teorema</span> </button> <ul id="toc-Dimostrazione_del_teorema-sublist" class="vector-toc-list"> <li id="toc-Dimostrazione_dell'esistenza" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimostrazione_dell'esistenza"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Dimostrazione dell'esistenza</span> </div> </a> <ul id="toc-Dimostrazione_dell'esistenza-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimostrazione_dell'unicità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimostrazione_dell'unicità"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Dimostrazione dell'unicità</span> </div> </a> <ul id="toc-Dimostrazione_dell'unicità-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-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-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 fondamentale dell'aritmetica</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 66 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-66" 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">66 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Fundamentalsatz_der_Arithmetik" title="Fundamentalsatz der Arithmetik - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Fundamentalsatz der Arithmetik" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A7%D9%84%D8%A3%D8%B3%D8%A7%D8%B3%D9%8A%D8%A9_%D9%81%D9%8A_%D8%A7%D9%84%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A%D8%A7%D8%AA" title="المبرهنة الأساسية في الحسابيات - arabo" lang="ar" hreflang="ar" data-title="المبرهنة الأساسية في الحسابيات" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teorema_fundamental_de_l%27aritm%C3%A9tica" title="Teorema fundamental de l'aritmética - asturiano" lang="ast" hreflang="ast" data-title="Teorema fundamental de l'aritmética" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%B0%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D1%82%D0%B0" title="Основна теорема на аритметиката - bulgaro" lang="bg" hreflang="bg" data-title="Основна теорема на аритметиката" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%9F%E0%A6%BF%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A7%87%E0%A6%B0_%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="পাটিগণিতের মৌলিক উপপাদ্য - bengalese" lang="bn" hreflang="bn" data-title="পাটিগণিতের মৌলিক উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_fonamental_de_l%27aritm%C3%A8tica" title="Teorema fonamental de l'aritmètica - catalano" lang="ca" hreflang="ca" data-title="Teorema fonamental de l'aritmètica" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Z%C3%A1kladn%C3%AD_v%C4%9Bta_aritmetiky" title="Základní věta aritmetiky - ceco" lang="cs" hreflang="cs" data-title="Základní věta aritmetiky" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%C4%83%D0%BD_%D1%82%C4%95%D0%BF_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B8" title="Арифметикăн тĕп теореми - ciuvascio" lang="cv" hreflang="cv" data-title="Арифметикăн тĕп теореми" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Aritmetikkens_fundamentals%C3%A6tning" title="Aritmetikkens fundamentalsætning - danese" lang="da" hreflang="da" data-title="Aritmetikkens fundamentalsætning" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Fundamentalsatz_der_Arithmetik" title="Fundamentalsatz der Arithmetik - tedesco" lang="de" hreflang="de" data-title="Fundamentalsatz der Arithmetik" 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%CE%BC%CE%B5%CE%BB%CE%B9%CF%8E%CE%B4%CE%B5%CF%82_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%B9%CE%BA%CE%AE%CF%82" title="Θεμελιώδες θεώρημα αριθμητικής - greco" lang="el" hreflang="el" data-title="Θεμελιώδες θεώρημα αριθμητικής" 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/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic - inglese" lang="en" hreflang="en" data-title="Fundamental theorem of arithmetic" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Fundamenta_teoremo_de_aritmetiko" title="Fundamenta teoremo de aritmetiko - esperanto" lang="eo" hreflang="eo" data-title="Fundamenta teoremo de aritmetiko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_fundamental_de_la_aritm%C3%A9tica" title="Teorema fundamental de la aritmética - spagnolo" lang="es" hreflang="es" data-title="Teorema fundamental de la aritmética" 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/Aritmetikaren_oinarrizko_teorema" title="Aritmetikaren oinarrizko teorema - basco" lang="eu" hreflang="eu" data-title="Aritmetikaren oinarrizko 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%D8%B3%D8%A7%D8%B3%DB%8C_%D8%AD%D8%B3%D8%A7%D8%A8" title="قضیه اساسی حساب - persiano" lang="fa" hreflang="fa" data-title="قضیه اساسی حساب" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aritmetiikan_peruslause" title="Aritmetiikan peruslause - finlandese" lang="fi" hreflang="fi" data-title="Aritmetiikan peruslause" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_fondamental_de_l%27arithm%C3%A9tique" title="Théorème fondamental de l'arithmétique - francese" lang="fr" hreflang="fr" data-title="Théorème fondamental de l'arithmétique" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Bunteoirim_na_huimhr%C3%ADochta" title="Bunteoirim na huimhríochta - irlandese" lang="ga" hreflang="ga" data-title="Bunteoirim na huimhríochta" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_fundamental_da_aritm%C3%A9tica" title="Teorema fundamental da aritmética - galiziano" lang="gl" hreflang="gl" data-title="Teorema fundamental da aritmética" 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%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="המשפט היסודי של האריתמטיקה - ebraico" lang="he" hreflang="he" data-title="המשפט היסודי של האריתמטיקה" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%99%E0%A5%8D%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4_%E0%A4%95%E0%A4%BE_%E0%A4%AE%E0%A5%82%E0%A4%B2%E0%A4%AD%E0%A5%82%E0%A4%A4_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="अङ्कगणित का मूलभूत प्रमेय - hindi" lang="hi" hreflang="hi" data-title="अङ्कगणित का मूलभूत प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Osnovni_teorem_aritmetike" title="Osnovni teorem aritmetike - croato" lang="hr" hreflang="hr" data-title="Osnovni teorem aritmetike" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/A_sz%C3%A1melm%C3%A9let_alapt%C3%A9tele" title="A számelmélet alaptétele - ungherese" lang="hu" hreflang="hu" data-title="A számelmélet alaptétele" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%BE%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%B0%D5%AB%D5%B4%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Թվաբանության հիմնական թեորեմ - armeno" lang="hy" hreflang="hy" data-title="Թվաբանության հիմնական թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_dasar_aritmetika" title="Teorema dasar aritmetika - indonesiano" lang="id" hreflang="id" data-title="Teorema dasar aritmetika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Undirst%C3%B6%C3%B0usetning_reikningslistarinnar" title="Undirstöðusetning reikningslistarinnar - islandese" lang="is" hreflang="is" data-title="Undirstöðusetning reikningslistarinnar" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AE%97%E8%A1%93%E3%81%AE%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="算術の基本定理 - giapponese" lang="ja" hreflang="ja" data-title="算術の基本定理" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%94%E1%83%A2%E1%83%98%E1%83%99%E1%83%98%E1%83%A1_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%93%E1%83%90%E1%83%9B%E1%83%94%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%90" title="არითმეტიკის ფუნდამენტური თეორემა - georgiano" lang="ka" hreflang="ka" data-title="არითმეტიკის ფუნდამენტური თეორემა" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%82%E0%B2%95%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4%E0%B2%A6_%E0%B2%AE%E0%B3%82%E0%B2%B2%E0%B2%AD%E0%B3%82%E0%B2%A4_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B3%87%E0%B2%AF" title="ಅಂಕಗಣಿತದ ಮೂಲಭೂತ ಪ್ರಮೇಯ - kannada" lang="kn" hreflang="kn" data-title="ಅಂಕಗಣಿತದ ಮೂಲಭೂತ ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%B0%EC%88%A0%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EC%A0%95%EB%A6%AC" title="산술의 기본 정리 - coreano" lang="ko" hreflang="ko" data-title="산술의 기본 정리" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BD%D1%8B%D0%BD_%D0%BD%D0%B5%D0%B3%D0%B8%D0%B7%D0%B3%D0%B8_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Арифметиканын негизги теориясы - kirghiso" lang="ky" hreflang="ky" data-title="Арифметиканын негизги теориясы" data-language-autonym="Кыргызча" data-language-local-name="kirghiso" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theorema_fundamentale_arithmeticae" title="Theorema fundamentale arithmeticae - latino" lang="la" hreflang="la" data-title="Theorema fundamentale arithmeticae" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Teorem_fundal_de_aritmetica" title="Teorem fundal de aritmetica - Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Teorem fundal de aritmetica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_fondamental_de_l%27aritmetica" title="Teorema fondamental de l'aritmetica - lombardo" lang="lmo" hreflang="lmo" data-title="Teorema fondamental de l'aritmetica" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pagrindin%C4%97_aritmetikos_teorema" title="Pagrindinė aritmetikos teorema - lituano" lang="lt" hreflang="lt" data-title="Pagrindinė aritmetikos teorema" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Aritm%C4%93tikas_pamatteor%C4%93ma" title="Aritmētikas pamatteorēma - lettone" lang="lv" hreflang="lv" data-title="Aritmētikas pamatteorēma" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%B0%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D1%82%D0%B0" title="Основна теорема на аритметиката - macedone" lang="mk" hreflang="mk" data-title="Основна теорема на аритметиката" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%B2%E0%B5%86_%E0%B4%85%E0%B4%9F%E0%B4%BF%E0%B4%B8%E0%B5%8D%E0%B4%A5%E0%B4%BE%E0%B4%A8_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="അങ്കഗണിതത്തിലെ അടിസ്ഥാന സിദ്ധാന്തം - malayalam" lang="ml" hreflang="ml" data-title="അങ്കഗണിതത്തിലെ അടിസ്ഥാന സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teorem_asas_aritmetik" title="Teorem asas aritmetik - malese" lang="ms" hreflang="ms" data-title="Teorem asas aritmetik" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%8F%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%81%8F_%E1%80%A1%E1%80%81%E1%80%BC%E1%80%B1%E1%80%81%E1%80%B6%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%99%E1%80%BA" title="ဂဏန်းသင်္ချာ၏ အခြေခံသီအိုရမ် - birmano" lang="my" hreflang="my" data-title="ဂဏန်းသင်္ချာ၏ အခြေခံသီအိုရမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoofdstelling_van_de_rekenkunde" title="Hoofdstelling van de rekenkunde - olandese" lang="nl" hreflang="nl" data-title="Hoofdstelling van de rekenkunde" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Aritmetikkens_fundamentalteorem" title="Aritmetikkens fundamentalteorem - norvegese bokmål" lang="nb" hreflang="nb" data-title="Aritmetikkens fundamentalteorem" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zasadnicze_twierdzenie_arytmetyki" title="Zasadnicze twierdzenie arytmetyki - polacco" lang="pl" hreflang="pl" data-title="Zasadnicze twierdzenie arytmetyki" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_fondamental_dl%27aritm%C3%A9tica" title="Teorema fondamental dl'aritmética - piemontese" lang="pms" hreflang="pms" data-title="Teorema fondamental dl'aritmética" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_fundamental_da_aritm%C3%A9tica" title="Teorema fundamental da aritmética - portoghese" lang="pt" hreflang="pt" data-title="Teorema fundamental da aritmética" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_fundamental%C4%83_a_aritmeticii" title="Teorema fundamentală a aritmeticii - rumeno" lang="ro" hreflang="ro" data-title="Teorema fundamentală a aritmeticii" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="voce consigliata"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B8" title="Основная теорема арифметики - russo" lang="ru" hreflang="ru" data-title="Основная теорема арифметики" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiur%C3%A8ma_funnamint%C3%A0li_di_l%27arittim%C3%A8tica" title="Tiurèma funnamintàli di l'arittimètica - siciliano" lang="scn" hreflang="scn" data-title="Tiurèma funnamintàli di l'arittimètica" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%82%E0%B6%9A_%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA%E0%B7%9A_%E0%B6%B8%E0%B7%96%E0%B6%BD%E0%B7%92%E0%B6%9A_%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%9A%E0%B6%BA%E0%B6%BA" title="අංක ගණිතයේ මූලික ප්‍රමේයය - singalese" lang="si" hreflang="si" data-title="අංක ගණිතයේ මූලික ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="singalese" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic - Simple English" lang="en-simple" hreflang="en-simple" data-title="Fundamental theorem of arithmetic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Z%C3%A1kladn%C3%A1_veta_aritmetiky" title="Základná veta aritmetiky - slovacco" lang="sk" hreflang="sk" data-title="Základná veta aritmetiky" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Osnovni_izrek_aritmetike" title="Osnovni izrek aritmetike - sloveno" lang="sl" hreflang="sl" data-title="Osnovni izrek aritmetike" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema_themelore_e_aritmetik%C3%ABs" title="Teorema themelore e aritmetikës - albanese" lang="sq" hreflang="sq" data-title="Teorema themelore e aritmetikës" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%B0%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B5" title="Основна теорема аритметике - serbo" lang="sr" hreflang="sr" data-title="Основна теорема аритметике" 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/Aritmetikens_fundamentalsats" title="Aritmetikens fundamentalsats - svedese" lang="sv" hreflang="sv" data-title="Aritmetikens fundamentalsats" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%85%E0%AE%9F%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%88%E0%AE%A4%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="எண்கணிதத்தின் அடிப்படைத் தேற்றம் - tamil" lang="ta" hreflang="ta" data-title="எண்கணிதத்தின் அடிப்படைத் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%A1%E0%B8%B9%E0%B8%A5%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="ทฤษฎีบทมูลฐานของเลขคณิต - thailandese" lang="th" hreflang="th" data-title="ทฤษฎีบทมูลฐานของเลขคณิต" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Aritmeti%C4%9Fin_temel_teoremi" title="Aritmetiğin temel teoremi - turco" lang="tr" hreflang="tr" data-title="Aritmetiğin temel 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%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B8" title="Основна теорема арифметики - ucraino" lang="uk" hreflang="uk" data-title="Основна теорема арифметики" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%DA%A9%D8%A7_%D8%A8%D9%86%DB%8C%D8%A7%D8%AF%DB%8C_%D9%85%D8%B3%D8%A6%D9%84%DB%81_%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA%DB%8C" title="حساب کا بنیادی مسئلہ اثباتی - urdu" lang="ur" hreflang="ur" data-title="حساب کا بنیادی مسئلہ اثباتی" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>Il <b>teorema fondamentale dell'aritmetica</b> afferma che: </p> <dl><dd><i>Ogni <a href="/wiki/Numero_naturale" title="Numero naturale">numero naturale</a> maggiore di 1 o è un <a href="/wiki/Numero_primo" title="Numero primo">numero primo</a> o si può esprimere come prodotto di <a href="/wiki/Numero_primo" title="Numero primo">numeri primi</a>. Tale rappresentazione è unica, se si prescinde dall'ordine in cui compaiono i fattori</i>.</dd></dl> <p>L'enunciato è facilmente verificabile per numeri naturali "piccoli": è facile scoprire 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 70}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>70</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 70}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5769c35091d646a29100b5f0c1df7fe3dd09c0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 70}"></span> è pari 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 2\times 5\times 7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 5\times 7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3d59584c305873b00682802c424b47634dad1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.168ex; height:2.176ex;" alt="{\displaystyle 2\times 5\times 7}"></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> equivale 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 2\times 2\times 5\times 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2\times 5\times 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c42ef70a56e22af4ba842dd8234f2cdf4bca065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.171ex; height:2.176ex;" alt="{\displaystyle 2\times 2\times 5\times 5}"></span> ovvero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}\times 5^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}\times 5^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f67c640dd44819a613770ea7cad86a73ec53ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.274ex; height:2.676ex;" alt="{\displaystyle 2^{2}\times 5^{2}}"></span>, ed è altrettanto facile verificare che per questi numeri non possono esistere altre scomposizioni in fattori primi. </p><p>Il teorema fu dimostrato per la prima volta, in un linguaggio matematico moderno, da <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> nelle <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i>;<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> <a href="/wiki/Euclide" title="Euclide">Euclide</a>, negli <i><a href="/wiki/Elementi_(Euclide)" title="Elementi (Euclide)">Elementi</a></i>, insieme all'esistenza della fattorizzazione<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>, aveva dimostrato una proposizione, oggi nota come <a href="/wiki/Lemma_di_Euclide" title="Lemma di Euclide">lemma di Euclide</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>, dalla quale si ricava la proprietà di fattorizzazione unica. </p><p>Nella <a href="/wiki/Teoria_degli_anelli" title="Teoria degli anelli">teoria degli anelli</a>, un analogo della proprietà espressa dal teorema costituisce la definizione stessa di <a href="/wiki/Dominio_a_fattorizzazione_unica" title="Dominio a fattorizzazione unica">anello a fattorizzazione unica</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Dimostrazione_del_teorema">Dimostrazione del teorema</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&veaction=edit&section=1" title="Modifica la sezione Dimostrazione del teorema" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&action=edit&section=1" title="Edit section's source code: Dimostrazione del teorema"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'enunciato del teorema asserisce l'esistenza di una fattorizzazione in numeri primi per ogni numero naturale, e successivamente la sua unicità. Dimostriamo separatamente le due affermazioni. </p> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione_dell'esistenza"><span id="Dimostrazione_dell.27esistenza"></span>Dimostrazione dell'esistenza</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&veaction=edit&section=2" title="Modifica la sezione Dimostrazione dell'esistenza" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&action=edit&section=2" title="Edit section's source code: Dimostrazione dell'esistenza"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalla definizione di <a href="/wiki/Numero_primo" title="Numero primo">numero primo</a> si deduce che ogni numero maggiore o uguale a 2 o è un numero primo oppure si può esprimere come prodotto di <a href="/wiki/Numero_primo" title="Numero primo">numeri primi</a>. Questo fatto si può dimostrare per <a href="/wiki/Principio_d%27induzione" title="Principio d'induzione">induzione</a>: </p> <ul><li><i>n</i>=2 è primo, quindi soddisfa quanto enunciato.</li> <li>Supponendo vero l'enunciato per tutti i numeri da 2 a <i>n</i>, dimostriamo che vale anche per <i>n</i>+1. Per <i>n</i>+1 ci sono due possibilità: o è primo oppure è divisibile per un numero <i>a</i> compreso tra 2 e <i>n</i>. Nel caso in cui <i>n</i>+1 sia divisibile per <i>a</i> per l'ipotesi induttiva o <i>a</i> è primo oppure <i>a</i> ha un divisore primo <i>p</i>. In quest'ultimo caso (per la <a href="/wiki/Relazione_transitiva" title="Relazione transitiva">proprietà transitiva</a> della divisibilità) <i>p</i> è anche un divisore di <i>n</i>+1. In ogni caso dunque o <i>n</i>+1 è primo o è divisibile per un primo.</li></ul> <p>La dimostrazione dell'esistenza della fattorizzazione per ogni numero procede ancora per <a href="/wiki/Principio_d%27induzione" title="Principio d'induzione">induzione</a>: </p> <ul><li><i>n</i>=2 è primo e dunque è già banalmente fattorizzato.</li> <li>Supponiamo vera l'esistenza di una fattorizzazione per tutti i naturali compresi tra 2 e <i>n</i> e dimostriamola vera anche per <i>n</i>+1. Considerando <i>n</i>+1, abbiamo due casi: <i>n</i>+1 è primo (e quindi è già fattorizzato) oppure <i>n</i>+1 è divisibile per un primo <i>p</i> (come dimostrato nella prima parte); in quest'ultimo caso il numero <i>m</i>=(<i>n</i>+1)/<i>p</i> è minore di <i>n</i>+1, e quindi verifica l'ipotesi induttiva, ovvero esiste una fattorizzazione di <i>m</i>. Ma allora <i>n</i>+1=<i>mp</i> cioè <i>n</i>+1 è fattorizzabile (è il prodotto di <i>m</i> e <i>p</i>).</li></ul> <p>Quindi l'esistenza di una fattorizzazione è dimostrata per ogni numero naturale <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione_dell'unicità"><span id="Dimostrazione_dell.27unicit.C3.A0"></span>Dimostrazione dell'unicità</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&veaction=edit&section=3" title="Modifica la sezione Dimostrazione dell'unicità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&action=edit&section=3" title="Edit section's source code: Dimostrazione dell'unicità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dimostriamo che se un numero ammette una <a href="/wiki/Fattorizzazione" title="Fattorizzazione">fattorizzazione</a> in <a href="/wiki/Numero_primo" title="Numero primo">numeri primi</a> questa è unica. </p><p><a href="/wiki/Dimostrazione_per_assurdo" title="Dimostrazione per assurdo">Per assurdo</a>: Si supponga che esistano dei numeri scomponibili in fattori primi in più di un modo, e si chiami <i>m</i> il più piccolo (che esiste per il <a href="/wiki/Principio_del_buon_ordinamento" title="Principio del buon ordinamento">principio del buon ordinamento</a>). Innanzitutto si dimostra che, date due fattorizzazioni di <i>m</i>, i numeri primi che si presentano nella prima fattorizzazione sono tutti distinti da quelli della seconda fattorizzazione. Siano infatti [1] e [2] le due diverse fattorizzazioni di <i>m</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1\right]\quad m=p_{1}p_{2}p_{3}\dots p_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>1</mn> <mo>]</mo> </mrow> <mspace width="1em" /> <mi>m</mi> <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> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1\right]\quad m=p_{1}p_{2}p_{3}\dots p_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f5811db1084e7aecef57ad70886b3ce9035736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.646ex; height:2.843ex;" alt="{\displaystyle \left[1\right]\quad m=p_{1}p_{2}p_{3}\dots p_{s}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[2\right]\quad m=q_{1}q_{2}q_{3}\dots q_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>2</mn> <mo>]</mo> </mrow> <mspace width="1em" /> <mi>m</mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[2\right]\quad m=q_{1}q_{2}q_{3}\dots q_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136d1cc333c4cb2eda1ee8318b722ee369aed173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.939ex; height:2.843ex;" alt="{\displaystyle \left[2\right]\quad m=q_{1}q_{2}q_{3}\dots q_{t}}"></span></dd></dl> <p>dove i <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> e i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d567ac2d170501680d2efa4c1d71d6a8569ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.947ex; height:2.343ex;" alt="{\displaystyle q_{j}}"></span> sono primi ma differenti tra loro, ovvero <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 \forall i,j\;\;p_{i}\neq q_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i,j\;\;p_{i}\neq q_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4838639ef6c2caf5e3520844b1e6c002221ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.392ex; height:2.843ex;" alt="{\displaystyle \forall i,j\;\;p_{i}\neq q_{j}}"></span> (se ci fosse un fattore identico <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_{h}=q_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{h}=q_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef95fdd3f2026b6fd543ba9f2e234adb5c7f3320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.662ex; height:2.009ex;" alt="{\displaystyle p_{h}=q_{k}}"></span> possiamo ricondurci al caso indicato 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> per tale fattore e ottenendo un numero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ea8347f7588b19652c2098395f059d76b12b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.725ex; height:2.509ex;" alt="{\displaystyle m'}"></span> a scomposizione multipla per cui vale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m'<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m'<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b938155755d2a56cb5b1a9531ce0036fffa74796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.864ex; height:2.509ex;" alt="{\displaystyle m'<m}"></span>, contraddicendo l'ipotesi 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> sia il più piccolo tra i numeri a scomposizione multipla). All'interno di ogni fattorizzazione ci possono comunque essere fattori ripetuti: ad esempio, 100 = 2×2×5×5. </p><p>A questo punto sappiamo 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> è diverso 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 q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"></span>; senza perdita di generalità possiamo supporre 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_{1}<q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}<q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444dc781f3ae4d657457c790eac5a0fec9b12017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.503ex; height:2.176ex;" alt="{\displaystyle p_{1}<q_{1}}"></span>. Poniamo allora </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[3\right]\quad n=(q_{1}-p_{1})q_{2}q_{3}\dots q_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>3</mn> <mo>]</mo> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[3\right]\quad n=(q_{1}-p_{1})q_{2}q_{3}\dots q_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d880977bd889f15375475c4ed7e16d5b3772b3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.167ex; height:2.843ex;" alt="{\displaystyle \left[3\right]\quad n=(q_{1}-p_{1})q_{2}q_{3}\dots q_{t}}"></span></dd></dl> <p>Evidentemente, <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<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cff096773597d7223f9d90162eb2d780dfc18dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle n<m}"></span>, dato che la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[3\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>3</mn> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[3\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61bc70ee3f681fd025694462a42f4c59b51aa46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left[3\right]}"></span> si può scrivere come </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[4\right]\quad n=q_{1}q_{2}q_{3}\dots q_{t}-p_{1}q_{2}q_{3}\dots q_{t}=m-p_{1}q_{2}q_{3}\dots q_{t}<m\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>4</mn> <mo>]</mo> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo><</mo> <mi>m</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[4\right]\quad n=q_{1}q_{2}q_{3}\dots q_{t}-p_{1}q_{2}q_{3}\dots q_{t}=m-p_{1}q_{2}q_{3}\dots q_{t}<m\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb97dcf89e78c7578064fa52f540d293dfc32b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.43ex; height:2.843ex;" alt="{\displaystyle \left[4\right]\quad n=q_{1}q_{2}q_{3}\dots q_{t}-p_{1}q_{2}q_{3}\dots q_{t}=m-p_{1}q_{2}q_{3}\dots q_{t}<m\;}"></span>.</dd></dl> <p>Dimostriamo ora 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 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> ammette almeno due fattorizzazioni distinte. </p><p>Iniziamo considerando il primo fattore 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 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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}-p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}-p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c577001eda3739d87554590feb5cc8c77449ec1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.155ex; height:2.343ex;" alt="{\displaystyle q_{1}-p_{1}}"></span>. Esso può essere primo o meno; nel caso non lo fosse lo fattorizzeremo e la nuova fattorizzazione 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 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> così ottenuta non ammetterebbe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> tra i suoi fattori. Infatti, per la prima parte della dimostrazione sappiamo 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> è diverso 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 q_{2},q_{3},\dots q_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2},q_{3},\dots q_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa9a828b22facfd76e0bf94e064e7fa54c45943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.224ex; height:2.009ex;" alt="{\displaystyle q_{2},q_{3},\dots q_{t}}"></span> e non può comparire nella eventuale fattorizzazione 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 q_{1}-p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}-p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c577001eda3739d87554590feb5cc8c77449ec1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.155ex; height:2.343ex;" alt="{\displaystyle q_{1}-p_{1}}"></span>, poiché se ciò accadesse significherebbe 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 q_{1}-p_{1}=p_{1}\cdot b\Rightarrow q_{1}=p_{1}(1+b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">⇒</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}-p_{1}=p_{1}\cdot b\Rightarrow q_{1}=p_{1}(1+b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bed12a4a03415e30cbce2878b4feafe467669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.991ex; height:2.843ex;" alt="{\displaystyle q_{1}-p_{1}=p_{1}\cdot b\Rightarrow q_{1}=p_{1}(1+b)}"></span></dd></dl> <p>e quindi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"></span> sarebbe divisibile 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 p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span>, il che non è possibile in quanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"></span> è un numero primo. </p><p>Prendendo ora l'ultima uguaglianza della <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[4\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>4</mn> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[4\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7c69ef5f5921df16193e53d68b16078f2b770c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left[4\right]}"></span> e sostituendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> con la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>1</mn> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0c1a0dcb72f1f11b3d1fd2315fdab0530964fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left[1\right]}"></span> otteniamo </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[5\right]\quad n=p_{1}p_{2}p_{3}\dots p_{s}-p_{1}q_{2}q_{3}\dots q_{t}\Rightarrow n=p_{1}(p_{2}p_{3}\dots p_{s}-q_{2}q_{3}\dots q_{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>5</mn> <mo>]</mo> </mrow> <mspace width="1em" /> <mi>n</mi> <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> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[5\right]\quad n=p_{1}p_{2}p_{3}\dots p_{s}-p_{1}q_{2}q_{3}\dots q_{t}\Rightarrow n=p_{1}(p_{2}p_{3}\dots p_{s}-q_{2}q_{3}\dots q_{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10f8cea5171c3ca7a8da92cb73e0fcbee86467f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.248ex; height:2.843ex;" alt="{\displaystyle \left[5\right]\quad n=p_{1}p_{2}p_{3}\dots p_{s}-p_{1}q_{2}q_{3}\dots q_{t}\Rightarrow n=p_{1}(p_{2}p_{3}\dots p_{s}-q_{2}q_{3}\dots q_{t})}"></span></dd></dl> <p>In qualunque modo sia fattorizzabile il secondo fattore nella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[5\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>5</mn> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[5\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2da792f95e6491e2a393523e133352849450c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left[5\right]}"></span>, avremo ottenuto una fattorizzazione 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 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> che contiene <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> e che pertanto è diversa da quella nella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[3\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mn>3</mn> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[3\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61bc70ee3f681fd025694462a42f4c59b51aa46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left[3\right]}"></span>, contrariamente all'ipotesi 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> sia il numero più piccolo che ammette più di una fattorizzazione. </p><p>L'unicità è pertanto dimostrata. </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_fondamentale_dell%27aritmetica&veaction=edit&section=4" 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_fondamentale_dell%27aritmetica&action=edit&section=4" 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"><cite class="citation libro" style="font-style:normal"> <a href="/wiki/Carl_Boyer" title="Carl Boyer">Carl Benjamin Boyer</a>, <span style="font-style:italic;">Storia della matematica</span>, Milano, Mondadori, 1990, p. 582, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-04-33431-6" title="Speciale:RicercaISBN/978-88-04-33431-6">978-88-04-33431-6</a>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFEuclide">Euclide</a>, Libro VII, Proposizioni 31 e 32</cite>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFEuclide">Euclide</a>, Libro VII, proposizione 30</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_fondamentale_dell%27aritmetica&veaction=edit&section=5" 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_fondamentale_dell%27aritmetica&action=edit&section=5" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFEuclide" class="citation libro" style="font-style:normal"> <a href="/wiki/Euclide" title="Euclide">Euclide</a>, <span style="font-style:italic;"><a href="/wiki/Elementi_(Euclide)" title="Elementi (Euclide)">Elementi</a></span>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> <a href="/wiki/Harold_Davenport" title="Harold Davenport">Harold Davenport</a>, <span style="font-style:italic;">Aritmetica superiore – capitolo I</span>, Bologna, <a href="/wiki/Zanichelli" title="Zanichelli">Zanichelli</a>, 1994, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-08-09154-6" title="Speciale:RicercaISBN/88-08-09154-6">88-08-09154-6</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Tom M. Apostol</a>, <span style="font-style:italic;">Introduction to Analytic Number Theory – capitolo 1</span>, New York, <a href="/wiki/Springer_(azienda)" title="Springer (azienda)">Springer-Verlag</a>, 1976, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-90163-9" title="Speciale:RicercaISBN/0-387-90163-9">0-387-90163-9</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&veaction=edit&section=6" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_fondamentale_dell%27aritmetica&action=edit&section=6" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Fundamental theorem of arithmetic"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fundamental_theorem_of_arithmetic?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sul <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fundamental_theorem_of_arithmetic?uselang=it">teorema fondamentale dell'aritmetica</a></span></b></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_fondamentale_dell%27aritmetica&veaction=edit&section=7" 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_fondamentale_dell%27aritmetica&action=edit&section=7" 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="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/teorema-fondamentale-dell-aritmetica_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">aritmetica, teorema fondamentale dell'<span></span></span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q670235#P9621" 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> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/fundamental-theorem-of-arithmetic"><span style="font-style:italic;">fundamental theorem of arithmetic</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q670235#P1417" 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> <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/FundamentalTheoremofArithmetic.html"><span style="font-style:italic;">Fundamental Theorem of Arithmetic</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/Q670235#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-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 href="/wiki/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" title="Teorema dell'infinità dei numeri primi">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 class="mw-selflink selflink">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" alt="" width="1" height="1" style="border: none; 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