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<span>Arithmetic operations</span> </div> </a> <ul id="toc-Arithmetic_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Arithmetic functions</span> </div> </a> <ul id="toc-Arithmetic_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Proof</span> </div> </a> <button aria-controls="toc-Proof-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Proof subsection</span> </button> <ul id="toc-Proof-sublist" class="vector-toc-list"> <li id="toc-Existence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Existence</span> </div> </a> <ul id="toc-Existence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniqueness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Uniqueness</span> </div> </a> <ul id="toc-Uniqueness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniqueness_without_Euclid's_lemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness_without_Euclid's_lemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Uniqueness without Euclid's lemma</span> </div> </a> <ul id="toc-Uniqueness_without_Euclid's_lemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fundamental theorem of arithmetic</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 66 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 languages</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 – Alemannic" lang="gsw" hreflang="gsw" data-title="Fundamentalsatz der Arithmetik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" 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 – Arabic" lang="ar" hreflang="ar" data-title="المبرهنة الأساسية في الحسابيات" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Asturian" lang="ast" hreflang="ast" data-title="Teorema fundamental de l'aritmética" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</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 – Bangla" lang="bn" hreflang="bn" data-title="পাটিগণিতের মৌলিক উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Bulgarian" lang="bg" hreflang="bg" data-title="Основна теорема на аритметиката" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</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 – Catalan" lang="ca" hreflang="ca" data-title="Teorema fonamental de l'aritmètica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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 – Chuvash" lang="cv" hreflang="cv" data-title="Арифметикăн тĕп теореми" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</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 – Czech" lang="cs" hreflang="cs" data-title="Základní věta aritmetiky" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</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 – Danish" lang="da" hreflang="da" data-title="Aritmetikkens fundamentalsætning" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Primfaktorzerlegung#Fundamentalsatz_der_Arithmetik" title="Primfaktorzerlegung – German" lang="de" hreflang="de" data-title="Primfaktorzerlegung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 – Greek" lang="el" hreflang="el" data-title="Θεμελιώδες θεώρημα αριθμητικής" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_fundamental_de_la_aritm%C3%A9tica" title="Teorema fundamental de la aritmética – Spanish" lang="es" hreflang="es" data-title="Teorema fundamental de la aritmética" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aritmetikaren_oinarrizko_teorema" title="Aritmetikaren oinarrizko teorema – Basque" lang="eu" hreflang="eu" data-title="Aritmetikaren oinarrizko teorema" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="قضیه اساسی حساب" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_fondamental_de_l%27arithm%C3%A9tique" title="Théorème fondamental de l'arithmétique – French" lang="fr" hreflang="fr" data-title="Théorème fondamental de l'arithmétique" data-language-autonym="Français" data-language-local-name="French" 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 – Irish" lang="ga" hreflang="ga" data-title="Bunteoirim na huimhríochta" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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 – Galician" lang="gl" hreflang="gl" data-title="Teorema fundamental da aritmética" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%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 – Korean" lang="ko" hreflang="ko" data-title="산술의 기본 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%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 – Armenian" lang="hy" hreflang="hy" data-title="Թվաբանության հիմնական թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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 – Croatian" lang="hr" hreflang="hr" data-title="Osnovni teorem aritmetike" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</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 – Indonesian" lang="id" hreflang="id" data-title="Teorema dasar aritmetika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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 – Icelandic" lang="is" hreflang="is" data-title="Undirstöðusetning reikningslistarinnar" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica – Italian" lang="it" hreflang="it" data-title="Teorema fondamentale dell'aritmetica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="המשפט היסודי של האריתמטיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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-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 – Georgian" lang="ka" hreflang="ka" data-title="არითმეტიკის ფუნდამენტური თეორემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</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 – Kyrgyz" lang="ky" hreflang="ky" data-title="Арифметиканын негизги теориясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" 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 – Latin" lang="la" hreflang="la" data-title="Theorema fundamentale arithmeticae" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</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 – Latvian" lang="lv" hreflang="lv" data-title="Aritmētikas pamatteorēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</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 – Lithuanian" lang="lt" hreflang="lt" data-title="Pagrindinė aritmetikos teorema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</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 – Lombard" lang="lmo" hreflang="lmo" data-title="Teorema fondamental de l'aritmetica" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</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 – Hungarian" lang="hu" hreflang="hu" data-title="A számelmélet alaptétele" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Основна теорема на аритметиката" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Teorem asas aritmetik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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 – Burmese" lang="my" hreflang="my" data-title="ဂဏန်းသင်္ချာ၏ အခြေခံသီအိုရမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" 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 – Dutch" lang="nl" hreflang="nl" data-title="Hoofdstelling van de rekenkunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%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 – Japanese" lang="ja" hreflang="ja" data-title="算術の基本定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Aritmetikkens_fundamentalteorem" title="Aritmetikkens fundamentalteorem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Aritmetikkens fundamentalteorem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema fondamental dl'aritmética" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</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 – Polish" lang="pl" hreflang="pl" data-title="Zasadnicze twierdzenie arytmetyki" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</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 – Portuguese" lang="pt" hreflang="pt" data-title="Teorema fundamental da aritmética" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Teorema fundamentală a aritmeticii" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><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 – Russian" lang="ru" hreflang="ru" data-title="Основная теорема арифметики" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</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 – Albanian" lang="sq" hreflang="sq" data-title="Teorema themelore e aritmetikës" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</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 – Sicilian" lang="scn" hreflang="scn" data-title="Tiurèma funnamintàli di l'arittimètica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" 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 – Sinhala" lang="si" hreflang="si" data-title="අංක ගණිතයේ මූලික ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" 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 – Slovak" lang="sk" hreflang="sk" data-title="Základná veta aritmetiky" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Osnovni izrek aritmetike" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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 – Serbian" lang="sr" hreflang="sr" data-title="Основна теорема аритметике" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aritmetiikan_peruslause" title="Aritmetiikan peruslause – Finnish" lang="fi" hreflang="fi" data-title="Aritmetiikan peruslause" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Aritmetikens_fundamentalsats" title="Aritmetikens fundamentalsats – Swedish" lang="sv" hreflang="sv" data-title="Aritmetikens fundamentalsats" data-language-autonym="Svenska" data-language-local-name="Swedish" 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 – Thai" lang="th" hreflang="th" data-title="ทฤษฎีบทมูลฐานของเลขคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" 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 – Turkish" lang="tr" hreflang="tr" data-title="Aritmetiğin temel teoremi" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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 – Ukrainian" lang="uk" hreflang="uk" data-title="Основна теорема арифметики" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Integers have unique prime factorizations</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">Fundamental theorem of algebra</a> or <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Disqvisitiones-800.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/220px-Disqvisitiones-800.jpg" decoding="async" width="220" height="368" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/330px-Disqvisitiones-800.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/440px-Disqvisitiones-800.jpg 2x" data-file-width="478" data-file-height="800" /></a><figcaption>In <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i> (1801) <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> proved the unique factorization theorem <sup id="cite_ref-Gauss1801.loc=16_1-0" class="reference"><a href="#cite_note-Gauss1801.loc=16-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and used it to prove the <a href="/wiki/Law_of_quadratic_reciprocity" class="mw-redirect" title="Law of quadratic reciprocity">law of quadratic reciprocity</a>.<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></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>fundamental theorem of arithmetic</b>, also called the <b>unique factorization theorem</b> and <b>prime factorization theorem</b>, states that every <a href="/wiki/Integer" title="Integer">integer</a> greater than 1 can be represented uniquely as a product of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, <a href="/wiki/Up_to" title="Up to">up to</a> the order of the factors.<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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, </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 1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>⋅</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce70b65d42e425d1d3811fcce45bbab9b001d12e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.817ex; height:3.176ex;" alt="{\displaystyle 1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }"></span></dd></dl> <p>The theorem says two things about this example: first, that 1200 <em>can</em> be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. </p><p>The requirement that the factors be prime is necessary: factorizations containing <a href="/wiki/Composite_number" title="Composite number">composite numbers</a> may not be unique (for example, <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 12=2\cdot 6=3\cdot 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>6</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12=2\cdot 6=3\cdot 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1609052c6ed1ed98f9c149940f9679e31668f2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.53ex; height:2.176ex;" alt="{\displaystyle 12=2\cdot 6=3\cdot 4}"></span>). </p><p>This theorem is one of the main <a href="/wiki/Prime_number#Primality_of_one" title="Prime number">reasons why 1 is not considered a prime number</a>: if 1 were prime, then factorization into primes would not be unique; for example, <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\cdot 1=2\cdot 1\cdot 1=\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e235337dcfc0f8ac7cfec7a5e719a26149cd84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.031ex; height:2.176ex;" alt="{\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots }"></span> </p><p>The theorem generalizes to other <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> that are called <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a> and include <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a>, <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domains</a>, and <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. However, the theorem does not hold for <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This failure of unique factorization is one of the reasons for the difficulty of the proof of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat's</a> statement and <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i>. </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"> <p>If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. </p> <div class="templatequotecite">— <cite>Euclid, <a href="#CITEREFEuclidHeath1956">Elements Book VII</a>, Proposition 30</cite></div></blockquote> <p>(In modern terminology: if a prime <i>p</i> divides the product <i>ab</i>, then <i>p</i> divides either <i>a</i> or <i>b</i> or both.) Proposition 30 is referred to as <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a>, and it is the key in the proof of the fundamental theorem of arithmetic. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"> <p>Any composite number is measured by some prime number. </p> <div class="templatequotecite">— <cite>Euclid, <a href="#CITEREFEuclidHeath1956">Elements Book VII</a>, Proposition 31</cite></div></blockquote> <p>(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by <a href="/wiki/Proof_by_infinite_descent" title="Proof by infinite descent">infinite descent</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"> <p>Any number either is prime or is measured by some prime number. </p> <div class="templatequotecite">— <cite>Euclid, <a href="#CITEREFEuclidHeath1956">Elements Book VII</a>, Proposition 32</cite></div></blockquote> <p>Proposition 32 is derived from proposition 31, and proves that the decomposition is possible. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"> <p>If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it. </p> <div class="templatequotecite">— <cite>Euclid, <a href="#CITEREFEuclidHeath1956">Elements Book IX</a>, Proposition 14</cite></div></blockquote> <p>(In modern terminology: a <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. </p><p>While <a href="/wiki/Euclid" title="Euclid">Euclid</a> took the first step on the way to the existence of prime factorization, <a href="/wiki/Kam%C4%81l_al-D%C4%ABn_al-F%C4%81ris%C4%AB" title="Kamāl al-Dīn al-Fārisī">Kamāl al-Dīn al-Fārisī</a> took the final step<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and stated for the first time the fundamental theorem of arithmetic.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Article 16 of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>'s <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i> is an early modern statement and proof employing <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>.<sup id="cite_ref-Gauss1801.loc=16_1-1" class="reference"><a href="#cite_note-Gauss1801.loc=16-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Canonical_representation_of_a_positive_integer">Canonical representation of a positive integer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=3" title="Edit section: Canonical representation of a positive integer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every positive integer <span class="texhtml"><i>n</i> > 1</span> can be represented in exactly one way as a product of prime powers </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f0007be7ec06521f09c3ef78a33d558c7220ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.792ex; height:7.343ex;" alt="{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}},}"></span></dd></dl> <p>where <span class="texhtml"><i>p</i><sub>1</sub> < <i>p</i><sub>2</sub> < ... < <i>p</i><sub>k</sub></span> are primes and the <span class="texhtml"><i>n</i><sub><i>i</i></sub></span> are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the <a href="/wiki/Empty_product" title="Empty product">empty product</a> is equal to 1 (the empty product corresponds to <span class="texhtml"><i>k</i> = 0</span>). </p><p>This representation is called the <b>canonical representation</b><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> of <span class="texhtml"><i>n</i></span>, or the <b>standard form</b><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> of <i>n</i>. For example, </p> <dl><dd>999 = 3<sup>3</sup>×37,</dd> <dd>1000 = 2<sup>3</sup>×5<sup>3</sup>,</dd> <dd>1001 = 7×11×13.</dd></dl> <p>Factors <span class="texhtml"><i>p</i><sup>0</sup> = 1</span> may be inserted without changing the value of <span class="texhtml"><i>n</i></span> (for example, <span class="texhtml">1000 = 2<sup>3</sup>×3<sup>0</sup>×5<sup>3</sup></span>). In fact, any positive integer can be uniquely represented as an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> taken over all the positive prime numbers, as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96acf1f2e6f71cbd4675c76219fd7a519d03b64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.568ex; height:6.843ex;" alt="{\displaystyle n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},}"></span></dd></dl> <p>where a finite number of the <span class="texhtml"><i>n</i><sub><i>i</i></sub></span> are positive integers, and the others are zero. </p><p>Allowing negative exponents provides a canonical form for positive <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_operations">Arithmetic operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=4" title="Edit section: Arithmetic operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The canonical representations of the product, <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> (GCD), and <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> (LCM) of two numbers <i>a</i> and <i>b</i> can be expressed simply in terms of the canonical representations of <i>a</i> and <i>b</i> themselves: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∏</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋯</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∏</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋯</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∏</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81795d1887cf8a541bef2d06543998a828565a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:69.483ex; height:12.176ex;" alt="{\displaystyle {\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}}"></span></dd></dl> <p>However, <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>, especially of large numbers, is much more difficult than computing products, GCDs, or LCMs. So these formulas have limited use in practice. </p> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_functions">Arithmetic functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=5" title="Edit section: Arithmetic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic function</a></div> <p>Many arithmetic functions are defined using the canonical representation. In particular, the values of <a href="/wiki/Additive_function" title="Additive function">additive</a> and <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative</a> functions are determined by their values on the powers of prime numbers. </p> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=6" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The proof uses <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a> (<i>Elements</i> VII, 30): If a prime <a href="/wiki/Divisor" title="Divisor">divides</a> the product of two integers, then it must divide at least one of these integers. </p> <div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=7" title="Edit section: Existence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It must be shown that every integer greater than <span class="texhtml">1</span> is either prime or a product of primes. First, <span class="texhtml">2</span> is prime. Then, by <a href="/wiki/Strong_induction" class="mw-redirect" title="Strong induction">strong induction</a>, assume this is true for all numbers greater than <span class="texhtml">1</span> and less than <span class="texhtml"><i>n</i></span>. If <span class="texhtml"><i>n</i></span> is prime, there is nothing more to prove. Otherwise, there are integers <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span>, where <span class="texhtml"><i>n</i> = <i>a b</i></span>, and <span class="texhtml">1 < <i>a</i> ≤ <i>b</i> < <i>n</i></span>. By the induction hypothesis, <span class="texhtml"><i>a</i> = <i>p</i><sub>1</sub> <i>p</i><sub>2</sub> ⋅⋅⋅ <i>p</i><sub><i>j</i></sub></span> and <span class="texhtml"><i>b</i> = <i>q</i><sub>1</sub> <i>q</i><sub>2</sub> ⋅⋅⋅ <i>q</i><sub><i>k</i></sub></span> are products of primes. But then <span class="texhtml"><i>n</i> = <i>a b</i> = <i>p</i><sub>1</sub> <i>p</i><sub>2</sub> ⋅⋅⋅ <i>p</i><sub><i>j</i></sub> <i>q</i><sub>1</sub> <i>q</i><sub>2</sub> ⋅⋅⋅ <i>q</i><sub><i>k</i></sub></span> is a product of primes. </p> <div class="mw-heading mw-heading3"><h3 id="Uniqueness">Uniqueness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=8" title="Edit section: Uniqueness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let <span class="texhtml"><i>n</i></span> be the least such integer and write <span class="texhtml"><i>n</i> = <i>p</i><sub>1</sub> <i>p</i><sub>2</sub> ... <i>p</i><sub><i>j</i></sub> = <i>q</i><sub>1</sub> <i>q</i><sub>2</sub> ... <i>q</i><sub><i>k</i></sub></span>, where each <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> and <span class="texhtml"><i>q</i><sub><i>i</i></sub></span> is prime. We see that <span class="texhtml"><i>p</i><sub>1</sub></span> divides <span class="texhtml"><i>q</i><sub>1</sub> <i>q</i><sub>2</sub> ... <i>q</i><sub><i>k</i></sub></span>, so <span class="texhtml"><i>p</i><sub>1</sub></span> divides some <span class="texhtml"><i>q</i><sub><i>i</i></sub></span> by <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a>. Without loss of generality, say <span class="texhtml"><i>p</i><sub>1</sub></span> divides <span class="texhtml"><i>q</i><sub>1</sub></span>. Since <span class="texhtml"><i>p</i><sub>1</sub></span> and <span class="texhtml"><i>q</i><sub>1</sub></span> are both prime, it follows that <span class="texhtml"><i>p</i><sub>1</sub> = <i>q</i><sub>1</sub></span>. Returning to our factorizations of <span class="texhtml"><i>n</i></span>, we may cancel these two factors to conclude that <span class="texhtml"><i>p</i><sub>2</sub> ... <i>p</i><sub><i>j</i></sub> = <i>q</i><sub>2</sub> ... <i>q</i><sub><i>k</i></sub></span>. We now have two distinct prime factorizations of some integer strictly smaller than <span class="texhtml"><i>n</i></span>, which contradicts the minimality of <span class="texhtml"><i>n</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Uniqueness_without_Euclid's_lemma"><span id="Uniqueness_without_Euclid.27s_lemma"></span>Uniqueness without Euclid's lemma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=9" title="Edit section: Uniqueness without Euclid's lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental theorem of arithmetic can also be proved without using Euclid's lemma.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The proof that follows is inspired by Euclid's original version of the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>. </p><p>Assume that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, if it exists, must be a <a href="/wiki/Composite_number" title="Composite number">composite number</a> greater than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. Now, say </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></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> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></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> <mo>⋯</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00df449892db252de084f4d8351c752da1fc3b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.73ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}}"></span></dd></dl> <p>Every <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> must be distinct from every <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> <mo>.</mo> </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/78b6fee1054f3b69eb2d524b0b743dba56c2dd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.594ex; height:2.343ex;" alt="{\displaystyle q_{j}.}"></span> Otherwise, if say <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}=q_{j},}"> <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> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}=q_{j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29687cca9904769bdeb241fefbe3209f3be7db1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:7.751ex; height:2.343ex;" alt="{\displaystyle p_{i}=q_{j},}"></span> then there would exist some positive integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=s/p_{i}=s/q_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=s/p_{i}=s/q_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa98bbe2b3a9fbd785749dbcda0d1583955caf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.458ex; height:3.009ex;" alt="{\displaystyle t=s/p_{i}=s/q_{j}}"></span> that is smaller than <span class="texhtml mvar" style="font-style:italic;">s</span> and has two distinct prime factorizations. One may also suppose that <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> <mo>,</mo> </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/ad5bfe55bbb7f31f49a921d4e34af0638c16ce01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.15ex; height:2.176ex;" alt="{\displaystyle p_{1}<q_{1},}"></span> by exchanging the two factorizations, if needed. </p><p>Setting <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=p_{2}\cdots p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=p_{2}\cdots p_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3eda5760f6a8b76ba56ae16c60e96cfab5f0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.409ex; height:2.509ex;" alt="{\displaystyle P=p_{2}\cdots p_{m}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=q_{2}\cdots q_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <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"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=q_{2}\cdots q_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de389a303c3d64952ebb9393a8a6eca1c62a651e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.428ex; height:2.509ex;" alt="{\displaystyle Q=q_{2}\cdots q_{n},}"></span> one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=p_{1}P=q_{1}Q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=p_{1}P=q_{1}Q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c964c68b35f21dd26e00f80936a252c5543329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.833ex; height:2.509ex;" alt="{\displaystyle s=p_{1}P=q_{1}Q.}"></span> Also, since <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> <mo>,</mo> </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/ad5bfe55bbb7f31f49a921d4e34af0638c16ce01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.15ex; height:2.176ex;" alt="{\displaystyle p_{1}<q_{1},}"></span> one has <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<P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo><</mo> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q<P.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c0450857d7b4ca493fef3756f5bfde43daa3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.329ex; height:2.509ex;" alt="{\displaystyle Q<P.}"></span> It then follows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Q</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> <mi>Q</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>−</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c4d3c60b82ad68418303157a9b1e70f5c3a492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.286ex; height:2.843ex;" alt="{\displaystyle s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.}"></span></dd></dl> <p>As the positive integers less than <span class="texhtml mvar" style="font-style:italic;">s</span> have been supposed to have a unique prime factorization, <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> must occur in the factorization of either <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> or <span class="texhtml mvar" style="font-style:italic;">Q</span>. The latter case is impossible, as <span class="texhtml mvar" style="font-style:italic;">Q</span>, being smaller than <span class="texhtml mvar" style="font-style:italic;">s</span>, must have a unique prime factorization, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> differs from every <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> <mo>.</mo> </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/78b6fee1054f3b69eb2d524b0b743dba56c2dd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.594ex; height:2.343ex;" alt="{\displaystyle q_{j}.}"></span> The former case is also impossible, as, if <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> is a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <mo>,</mo> </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/91fce489ef9cd353bedfe6a297e8f0049b0cb77a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.802ex; height:2.343ex;" alt="{\displaystyle q_{1}-p_{1},}"></span> it must be also a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <mo>,</mo> </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/1e78d05d6875f534076e61e99c4e3a5c161f8fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.009ex;" alt="{\displaystyle q_{1},}"></span> which is impossible as <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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> are distinct primes. </p><p>Therefore, there cannot exist a smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, not factor into any prime. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=10" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Fundamental_theorem_of_arithmetic" title="Special:EditPage/Fundamental theorem of arithmetic">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">January 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The first generalization of the theorem is found in Gauss's second monograph (1832) on <a href="/wiki/Biquadratic_reciprocity" class="mw-redirect" title="Biquadratic reciprocity">biquadratic reciprocity</a>. This paper introduced what is now called the <a href="/wiki/Ring_theory" title="Ring theory">ring</a> of <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>, the set of all <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <i>a</i> + <i>bi</i> where <i>a</i> and <i>b</i> are integers. It is now denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [i].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03900897cdf515bbbc52879377653a871b9efc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.293ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i].}"></span> He showed that this ring has the four units ±1 and ±<i>i</i>, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes (<a href="/wiki/Up_to" title="Up to">up to</a> the order and multiplication by units).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Similarly, in 1844 while working on <a href="/wiki/Cubic_reciprocity" title="Cubic reciprocity">cubic reciprocity</a>, <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a> introduced the ring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\omega ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [\omega ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae955a9a0d0f342fc73aaafe28af604d23267f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\omega ]}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cb7e98520ab230df1687521fca5746e76ea62e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.876ex; height:4.176ex;" alt="{\textstyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}"></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 \omega ^{3}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{3}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb9d9d68cafe260162d4a9d60613ea9b841fb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.761ex; height:2.676ex;" alt="{\displaystyle \omega ^{3}=1}"></span> is a cube <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a>. This is the ring of <a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a>, and he proved it has the six units <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>ω</mi> <mo>,</mo> <mo>±</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ec390e7a53fb29473c21614c7ea0bf731ff743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.601ex; height:3.009ex;" alt="{\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}}"></span> and that it has unique factorization. </p><p>However, it was also discovered that unique factorization does not always hold. An example is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37499ef27234d8a67a65932184280bb17301312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.75ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}"></span>. In this ring one has<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </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 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c2c4ad5136d7aaa9ab490bad474c3b02668784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.863ex; height:3.176ex;" alt="{\displaystyle 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).}"></span></dd></dl> <p>Examples like this caused the notion of "prime" to be modified. In <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828485881660a382d22d27b7ddce858daa2ea760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.783ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"></span> it can be proven that if any of the factors above can be represented as a product, for example, 2 = <i>ab</i>, then one of <i>a</i> or <i>b</i> must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">−5</span></span>) nor (1 − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">−5</span></span>) even though it divides their product 6. In <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> 2 is called <a href="/wiki/Irreducible_element" title="Irreducible element">irreducible</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828485881660a382d22d27b7ddce858daa2ea760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.783ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"></span> (only divisible by itself or a unit) but not <a href="/wiki/Prime_element" title="Prime element">prime</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828485881660a382d22d27b7ddce858daa2ea760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.783ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"></span> (if it divides a product it must divide one of the factors). The mention of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828485881660a382d22d27b7ddce858daa2ea760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.783ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}"></span> is required because 2 is prime and irreducible in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}"></span> Using these definitions it can be proven that in any <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> every irreducible is prime". This is also true in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i]}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\omega ],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [\omega ],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0645ae396d9c6f03fd33508a10a838bd30741db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.937ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\omega ],}"></span> but not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {-5}}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [{\sqrt {-5}}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61cb2c7c704f5007aa548b630b3a05fa032f621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.397ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {-5}}].}"></span> </p><p>The rings in which factorization into irreducibles is essentially unique are called <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a>. Important examples are <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> over the integers or over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domains</a> and <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a>. </p><p>In 1843 <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer</a> introduced the concept of <a href="/wiki/Ideal_number" title="Ideal number">ideal number</a>, which was developed further by <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a> (1876) into the modern theory of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a>. </p><p>There is a version of <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">unique factorization for ordinals</a>, though it requires some additional conditions to ensure uniqueness. </p><p>Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative Möbius monoids. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a> – Decomposition of a number into a product</li> <li><a href="/wiki/Prime_signature" title="Prime signature">Prime signature</a> – Multiset of prime exponents in a prime factorization</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Gauss1801.loc=16-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gauss1801.loc=16_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gauss1801.loc=16_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGauss1986">Gauss (1986</a>, Art. 16)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFGauss1986">Gauss (1986</a>, Art. 131)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFLong1972">Long (1972</a>, p. 44)</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 53)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardyWright2008">Hardy & Wright (2008</a>, Thm 2)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">In a <a href="/wiki/Ring_of_algebraic_integers" class="mw-redirect" title="Ring of algebraic integers">ring of algebraic integers</a>, the factorization into prime elements may be non unique, but one can recover a unique factorization if one factors into <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil2007">Weil (2007</a>, p. 5): "Even in Euclid, we fail to find a general statement about the uniqueness of the factorization of an integer into primes; surely he may have been aware of it, but all he has is a statement (Eucl.IX.I4) about the l.c.m. of any number of given primes."</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFA._Goksel_Agargun_and_E._Mehmet_Özkan" class="citation journal cs1">A. Goksel Agargun and E. Mehmet Özkan. <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/82721726.pdf">"A Historical Survey of the Fundamental Theorem of Arithmetic"</a> <span class="cs1-format">(PDF)</span>. <i>Historia Mathematica</i>: 209. <q>One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first proposition.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=A+Historical+Survey+of+the+Fundamental+Theorem+of+Arithmetic&rft.pages=209&rft.au=A.+Goksel+Agargun+and+E.+Mehmet+%C3%96zkan&rft_id=https%3A%2F%2Fcore.ac.uk%2Fdownload%2Fpdf%2F82721726.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRashed2002" class="citation book cs1">Rashed, Roshdi (2002-09-11). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7veIAgAAQBAJ&q=fundamental+theorem+of+arithmetic+discovered+al-farisi&pg=PA385"><i>Encyclopedia of the History of Arabic Science</i></a>. Routledge. p. 385. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781134977246" title="Special:BookSources/9781134977246"><bdi>9781134977246</bdi></a>. <q>The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first time the fundamental theorem of arithmetic.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+the+History+of+Arabic+Science&rft.pages=385&rft.pub=Routledge&rft.date=2002-09-11&rft.isbn=9781134977246&rft.aulast=Rashed&rft.aufirst=Roshdi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7veIAgAAQBAJ%26q%3Dfundamental%2Btheorem%2Bof%2Barithmetic%2Bdiscovered%2Bal-farisi%26pg%3DPA385&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFLong1972">Long (1972</a>, p. 45)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 55)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardyWright2008">Hardy & Wright (2008</a>, § 1.2)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDawson2015" class="citation cs2">Dawson, John W. (2015), <i>Why Prove it Again? Alternative Proofs in Mathematical Practice.</i>, Springer, p. 45, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783319173689" title="Special:BookSources/9783319173689"><bdi>9783319173689</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Why+Prove+it+Again%3F+Alternative+Proofs+in+Mathematical+Practice.&rft.pages=45&rft.pub=Springer&rft.date=2015&rft.isbn=9783319173689&rft.aulast=Dawson&rft.aufirst=John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFGauss1832">Gauss, BQ, §§ 31–34</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardyWright2008">Hardy & Wright (2008</a>, § 14.6)</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i> has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1986" class="citation cs2">Gauss, Carl Friedrich (1986), <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/algebra/book/978-0-387-96254-2"><i>Disquisitiones Arithemeticae (Second, corrected edition)</i></a>, translated by Clarke, Arthur A., New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96254-2" title="Special:BookSources/978-0-387-96254-2"><bdi>978-0-387-96254-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Disquisitiones+Arithemeticae+%28Second%2C+corrected+edition%29&rft.place=New+York&rft.pub=Springer&rft.date=1986&rft.isbn=978-0-387-96254-2&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Falgebra%2Fbook%2F978-0-387-96254-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1965" class="citation cs2 cs1-prop-foreign-lang-source">Gauss, Carl Friedrich (1965), <i>Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)</i> (in German), translated by Maser, H., New York: Chelsea, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8284-0191-8" title="Special:BookSources/0-8284-0191-8"><bdi>0-8284-0191-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Untersuchungen+%C3%BCber+hohere+Arithmetik+%28Disquisitiones+Arithemeticae+%26+other+papers+on+number+theory%29+%28Second+edition%29&rft.place=New+York&rft.pub=Chelsea&rft.date=1965&rft.isbn=0-8284-0191-8&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li></ul> <p>The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § <i>n</i>". Footnotes referencing the <i>Disquisitiones Arithmeticae</i> are of the form "Gauss, DA, Art. <i>n</i>". </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1828" class="citation cs2">Gauss, Carl Friedrich (1828), <i>Theoria residuorum biquadraticorum, Commentatio prima</i>, Göttingen: Comment. Soc. regiae sci, Göttingen 6</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoria+residuorum+biquadraticorum%2C+Commentatio+prima&rft.place=G%C3%B6ttingen&rft.pub=Comment.+Soc.+regiae+sci%2C+G%C3%B6ttingen+6&rft.date=1828&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1832" class="citation cs2">Gauss, Carl Friedrich (1832), <i>Theoria residuorum biquadraticorum, Commentatio secunda</i>, Göttingen: Comment. Soc. regiae sci, Göttingen 7</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoria+residuorum+biquadraticorum%2C+Commentatio+secunda&rft.place=G%C3%B6ttingen&rft.pub=Comment.+Soc.+regiae+sci%2C+G%C3%B6ttingen+7&rft.date=1832&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li></ul> <p>These are in Gauss's <i>Werke</i>, Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the <i>Disquisitiones</i>. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuclidHeath1956" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Euclid" title="Euclid">Euclid</a> (1956), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbooksofe00eucl"><i>The thirteen books of the Elements</i></a></span>, vol. 2 (Books III-IX), Translated by <a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Thomas Little Heath</a> (Second Edition Unabridged ed.), New York: <a href="/wiki/Dover" title="Dover">Dover</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60089-5" title="Special:BookSources/978-0-486-60089-5"><bdi>978-0-486-60089-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+thirteen+books+of+the+Elements&rft.place=New+York&rft.edition=Second+Edition+Unabridged&rft.pub=Dover&rft.date=1956&rft.isbn=978-0-486-60089-5&rft.au=Euclid&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthirteenbooksofe00eucl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright2008" class="citation book cs2"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (2008) [1938], <i>An Introduction to the Theory of Numbers</i>, Revised by <a href="/wiki/Roger_Heath-Brown" title="Roger Heath-Brown">D. R. Heath-Brown</a> and <a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">J. H. Silverman</a>. Foreword by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>. (6th ed.), Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921986-5" title="Special:BookSources/978-0-19-921986-5"><bdi>978-0-19-921986-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243">2445243</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1159.11001">1159.11001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.place=Oxford&rft.edition=6th&rft.pub=Oxford+University+Press&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1159.11001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2445243%23id-name%3DMR&rft.isbn=978-0-19-921986-5&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLong1972" class="citation cs2">Long, Calvin T. (1972), <i>Elementary Introduction to Number Theory</i> (2nd ed.), Lexington: <a href="/wiki/D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77-171950">77-171950</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Introduction+to+Number+Theory&rft.place=Lexington&rft.edition=2nd&rft.pub=D.+C.+Heath+and+Company&rft.date=1972&rft_id=info%3Alccn%2F77-171950&rft.aulast=Long&rft.aufirst=Calvin+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPettofrezzoByrkit1970" class="citation cs2">Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), <i>Elements of Number Theory</i>, Englewood Cliffs: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77-81766">77-81766</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Number+Theory&rft.place=Englewood+Cliffs&rft.pub=Prentice+Hall&rft.date=1970&rft_id=info%3Alccn%2F77-81766&rft.aulast=Pettofrezzo&rft.aufirst=Anthony+J.&rft.au=Byrkit%2C+Donald+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiesel1994" class="citation cs2">Riesel, Hans (1994), <i>Prime Numbers and Computer Methods for Factorization (second edition)</i>, Boston: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3743-5" title="Special:BookSources/0-8176-3743-5"><bdi>0-8176-3743-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers+and+Computer+Methods+for+Factorization+%28second+edition%29&rft.place=Boston&rft.pub=Birkh%C3%A4user&rft.date=1994&rft.isbn=0-8176-3743-5&rft.aulast=Riesel&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeil2007" class="citation cs2">Weil, André (2007) [1984], <i><a href="/wiki/Number_Theory:_An_Approach_through_History_from_Hammurapi_to_Legendre" class="mw-redirect" title="Number Theory: An Approach through History from Hammurapi to Legendre">Number Theory: An Approach through History from Hammurapi to Legendre</a></i>, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-817-64565-6" title="Special:BookSources/978-0-817-64565-6"><bdi>978-0-817-64565-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory%3A+An+Approach+through+History+from+Hammurapi+to+Legendre&rft.place=Boston%2C+MA&rft.series=Modern+Birkh%C3%A4user+Classics&rft.pub=Birkh%C3%A4user&rft.date=2007&rft.isbn=978-0-817-64565-6&rft.aulast=Weil&rft.aufirst=Andr%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fundamental_theorem_of_arithmetic&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://gowers.wordpress.com/2011/11/13/why-isnt-the-fundamental-theorem-of-arithmetic-obvious">Why isn’t the fundamental theorem of arithmetic obvious?</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/blue/gcd_fta.shtml">GCD and the Fundamental Theorem of Arithmetic</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a>.</li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/fundamentaltheoremofarithmeticproofofthe">PlanetMath: Proof of fundamental theorem of arithmetic</a></li> <li><a rel="nofollow" class="external text" href="http://fermatslasttheorem.blogspot.com/2005/06/unique-factorization.html">Fermat's Last Theorem Blog: Unique Factorization</a>, a blog that covers the history of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> from <a href="/wiki/Diophantus_of_Alexandria" class="mw-redirect" title="Diophantus of Alexandria">Diophantus of Alexandria</a> to the proof by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>.</li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/FundamentalTheoremOfArithmetic/">"Fundamental Theorem of Arithmetic"</a> by Hector Zenil, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrime" class="citation cs2">Grime, James, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=IQofiPqhJ_s">"1 and Prime Numbers"</a>, <i>Numberphile</i>, <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/IQofiPqhJ_s">archived</a> from the original on 2021-12-11</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numberphile&rft.atitle=1+and+Prime+Numbers&rft.aulast=Grime&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DIQofiPqhJ_s&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFundamental+theorem+of+arithmetic" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist 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]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Divisor_classes" title="Template:Divisor classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Divisor_classes" title="Template talk:Divisor classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Divisor_classes" title="Special:EditPage/Template:Divisor classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Divisibility-based_sets_of_integers" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function</a></li> <li><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li> <li><a class="mw-selflink selflink">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prime_number" title="Prime number">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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