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class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Uvod</div> </a> </li> <li id="toc-Računanje_Eulerjeve_funkcije_φ(n)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Računanje_Eulerjeve_funkcije_φ(n)"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Računanje Eulerjeve funkcije φ(<i>n</i>)</span> </div> </a> <button aria-controls="toc-Računanje_Eulerjeve_funkcije_φ(n)-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>Vklopi podrazdelek Računanje Eulerjeve funkcije φ(<i>n</i>)</span> </button> <ul id="toc-Računanje_Eulerjeve_funkcije_φ(n)-sublist" class="vector-toc-list"> <li id="toc-Praštevila" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Praštevila"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Praštevila</span> </div> </a> <ul id="toc-Praštevila-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Potence_praštevil" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Potence_praštevil"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Potence praštevil</span> </div> </a> <ul id="toc-Potence_praštevil-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tuja_števila" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tuja_števila"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Tuja števila</span> </div> </a> <ul id="toc-Tuja_števila-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Druge_značilnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Druge_značilnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Druge značilnosti</span> </div> </a> <ul id="toc-Druge_značilnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rodovna_funkcija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rodovna_funkcija"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rodovna funkcija</span> </div> </a> <ul id="toc-Rodovna_funkcija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obnašanje_funkcije" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Obnašanje_funkcije"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Obnašanje funkcije</span> </div> </a> <ul id="toc-Obnašanje_funkcije-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nekatere_vrednosti_funkcije" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nekatere_vrednosti_funkcije"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Nekatere vrednosti funkcije</span> </div> </a> <ul id="toc-Nekatere_vrednosti_funkcije-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zunanje_povezave" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zunanje_povezave"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Zunanje povezave</span> </div> </a> <ul id="toc-Zunanje_povezave-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="Vsebina" 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="Vklopi kazalo vsebine" > <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">Vklopi kazalo vsebine</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">Eulerjeva funkcija fi</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="P9jdi na članek v drugem jeziku. Na voljo v 41 jezikih." > <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-41" 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">41 jezikov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A4%D8%B4%D8%B1_%D8%A3%D9%88%D9%8A%D9%84%D8%B1" title="مؤشر أويلر – arabščina" lang="ar" hreflang="ar" data-title="مؤشر أويلر" data-language-autonym="العربية" data-language-local-name="arabščina" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Функция на Ойлер – bolgarščina" lang="bg" hreflang="bg" data-title="Функция на Ойлер" data-language-autonym="Български" data-language-local-name="bolgarščina" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0_%E0%A6%9F%E0%A7%8B%E0%A6%B6%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%9F_%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8" title="অয়লার টোশেন্ট ফাংশন – bengalščina" lang="bn" hreflang="bn" data-title="অয়লার টোশেন্ট ফাংশন" data-language-autonym="বাংলা" data-language-local-name="bengalščina" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_%CF%86_d%27Euler" title="Funció φ d'Euler – katalonščina" lang="ca" hreflang="ca" data-title="Funció φ d'Euler" data-language-autonym="Català" data-language-local-name="katalonščina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eulerova_funkce" title="Eulerova funkce – češčina" lang="cs" hreflang="cs" data-title="Eulerova funkce" data-language-autonym="Čeština" data-language-local-name="češčina" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiant_%CF%86_Euler" title="Ffwythiant φ Euler – valižanščina" lang="cy" hreflang="cy" data-title="Ffwythiant φ Euler" data-language-autonym="Cymraeg" data-language-local-name="valižanščina" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Eulers_totientfunktion" title="Eulers totientfunktion – danščina" lang="da" hreflang="da" data-title="Eulers totientfunktion" data-language-autonym="Dansk" data-language-local-name="danščina" 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/Eulersche_Phi-Funktion" title="Eulersche Phi-Funktion – nemščina" lang="de" hreflang="de" data-title="Eulersche Phi-Funktion" data-language-autonym="Deutsch" data-language-local-name="nemščina" 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%A3%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Συνάρτηση Όιλερ – grščina" lang="el" hreflang="el" data-title="Συνάρτηση Όιλερ" data-language-autonym="Ελληνικά" data-language-local-name="grščina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function" title="Euler's totient function – angleščina" lang="en" hreflang="en" data-title="Euler's totient function" data-language-autonym="English" data-language-local-name="angleščina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Funkcio_%CF%86" title="Funkcio φ – esperanto" lang="eo" hreflang="eo" data-title="Funkcio φ" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_%CF%86_de_Euler" title="Función φ de Euler – španščina" lang="es" hreflang="es" data-title="Función φ de Euler" data-language-autonym="Español" data-language-local-name="španščina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eulerren_%CF%86_funtzioa" title="Eulerren φ funtzioa – baskovščina" lang="eu" hreflang="eu" data-title="Eulerren φ funtzioa" data-language-autonym="Euskara" data-language-local-name="baskovščina" 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/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%81%DB%8C_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="تابع فی اویلر – perzijščina" lang="fa" hreflang="fa" data-title="تابع فی اویلر" data-language-autonym="فارسی" data-language-local-name="perzijščina" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Eulerin_%CF%86-funktio" title="Eulerin φ-funktio – finščina" lang="fi" hreflang="fi" data-title="Eulerin φ-funktio" data-language-autonym="Suomi" data-language-local-name="finščina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Indicatrice_d%27Euler" title="Indicatrice d'Euler – francoščina" lang="fr" hreflang="fr" data-title="Indicatrice d'Euler" data-language-autonym="Français" data-language-local-name="francoščina" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_totiente_de_Euler" title="Función totiente de Euler – galicijščina" lang="gl" hreflang="gl" data-title="Función totiente de Euler" data-language-autonym="Galego" data-language-local-name="galicijščina" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8" title="פונקציית אוילר – hebrejščina" lang="he" hreflang="he" data-title="פונקציית אוילר" data-language-autonym="עברית" data-language-local-name="hebrejščina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Eulerova_funkcija" title="Eulerova funkcija – hrvaščina" lang="hr" hreflang="hr" data-title="Eulerova funkcija" data-language-autonym="Hrvatski" data-language-local-name="hrvaščina" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Fonksyon_phi_Euler" title="Fonksyon phi Euler – haitijska kreolščina" lang="ht" hreflang="ht" data-title="Fonksyon phi Euler" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitijska kreolščina" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-f%C3%BCggv%C3%A9ny" title="Euler-függvény – madžarščina" lang="hu" hreflang="hu" data-title="Euler-függvény" data-language-autonym="Magyar" data-language-local-name="madžarščina" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_phi_Euler" title="Fungsi phi Euler – indonezijščina" lang="id" hreflang="id" data-title="Fungsi phi Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijščina" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero – italijanščina" lang="it" hreflang="it" data-title="Funzione φ di Eulero" data-language-autonym="Italiano" data-language-local-name="italijanščina" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%CF%86%E9%96%A2%E6%95%B0" title="オイラーのφ関数 – japonščina" lang="ja" hreflang="ja" data-title="オイラーのφ関数" data-language-autonym="日本語" data-language-local-name="japonščina" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D1%81%D1%8B" title="Эйлер функциясы – kazaščina" lang="kk" hreflang="kk" data-title="Эйлер функциясы" data-language-autonym="Қазақша" data-language-local-name="kazaščina" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%94%BC_%ED%95%A8%EC%88%98" title="오일러 피 함수 – korejščina" lang="ko" hreflang="ko" data-title="오일러 피 함수" data-language-autonym="한국어" data-language-local-name="korejščina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%93%E0%B4%AF%E0%B5%8D%E0%B4%B2%E0%B4%B1%E0%B5%81%E0%B4%9F%E0%B5%86_%E0%B4%9F%E0%B5%8B%E0%B4%B7%E0%B5%8D%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D_%E0%B4%AB%E0%B4%B2%E0%B4%A8%E0%B4%82" title="ഓയ്ലറുടെ ടോഷ്യന്റ് ഫലനം – malajalamščina" lang="ml" hreflang="ml" data-title="ഓയ്ലറുടെ ടോഷ്യന്റ് ഫലനം" data-language-autonym="മലയാളം" data-language-local-name="malajalamščina" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indicator_(getaltheorie)" title="Indicator (getaltheorie) – nizozemščina" lang="nl" hreflang="nl" data-title="Indicator (getaltheorie)" data-language-autonym="Nederlands" data-language-local-name="nizozemščina" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eulers_totientfunksjon" title="Eulers totientfunksjon – knjižna norveščina" lang="nb" hreflang="nb" data-title="Eulers totientfunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="knjižna norveščina" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_%CF%86" title="Funkcja φ – poljščina" lang="pl" hreflang="pl" data-title="Funkcja φ" data-language-autonym="Polski" data-language-local-name="poljščina" 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/Fun%C3%A7%C3%A3o_totiente_de_Euler" title="Função totiente de Euler – portugalščina" lang="pt" hreflang="pt" data-title="Função totiente de Euler" data-language-autonym="Português" data-language-local-name="portugalščina" 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/Indicatorul_lui_Euler" title="Indicatorul lui Euler – romunščina" lang="ro" hreflang="ro" data-title="Indicatorul lui Euler" data-language-autonym="Română" data-language-local-name="romunščina" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Функция Эйлера – ruščina" lang="ru" hreflang="ru" data-title="Функция Эйлера" data-language-autonym="Русский" data-language-local-name="ruščina" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euler%27s_totient_function" title="Euler's totient function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euler's totient function" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%84%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Ојлерова фи функција – srbščina" lang="sr" hreflang="sr" data-title="Ојлерова фи функција" data-language-autonym="Српски / srpski" data-language-local-name="srbščina" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link 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id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sl" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Slika:EulerPhi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/250px-EulerPhi.svg.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/375px-EulerPhi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/EulerPhi.svg/500px-EulerPhi.svg.png 2x" data-file-width="731" data-file-height="551" /></a><figcaption>Graf prvih tisoč vrednosti funkcije <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 \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}"></span></figcaption></figure> <p><b>Eulerjeva fúnkcija φ(<i>n</i>)</b> [òjlerjeva ~ fí] je v <a href="/wiki/Teorija_%C5%A1tevil" title="Teorija števil">teoriji števil</a> <a href="/wiki/Multiplikativna_funkcija" title="Multiplikativna funkcija">multiplikativna</a> <a href="/wiki/Aritmeti%C4%8Dna_funkcija" title="Aritmetična funkcija">aritmetična funkcija</a> poljubnega <a href="/wiki/Pozitivno_%C5%A1tevilo" title="Pozitivno število">pozitivnega</a> <a href="/wiki/Celo_%C5%A1tevilo" title="Celo število">celega</a> <a href="/wiki/%C5%A0tevilo" title="Število">števila</a> <i>n</i> in da skupno število pozitivnih celih števil, ki ne presegajo <i>n</i>, in so <i>n</i> <a href="/wiki/Tuje_%C5%A1tevilo" title="Tuje število">tuja</a>. Ali drugače rečeno, ki so manjša od <i>n</i> in so <i>n</i> relativno <a href="/wiki/Pra%C5%A1tevilo" title="Praštevilo">praštevila</a>. Na primer, φ(<a href="/wiki/8_(%C5%A1tevilo)" title="8 (število)">8</a>) = 4, ker so štiri števila, <a href="/wiki/1_(%C5%A1tevilo)" title="1 (število)">1</a>, <a href="/wiki/3_(%C5%A1tevilo)" title="3 (število)">3</a>, <a href="/wiki/5_(%C5%A1tevilo)" title="5 (število)">5</a> in <a href="/wiki/7_(%C5%A1tevilo)" title="7 (število)">7</a> tuja številu 8. <a href="/wiki/Funkcija" class="mw-redirect" title="Funkcija">Funkcijo</a> je uvedel in raziskoval <a href="/wiki/%C5%A0vicarji" title="Švicarji">švicarski</a> <a href="/wiki/Matematik" class="mw-redirect" title="Matematik">matematik</a> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in se imenuje po njem. Funkciji rečejo tudi kar <b>Eulerjeva funkcija</b>. Angleški matematik <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a> je zanjo leta 1882 uvedel ime <i>»totientna funkcija«</i>, kjer je »totient« sestavljenka iz <i>tot</i>alni in kvoc<i>ient</i>. </p><p>Eulerjeva funkcija fi je pomembna predvsem, ker da velikost multiplikativne <a href="/wiki/Grupa_(matematika)" class="mw-redirect" title="Grupa (matematika)">grupe</a> celih števil po <a href="/wiki/Modularna_aritmetika" title="Modularna aritmetika">modulu</a> <i>n</i>, oziroma natančneje, φ(<i>n</i>) je <a href="/wiki/Kardinalno_%C5%A1tevilo" title="Kardinalno število">kardinalno število</a> enotskih grup <a href="/wiki/Kolobar" title="Kolobar">kolobarja</a> <b>Z</b>/<i>n</i><b>Z</b>. To dejstvo skupaj z <a href="/w/index.php?title=Lagrangeev_izrek&action=edit&redlink=1" class="new" title="Lagrangeev izrek (stran ne obstaja)">Lagrangeevim izrekom</a> zagotovi dokaz <a href="/wiki/Eulerjev_izrek" title="Eulerjev izrek">Eulerjevega izreka</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Računanje_Eulerjeve_funkcije_φ(n)"><span id="Ra.C4.8Dunanje_Eulerjeve_funkcije_.CF.86.28n.29"></span>Računanje Eulerjeve funkcije φ(<i>n</i>)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=1" title="Uredi razdelek: Računanje Eulerjeve funkcije φ(n)" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=1" title="Urejanje izvorne kode razdelka: Računanje Eulerjeve funkcije φ(n)"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Praštevila"><span id="Pra.C5.A1tevila"></span>Praštevila</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=2" title="Uredi razdelek: Praštevila" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=2" title="Urejanje izvorne kode razdelka: Praštevila"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Če je <i>n</i> sámo praštevilo <i>p</i>, velja φ(<i>p</i>) = <i>p</i> - 1. </p><p>Primer: φ(19) = 19 - 1 = 18. </p> <div class="mw-heading mw-heading3"><h3 id="Potence_praštevil"><span id="Potence_pra.C5.A1tevil"></span>Potence praštevil</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=3" title="Uredi razdelek: Potence praštevil" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=3" title="Urejanje izvorne kode razdelka: Potence praštevil"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Če je <i>n</i> = <i>p</i><sup><i>m</i></sup> (<i>m</i> ≥ 1) <a href="/wiki/Potenca" class="mw-redirect" title="Potenca">potenca</a> kakega praštevila, velja: </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 \varphi (n)=\varphi (p^{m})=p^{m}-p^{m-1}=p^{m-1}(p-1)\!\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=\varphi (p^{m})=p^{m}-p^{m-1}=p^{m-1}(p-1)\!\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faec439de41fdf31cf7c418f85a2a95dc34ab78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.397ex; height:3.176ex;" alt="{\displaystyle \varphi (n)=\varphi (p^{m})=p^{m}-p^{m-1}=p^{m-1}(p-1)\!\,.}"></span></dd></dl> <p>Primer: φ(16) = φ( 2<sup>4</sup>) = 2<sup>4</sup> - 2<sup>3</sup> = 16 - 8 = 8, oziroma: φ(16) = φ(2<sup>4</sup>) = 2<sup>3</sup> · (2 - 1) = 8 · 1 = 8. </p> <div class="mw-heading mw-heading3"><h3 id="Tuja_števila"><span id="Tuja_.C5.A1tevila"></span>Tuja števila</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=4" title="Uredi razdelek: Tuja števila" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=4" title="Urejanje izvorne kode razdelka: Tuja števila"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Če sta si <i>a</i> in <i>b</i> tuji števili, je funkcija multiplikativna in velja: </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 \varphi (ab)=\varphi (a)\varphi (b)\!\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (ab)=\varphi (a)\varphi (b)\!\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c6850c5a945719e74aa4252af3c5a49bffdd5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.188ex; height:2.843ex;" alt="{\displaystyle \varphi (ab)=\varphi (a)\varphi (b)\!\,.}"></span></dd></dl> <p>Primer: φ(15) = φ(3) · φ(5) = 2 · 4 = 8. </p><p>Obe zgornji značilnosti lahko združimo v: </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 \varphi (n)=n\prod _{p|n}\left(1-{1 \over p}\right)\!\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)=n\prod _{p|n}\left(1-{1 \over p}\right)\!\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb9ea57d0876bdf2055674d4743d48a15375899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.424ex; height:7.176ex;" alt="{\displaystyle \varphi (n)=n\prod _{p|n}\left(1-{1 \over p}\right)\!\,.}"></span></dd></dl> <p>Primer: φ(2004) = φ(2<sup>2</sup> · 3 · 167) = 2004 · (1 - 1/2) · (1 - 1/3) · (1 - 1/167) = 664; </p> <div class="mw-heading mw-heading2"><h2 id="Druge_značilnosti"><span id="Druge_zna.C4.8Dilnosti"></span>Druge značilnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=5" title="Uredi razdelek: Druge značilnosti" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=5" title="Urejanje izvorne kode razdelka: Druge značilnosti"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Rodovna_funkcija">Rodovna funkcija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=6" title="Uredi razdelek: Rodovna funkcija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=6" title="Urejanje izvorne kode razdelka: Rodovna funkcija"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Obnašanje_funkcije"><span id="Obna.C5.A1anje_funkcije"></span>Obnašanje funkcije</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=7" title="Uredi razdelek: Obnašanje funkcije" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=7" title="Urejanje izvorne kode razdelka: Obnašanje funkcije"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Nekatere_vrednosti_funkcije">Nekatere vrednosti funkcije</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=8" title="Uredi razdelek: Nekatere vrednosti funkcije" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=8" title="Urejanje izvorne kode razdelka: Nekatere vrednosti funkcije"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th><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 \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}"></span> </th> <th>+0</th> <th>+1</th> <th>+2</th> <th>+3</th> <th>+4</th> <th>+5</th> <th>+6</th> <th>+7</th> <th>+8</th> <th>+9 </th></tr> <tr> <th>0+ </th> <td> </td> <td>1</td> <td>1</td> <td>2</td> <td>2</td> <td>4</td> <td>2</td> <td>6</td> <td>4</td> <td>6 </td></tr> <tr> <th>10+ </th> <td>4</td> <td>10</td> <td>4</td> <td>12</td> <td>6</td> <td>8</td> <td>8</td> <td>16</td> <td>6</td> <td>18 </td></tr> <tr> <th>20+ </th> <td>8</td> <td>12</td> <td>10</td> <td>22</td> <td>8</td> <td>20</td> <td>12</td> <td>18</td> <td>12</td> <td>28 </td></tr> <tr> <th>30+ </th> <td>8</td> <td>30</td> <td>16</td> <td>20</td> <td>16</td> <td>24</td> <td>12</td> <td>36</td> <td>18</td> <td>24 </td></tr> <tr> <th>40+ </th> <td>16</td> <td>40</td> <td>12</td> <td>42</td> <td>20</td> <td>24</td> <td>22</td> <td>46</td> <td>16</td> <td>42 </td></tr> <tr> <th>50+ </th> <td>20</td> <td>32</td> <td>24</td> <td>52</td> <td>18</td> <td>40</td> <td>24</td> <td>36</td> <td>28</td> <td>58 </td></tr> <tr> <th>60+ </th> <td>16</td> <td>60</td> <td>30</td> <td>36</td> <td>32</td> <td>48</td> <td>20</td> <td>66</td> <td>32</td> <td>44 </td></tr> <tr> <th>70+ </th> <td>24</td> <td>70</td> <td>24</td> <td>72</td> <td>36</td> <td>40</td> <td>36</td> <td>60</td> <td>24</td> <td>78 </td></tr> <tr> <th>80+ </th> <td>32</td> <td>54</td> <td>40</td> <td>82</td> <td>24</td> <td>64</td> <td>42</td> <td>56</td> <td>40</td> <td>88 </td></tr> <tr> <th>90+ </th> <td>24</td> <td>72</td> <td>44</td> <td>60</td> <td>46</td> <td>72</td> <td>32</td> <td>96</td> <td>42</td> <td>60 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Zunanje_povezave">Zunanje povezave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&veaction=edit&section=9" title="Uredi razdelek: Zunanje povezave" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulerjeva_funkcija_fi&action=edit&section=9" title="Urejanje izvorne kode razdelka: Zunanje povezave"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Totient_Function"><style data-mw-deduplicate="TemplateStyles:r5980307">.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 a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .citation .cs1-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}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.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}</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/w/index.php?title=Eric_Wolfgang_Weisstein&action=edit&redlink=1" class="new" title="Eric Wolfgang Weisstein (stran ne obstaja)">Weisstein, Eric Wolfgang</a>. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TotientFunction.html">»Totient Function«</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=neznano&rft.jtitle=MathWorld&rft.atitle=Totient+Function&rft.au=Weisstein%2C+Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTotientFunction.html&rfr_id=info%3Asid%2Fsl.wikipedia.org%3AEulerjeva+funkcija+fi" class="Z3988"></span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r5570670">.mw-parser-output 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