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skin-invert"> <img class="mw-logo-wordmark" alt="Уикипедия" src="/static/images/mobile/copyright/wikipedia-wordmark-bg.svg" style="width: 7.5em; height: 1.3125em;"> <img class="mw-logo-tagline" alt="Свободната енциклопедия" src="/static/images/mobile/copyright/wikipedia-tagline-bg.svg" width="120" height="12" style="width: 7.5em; height: 0.75em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A2%D1%8A%D1%80%D1%81%D0%B5%D0%BD%D0%B5" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Претърсване на Уикипедия [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Търсене</span> </a> <div 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class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Облик"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Облик" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Облик</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_bg.wikipedia.org&uselang=bg" class=""><span>Направете дарение</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A1%D1%8A%D0%B7%D0%B4%D0%B0%D0%B2%D0%B0%D0%BD%D0%B5_%D0%BD%D0%B0_%D1%81%D0%BC%D0%B5%D1%82%D0%BA%D0%B0&returnto=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE+%D1%87%D0%B8%D1%81%D0%BB%D0%BE&returntoquery=oldid%3D11992043" title="Насърчаваме Ви да си създадете сметка и да влезете, въпреки че не е задължително." class=""><span>Създаване на сметка</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A0%D0%B5%D0%B3%D0%B8%D1%81%D1%82%D1%80%D0%B8%D1%80%D0%B0%D0%BD%D0%B5_%D0%B8%D0%BB%D0%B8_%D0%B2%D0%BB%D0%B8%D0%B7%D0%B0%D0%BD%D0%B5&returnto=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE+%D1%87%D0%B8%D1%81%D0%BB%D0%BE&returntoquery=oldid%3D11992043" title="Препоръчваме Ви да влезете, въпреки, че не е задължително. [o]" accesskey="o" class=""><span>Влизане</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Допълнителни опции" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Лични инструменти" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Лични инструменти</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Потребителско меню" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_bg.wikipedia.org&uselang=bg"><span>Направете дарение</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A1%D1%8A%D0%B7%D0%B4%D0%B0%D0%B2%D0%B0%D0%BD%D0%B5_%D0%BD%D0%B0_%D1%81%D0%BC%D0%B5%D1%82%D0%BA%D0%B0&returnto=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE+%D1%87%D0%B8%D1%81%D0%BB%D0%BE&returntoquery=oldid%3D11992043" title="Насърчаваме Ви да си създадете сметка и да влезете, въпреки че не е задължително."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Създаване на сметка</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A0%D0%B5%D0%B3%D0%B8%D1%81%D1%82%D1%80%D0%B8%D1%80%D0%B0%D0%BD%D0%B5_%D0%B8%D0%BB%D0%B8_%D0%B2%D0%BB%D0%B8%D0%B7%D0%B0%D0%BD%D0%B5&returnto=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE+%D1%87%D0%B8%D1%81%D0%BB%D0%BE&returntoquery=oldid%3D11992043" title="Препоръчваме Ви да влезете, въпреки, че не е задължително. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Влизане</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Страници за излезли от системата редактори <a href="/wiki/%D0%9F%D0%BE%D0%BC%D0%BE%D1%89:%D0%92%D1%8A%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" aria-label="Научете повече за редактирането"><span>научете повече</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9C%D0%BE%D0%B8%D1%82%D0%B5_%D0%BF%D1%80%D0%B8%D0%BD%D0%BE%D1%81%D0%B8" title="Списък на промените, направени от този IP адрес [y]" accesskey="y"><span>Приноси</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9C%D0%BE%D1%8F%D1%82%D0%B0_%D0%B1%D0%B5%D1%81%D0%B5%D0%B4%D0%B0" title="Дискусия относно редакциите от този адрес [n]" accesskey="n"><span>Беседа</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Съдържание" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Съдържание</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">преместване към страничната лента</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">скриване</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Начало</div> </a> </li> <li id="toc-От_Евклид_до_Ойлер" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#От_Евклид_до_Ойлер"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>От Евклид до Ойлер</span> </div> </a> <ul id="toc-От_Евклид_до_Ойлер-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Съвременно_състояние_на_проблема" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Съвременно_състояние_на_проблема"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Съвременно състояние на проблема</span> </div> </a> <ul id="toc-Съвременно_състояние_на_проблема-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Първите_десет_съвършени_числа" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Първите_десет_съвършени_числа"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Първите десет съвършени числа</span> </div> </a> <ul id="toc-Първите_десет_съвършени_числа-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Вижте_също" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Вижте_също"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Вижте също</span> </div> </a> <ul id="toc-Вижте_също-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Източници" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Източници"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Източници</span> </div> </a> <ul id="toc-Източници-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Външни_препратки" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Външни_препратки"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Външни препратки</span> </div> </a> <ul id="toc-Външни_препратки-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="Съдържание" 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="Скриване/показване на съдържанието" > <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">Скриване/показване на съдържанието</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">Съвършено число</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="Към статията на друг език. Налична на 64 езика" > <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-64" 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">64 езика</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/%D8%B9%D8%AF%D8%AF_%D8%AA%D8%A7%D9%85" title="عدد تام – арабски" lang="ar" hreflang="ar" data-title="عدد تام" data-language-autonym="العربية" data-language-local-name="арабски" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D1%81%D0%BA%D0%B0%D0%BD%D0%B0%D0%BB%D1%8B_%D0%BB%D1%96%D0%BA" title="Дасканалы лік – беларуски" lang="be" hreflang="be" data-title="Дасканалы лік" data-language-autonym="Беларуская" data-language-local-name="беларуски" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%96%E0%A7%81%E0%A6%81%E0%A6%A4_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="নিখুঁত সংখ্যা – бенгалски" lang="bn" hreflang="bn" data-title="নিখুঁত সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="бенгалски" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_peurvat" title="Niver peurvat – бретонски" lang="br" hreflang="br" data-title="Niver peurvat" data-language-autonym="Brezhoneg" data-language-local-name="бретонски" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_perfecte" title="Nombre perfecte – каталонски" lang="ca" hreflang="ca" data-title="Nombre perfecte" data-language-autonym="Català" data-language-local-name="каталонски" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%A9%D8%A7%D9%85%DA%B5" title="ژمارەی کامڵ – кюрдски (централен)" lang="ckb" hreflang="ckb" data-title="ژمارەی کامڵ" data-language-autonym="کوردی" data-language-local-name="кюрдски (централен)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dokonal%C3%A9_%C4%8D%C3%ADslo" title="Dokonalé číslo – чешки" lang="cs" hreflang="cs" data-title="Dokonalé číslo" data-language-autonym="Čeština" data-language-local-name="чешки" 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/Fuldkomne_tal" title="Fuldkomne tal – датски" lang="da" hreflang="da" data-title="Fuldkomne tal" data-language-autonym="Dansk" data-language-local-name="датски" 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/Vollkommene_Zahl" title="Vollkommene Zahl – немски" lang="de" hreflang="de" data-title="Vollkommene Zahl" data-language-autonym="Deutsch" data-language-local-name="немски" 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%A4%CE%AD%CE%BB%CE%B5%CE%B9%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Τέλειος αριθμός – гръцки" lang="el" hreflang="el" data-title="Τέλειος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="гръцки" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_perf%C3%A8t" title="Nùmer perfèt – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer perfèt" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Perfect_number" title="Perfect number – английски" lang="en" hreflang="en" data-title="Perfect number" data-language-autonym="English" data-language-local-name="английски" 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/Perfekta_nombro" title="Perfekta nombro – есперанто" lang="eo" hreflang="eo" data-title="Perfekta nombro" data-language-autonym="Esperanto" data-language-local-name="есперанто" 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/N%C3%BAmero_perfecto" title="Número perfecto – испански" lang="es" hreflang="es" data-title="Número perfecto" data-language-autonym="Español" data-language-local-name="испански" 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/Zenbaki_perfektu" title="Zenbaki perfektu – баски" lang="eu" hreflang="eu" data-title="Zenbaki perfektu" data-language-autonym="Euskara" data-language-local-name="баски" 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%B9%D8%AF%D8%AF_%D8%AA%D8%A7%D9%85" title="عدد تام – персийски" lang="fa" hreflang="fa" data-title="عدد تام" data-language-autonym="فارسی" data-language-local-name="персийски" 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/T%C3%A4ydellinen_luku" title="Täydellinen luku – фински" lang="fi" hreflang="fi" data-title="Täydellinen luku" data-language-autonym="Suomi" data-language-local-name="фински" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/T%C3%A4vvelidseq_arvoq" title="Tävvelidseq arvoq – Võro" lang="vro" hreflang="vro" data-title="Tävvelidseq arvoq" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_parfait" title="Nombre parfait – френски" lang="fr" hreflang="fr" data-title="Nombre parfait" data-language-autonym="Français" data-language-local-name="френски" 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/Uimhir_fhoirfe" title="Uimhir fhoirfe – ирландски" lang="ga" hreflang="ga" data-title="Uimhir fhoirfe" data-language-autonym="Gaeilge" data-language-local-name="ирландски" 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/N%C3%BAmero_perfecto" title="Número perfecto – галисийски" lang="gl" hreflang="gl" data-title="Número perfecto" data-language-autonym="Galego" data-language-local-name="галисийски" 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%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מספר משוכלל – иврит" lang="he" hreflang="he" data-title="מספר משוכלל" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6k%C3%A9letes_sz%C3%A1mok" title="Tökéletes számok – унгарски" lang="hu" hreflang="hu" data-title="Tökéletes számok" data-language-autonym="Magyar" data-language-local-name="унгарски" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BF%D5%A1%D6%80%D5%B5%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Կատարյալ թիվ – арменски" lang="hy" hreflang="hy" data-title="Կատարյալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="арменски" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_perfecte" title="Numero perfecte – интерлингва" lang="ia" hreflang="ia" data-title="Numero perfecte" data-language-autonym="Interlingua" data-language-local-name="интерлингва" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_sempurna" title="Bilangan sempurna – индонезийски" lang="id" hreflang="id" data-title="Bilangan sempurna" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезийски" 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/Fullkomin_tala" title="Fullkomin tala – исландски" lang="is" hreflang="is" data-title="Fullkomin tala" data-language-autonym="Íslenska" data-language-local-name="исландски" 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/Numero_perfetto" title="Numero perfetto – италиански" lang="it" hreflang="it" data-title="Numero perfetto" data-language-autonym="Italiano" data-language-local-name="италиански" 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/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – японски" lang="ja" hreflang="ja" data-title="完全数" data-language-autonym="日本語" data-language-local-name="японски" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수 – корейски" lang="ko" hreflang="ko" data-title="완전수" data-language-autonym="한국어" data-language-local-name="корейски" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_nuwaze" title="Hejmarên nuwaze – кюрдски" lang="ku" hreflang="ku" data-title="Hejmarên nuwaze" data-language-autonym="Kurdî" data-language-local-name="кюрдски" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_perfectus" title="Numerus perfectus – латински" lang="la" hreflang="la" data-title="Numerus perfectus" data-language-autonym="Latina" data-language-local-name="латински" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_parfett" title="Nümar parfett – ломбардски" lang="lmo" hreflang="lmo" data-title="Nümar parfett" data-language-autonym="Lombard" data-language-local-name="ломбардски" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tobulasis_skai%C4%8Dius" title="Tobulasis skaičius – литовски" lang="lt" hreflang="lt" data-title="Tobulasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="литовски" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BE%D0%B2%D1%80%D1%88%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Совршен број – македонски" lang="mk" hreflang="mk" data-title="Совршен број" data-language-autonym="Македонски" data-language-local-name="македонски" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Nummero_perfetto" title="Nummero perfetto – неаполитански" lang="nap" hreflang="nap" data-title="Nummero perfetto" data-language-autonym="Napulitano" data-language-local-name="неаполитански" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Perfect_getal" title="Perfect getal – нидерландски" lang="nl" hreflang="nl" data-title="Perfect getal" data-language-autonym="Nederlands" data-language-local-name="нидерландски" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fulkomne_tal" title="Fulkomne tal – норвежки (нюношк)" lang="nn" hreflang="nn" data-title="Fulkomne tal" data-language-autonym="Norsk nynorsk" data-language-local-name="норвежки (нюношк)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Perfekt_tall" title="Perfekt tall – норвежки (букмол)" lang="nb" hreflang="nb" data-title="Perfekt tall" data-language-autonym="Norsk bokmål" data-language-local-name="норвежки (букмол)" 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/Liczby_doskona%C5%82e" title="Liczby doskonałe – полски" lang="pl" hreflang="pl" data-title="Liczby doskonałe" data-language-autonym="Polski" data-language-local-name="полски" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_p%C3%ABrfet" title="Nùmer përfet – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer përfet" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_perfeito" title="Número perfeito – португалски" lang="pt" hreflang="pt" data-title="Número perfeito" data-language-autonym="Português" data-language-local-name="португалски" 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/Num%C4%83r_perfect" title="Număr perfect – румънски" lang="ro" hreflang="ro" data-title="Număr perfect" data-language-autonym="Română" data-language-local-name="румънски" 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%A1%D0%BE%D0%B2%D0%B5%D1%80%D1%88%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Совершенное число – руски" lang="ru" hreflang="ru" data-title="Совершенное число" data-language-autonym="Русский" data-language-local-name="руски" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_pirfettu" title="Nùmmuru pirfettu – сицилиански" lang="scn" hreflang="scn" data-title="Nùmmuru pirfettu" data-language-autonym="Sicilianu" data-language-local-name="сицилиански" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Perfect_number" title="Perfect number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Perfect number" 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/Dokonal%C3%A9_%C4%8D%C3%ADslo" title="Dokonalé číslo – словашки" lang="sk" hreflang="sk" data-title="Dokonalé číslo" data-language-autonym="Slovenčina" data-language-local-name="словашки" 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/Popolno_%C5%A1tevilo" title="Popolno število – словенски" lang="sl" hreflang="sl" data-title="Popolno število" data-language-autonym="Slovenščina" data-language-local-name="словенски" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_p%C3%ABrsosur" title="Numrat e përsosur – албански" lang="sq" hreflang="sq" data-title="Numrat e përsosur" data-language-autonym="Shqip" data-language-local-name="албански" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B0%D0%B2%D1%80%D1%88%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Савршен број – сръбски" lang="sr" hreflang="sr" data-title="Савршен број" data-language-autonym="Српски / srpski" data-language-local-name="сръбски" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Perfekt_tal" title="Perfekt tal – шведски" lang="sv" hreflang="sv" data-title="Perfekt tal" data-language-autonym="Svenska" data-language-local-name="шведски" 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%A8%E0%AE%BF%E0%AE%B1%E0%AF%88%E0%AE%B5%E0%AF%86%E0%AE%A3%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="நிறைவெண் (கணிதம்) – тамилски" lang="ta" hreflang="ta" data-title="நிறைவெண் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="тамилски" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B0%B0%E0%B0%BF%E0%B0%AA%E0%B1%82%E0%B0%B0%E0%B1%8D%E0%B0%A3%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="పరిపూర్ణసంఖ్య – телугу" lang="te" hreflang="te" data-title="పరిపూర్ణసంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="телугу" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%BC%D1%83%D0%BA%D0%B0%D0%BC%D0%BC%D0%B0%D0%BB" title="Адади мукаммал – таджикски" lang="tg" hreflang="tg" data-title="Адади мукаммал" data-language-autonym="Тоҷикӣ" data-language-local-name="таджикски" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AA%E0%B8%A1%E0%B8%9A%E0%B8%B9%E0%B8%A3%E0%B8%93%E0%B9%8C" title="จำนวนสมบูรณ์ – тайски" lang="th" hreflang="th" data-title="จำนวนสมบูรณ์" data-language-autonym="ไทย" data-language-local-name="тайски" 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/M%C3%BCkemmel_say%C4%B1" title="Mükemmel sayı – турски" lang="tr" hreflang="tr" data-title="Mükemmel sayı" data-language-autonym="Türkçe" data-language-local-name="турски" 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%94%D0%BE%D1%81%D0%BA%D0%BE%D0%BD%D0%B0%D0%BB%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Досконале число – украински" lang="uk" hreflang="uk" data-title="Досконале число" data-language-autonym="Українська" data-language-local-name="украински" 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/%DA%A9%D8%A7%D9%85%D9%84_%D8%B9%D8%AF%D8%AF" title="کامل عدد – урду" lang="ur" hreflang="ur" data-title="کامل عدد" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mukammal_son" title="Mukammal son – узбекски" lang="uz" hreflang="uz" data-title="Mukammal son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="узбекски" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n" title="Số hoàn thiện – виетнамски" lang="vi" hreflang="vi" data-title="Số hoàn thiện" data-language-autonym="Tiếng Việt" data-language-local-name="виетнамски" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – ву китайски" lang="wuu" hreflang="wuu" data-title="完全数" data-language-autonym="吴语" data-language-local-name="ву китайски" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – китайски" lang="zh" hreflang="zh" data-title="完全数" data-language-autonym="中文" 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aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">от Уикипедия, свободната енциклопедия</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="cdx-message cdx-message--block cdx-message--warning mw-revision"><span class="cdx-message__icon"></span><div class="cdx-message__content"><div id="mw-revision-info">Версия от 15:45, 16 октомври 2023 на <a href="/wiki/%D0%9F%D0%BE%D1%82%D1%80%D0%B5%D0%B1%D0%B8%D1%82%D0%B5%D0%BB:InternetArchiveBot" class="mw-userlink" title="Потребител:InternetArchiveBot" data-mw-revid="11992043"><bdi>InternetArchiveBot</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/%D0%9F%D0%BE%D1%82%D1%80%D0%B5%D0%B1%D0%B8%D1%82%D0%B5%D0%BB_%D0%B1%D0%B5%D1%81%D0%B5%D0%B4%D0%B0:InternetArchiveBot" class="mw-redirect mw-usertoollinks-talk" title="Потребител беседа:InternetArchiveBot">беседа</a> | <a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9F%D1%80%D0%B8%D0%BD%D0%BE%D1%81%D0%B8/InternetArchiveBot" class="mw-usertoollinks-contribs" title="Специални:Приноси/InternetArchiveBot">приноси</a>)</span> <span class="comment">(Rescuing 0 sources and tagging 1 as dead.) #IABot (v2.0.9.5)</span></div><div id="mw-revision-nav">(<a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&diff=prev&oldid=11992043" title="Съвършено число">разл</a>) <a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&direction=prev&oldid=11992043" title="Съвършено число">← По-стара версия</a> | Текуща версия (разл) | По-нова версия → (разл)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="bg" dir="ltr"><p><b>Съвършено число</b> в <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математиката</a> (на <a href="/wiki/%D0%A1%D1%82%D0%B0%D1%80%D0%BE%D0%B3%D1%80%D1%8A%D1%86%D0%BA%D0%B8_%D0%B5%D0%B7%D0%B8%D0%BA" title="Старогръцки език">старогръцки</a>: <span lang="grc" dir="ltr" style="font-style:italic">ἀριθμὸς τέλειος</span>) се нарича <a href="/wiki/%D0%95%D1%81%D1%82%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Естествено число">естествено число</a>, което е равно на сумата от своите по-малки <a href="/wiki/%D0%94%D0%B5%D0%BB%D0%B8%D1%82%D0%B5%D0%BB" class="mw-redirect" title="Делител">делители</a> (т.е. различни от самото число). Например най-малкото такова число е 6 – делителите му са 1, 2 и 3 и е изпълнено свойството: 1 + 2 + 3 = 6. </p><p>Най-малките познати от античността съвършени числа са <a href="/wiki/6_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="6 (число)">6</a>, <a href="/wiki/28_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="28 (число)">28</a>, 496 и 8128. </p> <ul><li>За 28 делителите са 1, 2, 4, 7 и 14, така че <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+2+4+7+14=28}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>7</mn> <mo>+</mo> <mn>14</mn> <mo>=</mo> <mn>28</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2+4+7+14=28}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1db3b3ed47b848f87e2dcb6f7e2807b616851f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.76ex; height:2.343ex;" alt="{\displaystyle 1+2+4+7+14=28}"></span>.</li></ul> <p>По-нататък тъй като естествените числа растат, съвършените числа се срещат все по-рядко. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="От_Евклид_до_Ойлер"><span id=".D0.9E.D1.82_.D0.95.D0.B2.D0.BA.D0.BB.D0.B8.D0.B4_.D0.B4.D0.BE_.D0.9E.D0.B9.D0.BB.D0.B5.D1.80"></span>От Евклид до Ойлер</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=1" title="Редактиране на раздел: От Евклид до Ойлер" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=1" title="Edit section's source code: От Евклид до Ойлер"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Още <a href="/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4" title="Евклид">Евклид</a> в своите „Елементи“ пише, че първите четири съвършени числа могат да се пресметнат по формулата: </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 2^{n-1}(2^{n}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}(2^{n}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd65f770559bac921e8eed985ce1e8e2a2afe5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.675ex; height:3.176ex;" alt="{\displaystyle 2^{n-1}(2^{n}-1)}"></span>,</dd></dl> <p>където <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^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"></span> е <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Просто число">просто число</a>. </p> <ul><li>За <i>n</i> = 2: <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^{1}(2^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1}(2^{2}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79039a72f1c467217c24b1a6b0e16129974e9610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:3.176ex;" alt="{\displaystyle 2^{1}(2^{2}-1)}"></span> = 6 = 1 + 2 + 3</li> <li>За <i>n</i> = 3: <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}(2^{3}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}(2^{3}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0bba4ea263b7b2596a1b2d7e75d3f2e81e3653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:3.176ex;" alt="{\displaystyle 2^{2}(2^{3}-1)}"></span> = 28 = 1 + 2 + 4 + 7 + 14</li> <li>За <i>n</i> = 5: <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^{4}(2^{5}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}(2^{5}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d8a09d189889868abe3e0770a27ecc01f056a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:3.176ex;" alt="{\displaystyle 2^{4}(2^{5}-1)}"></span> = 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248</li> <li>За <i>n</i> = 7: <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^{6}(2^{7}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{6}(2^{7}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a81e785dfa4f53c7539262f2fe834cd9852c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:3.176ex;" alt="{\displaystyle 2^{6}(2^{7}-1)}"></span> = 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064</li></ul> <p>Повече от хилядолетие след Евклид <a href="/wiki/%D0%98%D0%B1%D0%BD_%D0%B0%D0%BB-%D0%A5%D0%B0%D0%B9%D1%82%D0%B0%D0%BC" title="Ибн ал-Хайтам">Ибн ал-Хайтам</a> (<a href="/wiki/%D0%90%D0%BB%D1%85%D0%B0%D0%B7%D0%B5%D0%BD" class="mw-redirect" title="Алхазен">Алхазен</a>) твърди, че всяко четно съвършено число е от вида <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^{n-1}(2^{n}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}(2^{n}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd65f770559bac921e8eed985ce1e8e2a2afe5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.675ex; height:3.176ex;" alt="{\displaystyle 2^{n-1}(2^{n}-1)}"></span>, където <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^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"></span> е просто число, но не може да докаже този резултат. Едва през XVIII век <a href="/wiki/%D0%9B%D0%B5%D0%BE%D0%BD%D0%B0%D1%80%D0%B4_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Леонард Ойлер">Леонард Ойлер</a> доказва, че по тази формула се получават всички четни съвършени числа. Тук имаме взаимно еднозначно съответствие между четните съвършени числа и простите <a href="/wiki/%D0%9C%D0%B5%D1%80%D1%81%D0%B5%D0%BD%D0%BE%D0%B2%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" class="mw-redirect" title="Мерсеново число">мерсенови числа</a>, които са от вида <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^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"></span> при просто число <i>n</i>. Този резултат се посочва като <b>теорема на Евклид – Ойлер</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Съвременно_състояние_на_проблема"><span id=".D0.A1.D1.8A.D0.B2.D1.80.D0.B5.D0.BC.D0.B5.D0.BD.D0.BD.D0.BE_.D1.81.D1.8A.D1.81.D1.82.D0.BE.D1.8F.D0.BD.D0.B8.D0.B5_.D0.BD.D0.B0_.D0.BF.D1.80.D0.BE.D0.B1.D0.BB.D0.B5.D0.BC.D0.B0"></span>Съвременно състояние на проблема</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=2" title="Редактиране на раздел: Съвременно състояние на проблема" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=2" title="Edit section's source code: Съвременно състояние на проблема"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>До септември 2021 г. са известни само 51 прости мерсенови числа и следователно са известни <b>51 четни съвършени числа</b>. Те се получават за </p><p><i>n</i> = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11 213, 19 937, 21 701, 23 209, 44 497, 86 243, 110 503, 132 049, 216 091, 756 839, 859 433, 1 257 787, 1 398 269, 2 976 221, 3 021 377, 6 972 593, 13 466 917, 20 996 011, 24 036 583, 25 964 951, 30 402 457, 32 582 657... </p><p>2<sup>32 582 656</sup> × (2<sup>32 582 657</sup> − 1) е с 19 616 714 цифри. </p><p>Все още не е известно дали простите мерсенови числа, а съответно и четните съвършени числа са безбройно много или са краен брой. Този математически проблем остава <b>нерешен</b>, въпреки че търсенето на мерсенови числа се извършва с помощта на много мощни компютри. </p><p>Освен това всички познати съвършени числа < 10<sup>18</sup> са четни, а ако съществуват нечетни, то те следва да са > 10<sup>1500</sup>, без да е известно дали въобще има такива – <b>още един нерешен математически проблем</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Първите_десет_съвършени_числа"><span id=".D0.9F.D1.8A.D1.80.D0.B2.D0.B8.D1.82.D0.B5_.D0.B4.D0.B5.D1.81.D0.B5.D1.82_.D1.81.D1.8A.D0.B2.D1.8A.D1.80.D1.88.D0.B5.D0.BD.D0.B8_.D1.87.D0.B8.D1.81.D0.BB.D0.B0"></span>Първите десет съвършени числа</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=3" title="Редактиране на раздел: Първите десет съвършени числа" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=3" title="Edit section's source code: Първите десет съвършени числа"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>6</li> <li>28</li> <li>496</li> <li>8128</li> <li>33 550 336</li> <li>8 589 869 056</li> <li>137 438 691 328</li> <li>2 305 843 008 139 952 128</li> <li>2 658 455 991 569 831 744 654 692 615 953 842 176</li> <li>191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216</li></ul> <p>(последователност A000396 в OEIS) </p> <div class="mw-heading mw-heading2"><h2 id="Вижте_също"><span id=".D0.92.D0.B8.D0.B6.D1.82.D0.B5_.D1.81.D1.8A.D1.89.D0.BE"></span>Вижте също</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=4" title="Редактиране на раздел: Вижте също" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=4" title="Edit section's source code: Вижте също"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Просто число">Просто число</a></li> <li><a href="/wiki/%D0%95%D1%81%D1%82%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Естествено число">Естествено число</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Източници"><span id=".D0.98.D0.B7.D1.82.D0.BE.D1.87.D0.BD.D0.B8.D1.86.D0.B8"></span>Източници</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=5" title="Редактиране на раздел: Източници" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=5" title="Edit section's source code: Източници"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Совершенное число – статия от Уикипедия на руски език [12 март 2008]</li> <li>Perfect number – статия от Уикипедия на английски език [17 март 2008]</li></ul> <div class="mw-heading mw-heading2"><h2 id="Външни_препратки"><span id=".D0.92.D1.8A.D0.BD.D1.88.D0.BD.D0.B8_.D0.BF.D1.80.D0.B5.D0.BF.D1.80.D0.B0.D1.82.D0.BA.D0.B8"></span>Външни препратки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&veaction=edit&section=6" title="Редактиране на раздел: Външни препратки" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A1%D1%8A%D0%B2%D1%8A%D1%80%D1%88%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE&action=edit&section=6" title="Edit section's source code: Външни препратки"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="color:blue; 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