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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</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"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</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/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP này</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="Trang Web"> <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="Nội dung" 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">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</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">Đầu</div> </a> </li> <li id="toc-Định_nghĩa_số_hoàn_hảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Định_nghĩa_số_hoàn_hảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Định nghĩa số hoàn hảo</span> </div> </a> <ul id="toc-Định_nghĩa_số_hoàn_hảo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Các_số_hoàn_hảo_chẵn" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Các_số_hoàn_hảo_chẵn"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Các số hoàn hảo chẵn</span> </div> </a> <ul id="toc-Các_số_hoàn_hảo_chẵn-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Số_hoàn_hảo_lẻ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Số_hoàn_hảo_lẻ"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Số hoàn hảo lẻ</span> </div> </a> <ul id="toc-Số_hoàn_hảo_lẻ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Xem_thêm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Xem_thêm"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Xem thêm</span> </div> </a> <ul id="toc-Xem_thêm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ghi_chú" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ghi_chú"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ghi chú</span> </div> </a> <ul id="toc-Ghi_chú-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liên_kết_ngoài" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Liên_kết_ngoài"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Liên kết ngoài</span> </div> </a> <ul id="toc-Liên_kết_ngoài-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="Nội dung" 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="Đóng mở mục lục" > <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">Đóng mở mục lục</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">Số hoàn thiện</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="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 64 ngôn ngữ" > <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 ngôn ngữ</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 – Tiếng Ả Rập" lang="ar" hreflang="ar" data-title="عدد تام" data-language-autonym="العربية" data-language-local-name="Tiếng Ả Rập" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_sempurna" title="Bilangan sempurna – Tiếng Indonesia" lang="id" hreflang="id" data-title="Bilangan sempurna" data-language-autonym="Bahasa Indonesia" data-language-local-name="Tiếng Indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Tiếng Bangla" lang="bn" hreflang="bn" data-title="নিখুঁত সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Tiếng Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Tiếng Belarus" lang="be" hreflang="be" data-title="Дасканалы лік" data-language-autonym="Беларуская" data-language-local-name="Tiếng Belarus" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_peurvat" title="Niver peurvat – Tiếng Breton" lang="br" hreflang="br" data-title="Niver peurvat" data-language-autonym="Brezhoneg" data-language-local-name="Tiếng Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%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" title="Съвършено число – Tiếng Bulgaria" lang="bg" hreflang="bg" data-title="Съвършено число" data-language-autonym="Български" data-language-local-name="Tiếng Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_perfecte" title="Nombre perfecte – Tiếng Catalan" lang="ca" hreflang="ca" data-title="Nombre perfecte" data-language-autonym="Català" data-language-local-name="Tiếng Catalan" 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/Dokonal%C3%A9_%C4%8D%C3%ADslo" title="Dokonalé číslo – Tiếng Séc" lang="cs" hreflang="cs" data-title="Dokonalé číslo" data-language-autonym="Čeština" data-language-local-name="Tiếng Séc" 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 – Tiếng Đan Mạch" lang="da" hreflang="da" data-title="Fuldkomne tal" data-language-autonym="Dansk" data-language-local-name="Tiếng Đan Mạch" 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 – Tiếng Đức" lang="de" hreflang="de" data-title="Vollkommene Zahl" data-language-autonym="Deutsch" data-language-local-name="Tiếng Đức" 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 – Tiếng Hy Lạp" lang="el" hreflang="el" data-title="Τέλειος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Tiếng Hy Lạp" 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 – Tiếng Anh" lang="en" hreflang="en" data-title="Perfect number" data-language-autonym="English" data-language-local-name="Tiếng Anh" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_perfecto" title="Número perfecto – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Número perfecto" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Perfekta_nombro" title="Perfekta nombro – Tiếng Quốc Tế Ngữ" lang="eo" hreflang="eo" data-title="Perfekta nombro" data-language-autonym="Esperanto" data-language-local-name="Tiếng Quốc Tế Ngữ" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_perfektu" title="Zenbaki perfektu – Tiếng Basque" lang="eu" hreflang="eu" data-title="Zenbaki perfektu" data-language-autonym="Euskara" data-language-local-name="Tiếng Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AA%D8%A7%D9%85" title="عدد تام – Tiếng Ba Tư" lang="fa" hreflang="fa" data-title="عدد تام" data-language-autonym="فارسی" data-language-local-name="Tiếng Ba Tư" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_parfait" title="Nombre parfait – Tiếng Pháp" lang="fr" hreflang="fr" data-title="Nombre parfait" data-language-autonym="Français" data-language-local-name="Tiếng Pháp" 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 – Tiếng Ireland" lang="ga" hreflang="ga" data-title="Uimhir fhoirfe" data-language-autonym="Gaeilge" data-language-local-name="Tiếng Ireland" 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 – Tiếng Galician" lang="gl" hreflang="gl" data-title="Número perfecto" data-language-autonym="Galego" data-language-local-name="Tiếng Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수 – Tiếng Hàn" lang="ko" hreflang="ko" data-title="완전수" data-language-autonym="한국어" data-language-local-name="Tiếng Hàn" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BF%D5%A1%D6%80%D5%B5%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Կատարյալ թիվ – Tiếng Armenia" lang="hy" hreflang="hy" data-title="Կատարյալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Tiếng Armenia" 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 – Tiếng Khoa Học Quốc Tế" lang="ia" hreflang="ia" data-title="Numero perfecte" data-language-autonym="Interlingua" data-language-local-name="Tiếng Khoa Học Quốc Tế" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fullkomin_tala" title="Fullkomin tala – Tiếng Iceland" lang="is" hreflang="is" data-title="Fullkomin tala" data-language-autonym="Íslenska" data-language-local-name="Tiếng Iceland" 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 – Tiếng Italy" lang="it" hreflang="it" data-title="Numero perfetto" data-language-autonym="Italiano" data-language-local-name="Tiếng Italy" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מספר משוכלל – Tiếng Do Thái" lang="he" hreflang="he" data-title="מספר משוכלל" data-language-autonym="עברית" data-language-local-name="Tiếng Do Thái" class="interlanguage-link-target"><span>עברית</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 – Tiếng Kurd" lang="ku" hreflang="ku" data-title="Hejmarên nuwaze" data-language-autonym="Kurdî" data-language-local-name="Tiếng Kurd" 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 – Tiếng La-tinh" lang="la" hreflang="la" data-title="Numerus perfectus" data-language-autonym="Latina" data-language-local-name="Tiếng La-tinh" class="interlanguage-link-target"><span>Latina</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 – Tiếng Litva" lang="lt" hreflang="lt" data-title="Tobulasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Tiếng Litva" class="interlanguage-link-target"><span>Lietuvių</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 – Tiếng Lombard" lang="lmo" hreflang="lmo" data-title="Nümar parfett" data-language-autonym="Lombard" data-language-local-name="Tiếng Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6k%C3%A9letes_sz%C3%A1mok" title="Tökéletes számok – Tiếng Hungary" lang="hu" hreflang="hu" data-title="Tökéletes számok" data-language-autonym="Magyar" data-language-local-name="Tiếng Hungary" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BE%D0%B2%D1%80%D1%88%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Совршен број – Tiếng Macedonia" lang="mk" hreflang="mk" data-title="Совршен број" data-language-autonym="Македонски" data-language-local-name="Tiếng Macedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Perfect_getal" title="Perfect getal – Tiếng Hà Lan" lang="nl" hreflang="nl" data-title="Perfect getal" data-language-autonym="Nederlands" data-language-local-name="Tiếng Hà Lan" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – Tiếng Nhật" lang="ja" hreflang="ja" data-title="完全数" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Nummero_perfetto" title="Nummero perfetto – Tiếng Napoli" lang="nap" hreflang="nap" data-title="Nummero perfetto" data-language-autonym="Napulitano" data-language-local-name="Tiếng Napoli" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Perfekt_tall" title="Perfekt tall – Tiếng Na Uy (Bokmål)" lang="nb" hreflang="nb" data-title="Perfekt tall" data-language-autonym="Norsk bokmål" data-language-local-name="Tiếng Na Uy (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fulkomne_tal" title="Fulkomne tal – Tiếng Na Uy (Nynorsk)" lang="nn" hreflang="nn" data-title="Fulkomne tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Tiếng Na Uy (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mukammal_son" title="Mukammal son – Tiếng Uzbek" lang="uz" hreflang="uz" data-title="Mukammal son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Tiếng Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_doskona%C5%82e" title="Liczby doskonałe – Tiếng Ba Lan" lang="pl" hreflang="pl" data-title="Liczby doskonałe" data-language-autonym="Polski" data-language-local-name="Tiếng Ba Lan" 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/N%C3%BAmero_perfeito" title="Número perfeito – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Número perfeito" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" 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 – Tiếng Romania" lang="ro" hreflang="ro" data-title="Număr perfect" data-language-autonym="Română" data-language-local-name="Tiếng Romania" 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 – Tiếng Nga" lang="ru" hreflang="ru" data-title="Совершенное число" data-language-autonym="Русский" data-language-local-name="Tiếng Nga" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_p%C3%ABrsosur" title="Numrat e përsosur – Tiếng Albania" lang="sq" hreflang="sq" data-title="Numrat e përsosur" data-language-autonym="Shqip" data-language-local-name="Tiếng Albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_pirfettu" title="Nùmmuru pirfettu – Tiếng Sicilia" lang="scn" hreflang="scn" data-title="Nùmmuru pirfettu" data-language-autonym="Sicilianu" data-language-local-name="Tiếng Sicilia" 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 – Tiếng Slovak" lang="sk" hreflang="sk" data-title="Dokonalé číslo" data-language-autonym="Slovenčina" data-language-local-name="Tiếng Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Popolno_%C5%A1tevilo" title="Popolno število – Tiếng Slovenia" lang="sl" hreflang="sl" data-title="Popolno število" data-language-autonym="Slovenščina" data-language-local-name="Tiếng Slovenia" class="interlanguage-link-target"><span>Slovenščina</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 – Tiếng Kurd Miền Trung" lang="ckb" hreflang="ckb" data-title="ژمارەی کامڵ" data-language-autonym="کوردی" data-language-local-name="Tiếng Kurd Miền Trung" class="interlanguage-link-target"><span>کوردی</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 – Tiếng Serbia" lang="sr" hreflang="sr" data-title="Савршен број" data-language-autonym="Српски / srpski" data-language-local-name="Tiếng Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/T%C3%A4ydellinen_luku" title="Täydellinen luku – Tiếng Phần Lan" lang="fi" hreflang="fi" data-title="Täydellinen luku" data-language-autonym="Suomi" data-language-local-name="Tiếng Phần Lan" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Perfekt_tal" title="Perfekt tal – Tiếng Thụy Điển" lang="sv" hreflang="sv" data-title="Perfekt tal" data-language-autonym="Svenska" data-language-local-name="Tiếng Thụy Điển" 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) – Tiếng Tamil" lang="ta" hreflang="ta" data-title="நிறைவெண் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tiếng Tamil" 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 – Tiếng Telugu" lang="te" hreflang="te" data-title="పరిపూర్ణసంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Tiếng Telugu" class="interlanguage-link-target"><span>తెలుగు</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 – Tiếng Thái" lang="th" hreflang="th" data-title="จำนวนสมบูรณ์" data-language-autonym="ไทย" data-language-local-name="Tiếng Thái" class="interlanguage-link-target"><span>ไทย</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 – Tiếng Tajik" lang="tg" hreflang="tg" data-title="Адади мукаммал" data-language-autonym="Тоҷикӣ" data-language-local-name="Tiếng Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</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 – Tiếng Thổ Nhĩ Kỳ" lang="tr" hreflang="tr" data-title="Mükemmel sayı" data-language-autonym="Türkçe" data-language-local-name="Tiếng Thổ Nhĩ Kỳ" 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 – Tiếng Ukraina" lang="uk" hreflang="uk" data-title="Досконале число" data-language-autonym="Українська" data-language-local-name="Tiếng Ukraina" 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 – Tiếng Urdu" lang="ur" hreflang="ur" data-title="کامل عدد" data-language-autonym="اردو" data-language-local-name="Tiếng Urdu" class="interlanguage-link-target"><span>اردو</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-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B8" title="完全數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="完全數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</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 – Tiếng Ngô" lang="wuu" hreflang="wuu" data-title="完全数" data-language-autonym="吴语" data-language-local-name="Tiếng Ngô" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B8" title="完全數 – Tiếng Quảng Đông" lang="yue" hreflang="yue" data-title="完全數" data-language-autonym="粵語" data-language-local-name="Tiếng Quảng Đông" class="interlanguage-link-target"><span>粵語</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 – Tiếng Trung" lang="zh" hreflang="zh" data-title="完全数" data-language-autonym="中文" data-language-local-name="Tiếng Trung" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q170043#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Không gian tên"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n" title="Xem bài viết [c]" accesskey="c"><span>Bài viết</span></a></li><li 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aria-label="Công cụ trang"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Bách khoa toàn thư mở Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="vi" dir="ltr"><p><b>Số hoàn hảo</b> (hay còn gọi là <b>số hoàn chỉnh</b>, <b>số hoàn thiện</b> hoặc <b>số hoàn thành</b>) là một số nguyên dương mà tổng các ước nguyên dương thực sự của nó (các số nguyên dương bị nó chia hết ngoại trừ nó) bằng chính nó. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Định_nghĩa_số_hoàn_hảo"><span id=".C4.90.E1.BB.8Bnh_ngh.C4.A9a_s.E1.BB.91_ho.C3.A0n_h.E1.BA.A3o"></span>Định nghĩa số hoàn hảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=1" title="Sửa đổi phần “Định nghĩa số hoàn hảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&action=edit&section=1" title="Sửa mã nguồn tại đề mục: Định nghĩa số hoàn hảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Số hoàn hảo là các <a href="/wiki/S%E1%BB%91_t%E1%BB%B1_nhi%C3%AAn" title="Số tự nhiên">số nguyên dương</a> <i>n</i> sao cho: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=s(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=s(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4111abd25c75f9cb27334c108ba7356550a0cc56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.788ex; height:2.843ex;" alt="{\displaystyle n=s(n)}"></span></dd></dl> <p>trong đó, s(n) là <a href="/wiki/T%E1%BB%95ng_%C6%B0%E1%BB%9Bc_s%E1%BB%91_th%E1%BB%B1c_s%E1%BB%B1" title="Tổng ước số thực sự">hàm tổng các ước thực sự</a> của n. Ví dụ: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6=s(6)=1+2+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=s(6)=1+2+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d94be8bec9c35af67c93fad51fe5ceb187fb52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.59ex; height:2.843ex;" alt="{\displaystyle 6=s(6)=1+2+3}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 28=s(28)=1+2+4+7+14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>28</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>7</mn> <mo>+</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28=s(28)=1+2+4+7+14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3d15f63dcb49333d535d9501f9cfa43fbae1e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.083ex; height:2.843ex;" alt="{\displaystyle 28=s(28)=1+2+4+7+14}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 496=s(496)=1+2+4+8+16+31+62+124+248}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>496</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>8</mn> <mo>+</mo> <mn>16</mn> <mo>+</mo> <mn>31</mn> <mo>+</mo> <mn>62</mn> <mo>+</mo> <mn>124</mn> <mo>+</mo> <mn>248</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496=s(496)=1+2+4+8+16+31+62+124+248}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719f4386896b42e80a28a2eed3dd169f29e9f1e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.394ex; height:2.843ex;" alt="{\displaystyle 496=s(496)=1+2+4+8+16+31+62+124+248}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8128=s(8128)=1+2+4+8+16+32+64+127+254+508+1016+2032+4064}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8128</mn> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>8128</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>8</mn> <mo>+</mo> <mn>16</mn> <mo>+</mo> <mn>32</mn> <mo>+</mo> <mn>64</mn> <mo>+</mo> <mn>127</mn> <mo>+</mo> <mn>254</mn> <mo>+</mo> <mn>508</mn> <mo>+</mo> <mn>1016</mn> <mo>+</mo> <mn>2032</mn> <mo>+</mo> <mn>4064</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8128=s(8128)=1+2+4+8+16+32+64+127+254+508+1016+2032+4064}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62f727815d2f577a89f82bce7416f169df7e3875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.517ex; height:2.843ex;" alt="{\displaystyle 8128=s(8128)=1+2+4+8+16+32+64+127+254+508+1016+2032+4064}"></span> </p><p>Hoặc: </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 \sigma (n)=2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)=2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3627e031c536dcff0b99eff7778e0ace371361bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.189ex; height:2.843ex;" alt="{\displaystyle \sigma (n)=2n}"></span></dd></dl> <p>trong đó, <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 \sigma (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3213bbd075ed40d41d2b72dfaa4a190e031d3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle \sigma (n)}"></span> là <a href="/w/index.php?title=H%C3%A0m_t%E1%BB%95ng_c%C3%A1c_%C6%B0%E1%BB%9Bc&action=edit&redlink=1" class="new" title="Hàm tổng các ước (trang không tồn tại)">hàm tổng các ước</a> của <i>n</i>, bao gồm cả <i>n</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Các_số_hoàn_hảo_chẵn"><span id="C.C3.A1c_s.E1.BB.91_ho.C3.A0n_h.E1.BA.A3o_ch.E1.BA.B5n"></span>Các số hoàn hảo chẵn</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=2" title="Sửa đổi phần “Các số hoàn hảo chẵn”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&action=edit&section=2" title="Sửa mã nguồn tại đề mục: Các số hoàn hảo chẵn"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right; width:30%; margin:0em 1em 1em 1em; background:#eee; border:#ccc solid"> <tbody><tr> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/20px-Question%2C_Web_Fundamentals.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/30px-Question%2C_Web_Fundamentals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/40px-Question%2C_Web_Fundamentals.svg.png 2x" data-file-width="44" data-file-height="44" /></span></span></td> <td><b>Vấn đề mở trong toán học</b>:<br /><div style="padding:5px; font-style:italic;">Liệu có vô số số hoàn hảo?</div><small>(<a href="/wiki/Danh_s%C3%A1ch_v%E1%BA%A5n_%C4%91%E1%BB%81_m%E1%BB%9F_trong_to%C3%A1n_h%E1%BB%8Dc" title="Danh sách vấn đề mở trong toán học">các vấn đề mở khác trong toán học</a>)</small> </td></tr></tbody></table> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> đã khám phá ra 4 số hoàn hảo nhỏ nhất dưới dạng: 2<sup><i>p</i>−1</sup>(2<sup><i>p</i></sup> − 1): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2\Rightarrow 2^{2-1}(2^{2}-1)=2\times 3=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">⇒</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <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> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2\Rightarrow 2^{2-1}(2^{2}-1)=2\times 3=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456e92c7a65764cafcd8ac3021b3ffbf4fddf017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:34.005ex; height:3.176ex;" alt="{\displaystyle p=2\Rightarrow 2^{2-1}(2^{2}-1)=2\times 3=6}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=3\Rightarrow 2^{3-1}(2^{3}-1)=4\times 7=28}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">⇒</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</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> <mo>=</mo> <mn>4</mn> <mo>×</mo> <mn>7</mn> <mo>=</mo> <mn>28</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=3\Rightarrow 2^{3-1}(2^{3}-1)=4\times 7=28}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ea4c4348d451d74ed5e71090851f9b4656e85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:35.167ex; height:3.176ex;" alt="{\displaystyle p=3\Rightarrow 2^{3-1}(2^{3}-1)=4\times 7=28}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=5\Rightarrow 2^{5-1}(2^{5}-1)=16\times 31=496}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mo stretchy="false">⇒</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>1</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> <mo>=</mo> <mn>16</mn> <mo>×</mo> <mn>31</mn> <mo>=</mo> <mn>496</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=5\Rightarrow 2^{5-1}(2^{5}-1)=16\times 31=496}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5477416246db9e6c9edd87b34a19b60b48108721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:38.654ex; height:3.176ex;" alt="{\displaystyle p=5\Rightarrow 2^{5-1}(2^{5}-1)=16\times 31=496}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=7\Rightarrow 2^{7-1}(2^{7}-1)=64\times 127=8128}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>7</mn> <mo stretchy="false">⇒</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>−</mo> <mn>1</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> <mo>=</mo> <mn>64</mn> <mo>×</mo> <mn>127</mn> <mo>=</mo> <mn>8128</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=7\Rightarrow 2^{7-1}(2^{7}-1)=64\times 127=8128}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2227e802b7b346f5b101e840c041fe04a9a192b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:40.979ex; height:3.176ex;" alt="{\displaystyle p=7\Rightarrow 2^{7-1}(2^{7}-1)=64\times 127=8128}"></span></dd></dl> <p>Chú ý rằng: 2<sup>p</sup> − 1 đều là <a href="/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91" title="Số nguyên tố">số nguyên tố</a> trong mỗi ví dụ trên, Euclid chứng minh rằng công thức: 2<sup>p−1</sup>(2<sup>p</sup> − 1) sẽ cho ta một số hoàn hảo chẵn khi và chỉ khi 2<sup>p</sup> − 1 là <a href="/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91" title="Số nguyên tố">số nguyên tố</a> (<a href="/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91_Mersenne" title="Số nguyên tố Mersenne">số nguyên tố Mersenne</a>). </p><p>Các nhà toán học cổ đại chấp nhận đây là 4 số hoàn hảo nhỏ nhất mà họ biết, nhưng đa số những giả định trên đây đã không được chứng minh là đúng. Một trong số đó là nếu 2, 3, 5, 7 là bốn số nguyên tố đầu tiên thì nhất định sẽ có số hoàn thiện thứ năm khi p = 11, số nguyên tố thứ năm. Nhưng 2<sup>11</sup> − 1 = 2047 = 23 × 89 lại là hợp số, và thế là p = 11 không thu được số hoàn hảo. 2 sai lầm khác của họ là: </p><p>Số hoàn hảo thứ năm phải có năm chữ số theo hệ cơ số 10 vì bốn số hoàn hảo đầu tiên có lần lượt 1, 2, 3, 4 chữ số </p><p>Chữ số hàng đơn vị của số hoàn hảo phải là 6, 8, 6, 8 và cứ thế lặp lại. </p><p>Số hoàn hảo thứ năm là <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 33.550.336=2^{12}(2^{13}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>33.550.336</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 33.550.336=2^{12}(2^{13}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb592bceadc59d94b3911f15fe24833bb3282caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.581ex; height:3.176ex;" alt="{\displaystyle 33.550.336=2^{12}(2^{13}-1)}"></span> bao gồm 8 chữ số, vậy nhận định 1 đã sai, về nhận định thứ 2 thì số này tận cùng là 6. Tuy nhiên đến số hoàn hảo thứ sáu là <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 8.589.869.056=2^{16}(2^{17}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8.589.869.056</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8.589.869.056=2^{16}(2^{17}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cc7653055f104ca169d7e1cc846cc32681803c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.553ex; height:3.176ex;" alt="{\displaystyle 8.589.869.056=2^{16}(2^{17}-1)}"></span> thì cũng tận cùng là 6. Nói cách khác bất cứ số hoàn hảo chẵn nào cũng phải có chữ số tận cùng là 6 hoặc 8. </p><p>Để <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^{p}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c5977dbf385ba719fbb90f67b0a3d91e1da6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.224ex; height:2.509ex;" alt="{\displaystyle 2^{p}-1}"></span> là số nguyên tố thì điều kiện cần nhưng chưa đủ là p là số nguyên tố. Số nguyên tố có dạng 2<sup>p</sup> − 1 được gọi là <a href="/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91_Mersenne" title="Số nguyên tố Mersenne">Số nguyên tố Mersenne</a> sau khi được 1 nhà tu vào thế kỷ 17 là <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a>, người học <a href="/wiki/L%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91" title="Lý thuyết số">lý thuyết số</a> và số hoàn hảo tìm ra. </p><p>Hơn 1000 năm sau Euclid, <a href="/wiki/Ibn_al-Haytham" class="mw-redirect" title="Ibn al-Haytham">Ibn al-Haytham</a> <a href="/w/index.php?title=Alhazen_circa&action=edit&redlink=1" class="new" title="Alhazen circa (trang không tồn tại)">Alhazen circa</a> nhận ra rằng mọi số hoàn hảo chẵn đều phải có dạng 2<sup>p−1</sup>(2<sup>p</sup> − 1) khi 2<sup>p</sup> − 1 là số nguyên tố, nhưng ông ta không thể chứng minh được kết quả này.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Mãi tới thế kỷ 18 là <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> đã chứng minh công thức 2<sup>p−1</sup>(2<sup>p</sup> − 1) là sẽ tìm ra các số hoàn hảo chẵn. Đó là lý do dẫn tới sự liên hệ giữa số hoàn hảo và số nguyên tố Mersenne. Kết quả này thường được gọi là thuyết Euclid-Euler. Cho tới tháng 9 năm 2008, mới chỉ có 46 số Mersenne được tìm ra,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> có nghĩa đây là số hoàn hảo thứ 46 được biết, số lớn nhất là 2<sup>43.112.608</sup> × (2<sup>43.112.609</sup> − 1) với 25.956.377 chữ số. </p><p>39 số hoàn hảo chẵn đầu tiên có dạng 2<sup>p−1</sup>(2<sup>p</sup> − 1) khi </p> <dl><dd>p = <a href="/wiki/2_(s%E1%BB%91)" title="2 (số)">2</a>, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 (dãy số <span class="nowrap"><a href="//oeis.org/A000043" class="extiw" title="oeis:A000043">A000043</a></span> trong bảng <a href="/wiki/B%E1%BA%A3ng_tra_c%E1%BB%A9u_d%C3%A3y_s%E1%BB%91_nguy%C3%AAn_tr%E1%BB%B1c_tuy%E1%BA%BFn" title="Bảng tra cứu dãy số nguyên trực tuyến">OEIS</a>)</dd></dl> <p>7 số khác được biết là khi p = 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609. Chưa ai biết là có để sót số nào giữa chúng hay không </p><p>Cũng chưa ai biết chắc chắn là có <a href="/w/index.php?title=V%C3%B4_h%E1%BA%A1n_s%E1%BB%91&action=edit&redlink=1" class="new" title="Vô hạn số (trang không tồn tại)">vô hạn</a> số nguyên tố Mersenne và số hoàn hảo hay không. Việc tìm ra các số nguyên tố Mersenne mới được thực hiện bởi các <a href="/wiki/Si%C3%AAu_m%C3%A1y_t%C3%ADnh" title="Siêu máy tính">siêu máy tính</a> </p><p>Các số hoàn hảo đều là <a href="/wiki/S%E1%BB%91_tam_gi%C3%A1c" title="Số tam giác">số tam giác</a> thứ 2<sup>p</sup> − 1 (là tổng của tất cả các số tự nhiên từ 1 đến 2<sup>p</sup> − 1): </p> <dl><dd>p = 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 6=1+2+3\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=1+2+3\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0bf9e1e97d73a63b8948fa0bf2d94a4d9fa89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.816ex; height:2.343ex;" alt="{\displaystyle 6=1+2+3\,}"></span></dd> <dd>p = 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 28=1+2+3+4+5+6+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28=1+2+3+4+5+6+7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e597ca4532e768b9c319813148a5936fa94f2764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.603ex; height:2.343ex;" alt="{\displaystyle 28=1+2+3+4+5+6+7}"></span></dd> <dd>p = 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 496=1+2+3+...+29+30+31}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mn>29</mn> <mo>+</mo> <mn>30</mn> <mo>+</mo> <mn>31</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496=1+2+3+...+29+30+31}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4d2004161d81d216b21a25d92f8f07c4768eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:35.128ex; height:2.343ex;" alt="{\displaystyle 496=1+2+3+...+29+30+31}"></span></dd> <dd>p = 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 8128=1+2+3+...+125+126+127}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8128</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mn>125</mn> <mo>+</mo> <mn>126</mn> <mo>+</mo> <mn>127</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8128=1+2+3+...+125+126+127}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ff5c1f105f5bc00afcd64915b54294a4ff24f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:39.777ex; height:2.343ex;" alt="{\displaystyle 8128=1+2+3+...+125+126+127}"></span></dd></dl> <p>Các số hoàn hảo đều là tổ hợp chập 2 của 2<sup>p</sup>: </p> <dl><dd>p = 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 6=C_{4}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=C_{4}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16bcb61e5e2aca00183263b716c88e409d22f42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.113ex; height:3.176ex;" alt="{\displaystyle 6=C_{4}^{2}}"></span></dd> <dd>p = 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 28=C_{8}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28=C_{8}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5cfb8d727bd05f61269e38f2ead80e8f574720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.275ex; height:3.176ex;" alt="{\displaystyle 28=C_{8}^{2}}"></span></dd> <dd>p = 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 496=C_{32}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496=C_{32}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95cd6c08b328684d60a2e326bc3efb0d9678c118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.124ex; height:3.176ex;" alt="{\displaystyle 496=C_{32}^{2}}"></span></dd> <dd>p = 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 8128=C_{128}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8128</mn> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>128</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8128=C_{128}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb656b15fcfebc5d86db66e4cbaba35d3187e13d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.108ex; height:3.176ex;" alt="{\displaystyle 8128=C_{128}^{2}}"></span></dd></dl> <p>Các số hoàn hảo đều có tổng các nghịch đảo của các ước (kể cả chính nó) đúng bằng 2: </p> <dl><dd>6: <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+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39937f14f42d7394de2308630d1a5c0ee9b8c296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.94ex; height:5.176ex;" alt="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}=2}"></span></dd> <dd>28: <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+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{7}}+{\frac {1}{14}}+{\frac {1}{28}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>28</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{7}}+{\frac {1}{14}}+{\frac {1}{28}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103c079b3ab3d6d5e35133b132cf70a19dc85d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.943ex; height:5.343ex;" alt="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{7}}+{\frac {1}{14}}+{\frac {1}{28}}=2}"></span></dd> <dd>496: <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+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{31}}+{\frac {1}{62}}+{\frac {1}{124}}+{\frac {1}{248}}+{\frac {1}{496}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>31</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>62</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>124</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>248</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>496</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{31}}+{\frac {1}{62}}+{\frac {1}{124}}+{\frac {1}{248}}+{\frac {1}{496}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5306334fbf2277f0561e8ccfada3237c950514a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.436ex; height:5.343ex;" alt="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{31}}+{\frac {1}{62}}+{\frac {1}{124}}+{\frac {1}{248}}+{\frac {1}{496}}=2}"></span></dd> <dd>8128: <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+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{127}}+{\frac {1}{254}}+{\frac {1}{508}}+{\frac {1}{1016}}+{\frac {1}{2032}}+{\frac {1}{4064}}+{\frac {1}{8128}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>127</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>254</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>508</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1016</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2032</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4064</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8128</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{127}}+{\frac {1}{254}}+{\frac {1}{508}}+{\frac {1}{1016}}+{\frac {1}{2032}}+{\frac {1}{4064}}+{\frac {1}{8128}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d031e8362be3a23b5373a830cc077ae3650847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:92.742ex; height:5.343ex;" alt="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{127}}+{\frac {1}{254}}+{\frac {1}{508}}+{\frac {1}{1016}}+{\frac {1}{2032}}+{\frac {1}{4064}}+{\frac {1}{8128}}=2}"></span></dd></dl> <p>Số 6 là số tự nhiên duy nhất có tổng các ước bằng tích các ước (không kể chính nó): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6=1+2+3=1\times 2\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=1+2+3=1\times 2\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1494b6dafb91e3046b3c5f0e312b15e09ed15f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.696ex; height:2.343ex;" alt="{\displaystyle 6=1+2+3=1\times 2\times 3}"></span></dd></dl> <p>Trừ số 6, mọi số hoàn hảo đều là tổng của 2<sup>(p−1)/2</sup> số lập phương lẻ liên tiếp từ 1<sup>3</sup> đến (2<sup>(p+1)/2</sup> − 1)<sup>3</sup>: </p> <dl><dd>p = 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 28=1^{3}+3^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28=1^{3}+3^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e39db82fb710db62e5b63858a7dc798ce3744ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.084ex; height:2.843ex;" alt="{\displaystyle 28=1^{3}+3^{3}\,}"></span></dd> <dd>p = 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 496=1^{3}+3^{3}+5^{3}+7^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496=1^{3}+3^{3}+5^{3}+7^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4291afc245485511a8dca5d6d22ed847b1f1868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.361ex; height:2.843ex;" alt="{\displaystyle 496=1^{3}+3^{3}+5^{3}+7^{3}\,}"></span></dd> <dd>p = 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 8128=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8128</mn> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8128=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7bb718475cc836a8d24c5695ac267b6ff73a25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:49.239ex; height:2.843ex;" alt="{\displaystyle 8128=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\,}"></span></dd></dl> <p>Trừ số 6, mọi số hoàn hảo khi chia 9 thì đều thu được thương là số tam giác thứ (2<sup>p</sup> − 2)/3 và <a href="/wiki/S%E1%BB%91_d%C6%B0" title="Số dư">số dư</a> là 1: </p> <dl><dd>p = 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 28=9\times (1+2)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>=</mo> <mn>9</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28=9\times (1+2)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e758167533f90f2fce12d2e13c0563fb0a9508e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.404ex; height:2.843ex;" alt="{\displaystyle 28=9\times (1+2)+1}"></span></dd> <dd>p = 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 496=9\times (1+2+3+...+8+9+10)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mo>=</mo> <mn>9</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mn>8</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496=9\times (1+2+3+...+8+9+10)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffb5892d5cf647dd3d613ef3ccc6a668a91f296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.618ex; height:2.843ex;" alt="{\displaystyle 496=9\times (1+2+3+...+8+9+10)+1}"></span></dd> <dd>p = 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 8128=9\times (1+2+3+...+40+41+42)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8128</mn> <mo>=</mo> <mn>9</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mn>40</mn> <mo>+</mo> <mn>41</mn> <mo>+</mo> <mn>42</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8128=9\times (1+2+3+...+40+41+42)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2147b37cb15a4e94c9f59c6bcbc24ca41e1527e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.105ex; height:2.843ex;" alt="{\displaystyle 8128=9\times (1+2+3+...+40+41+42)+1}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Số_hoàn_hảo_lẻ"><span id="S.E1.BB.91_ho.C3.A0n_h.E1.BA.A3o_l.E1.BA.BB"></span>Số hoàn hảo lẻ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=3" title="Sửa đổi phần “Số hoàn hảo lẻ”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&action=edit&section=3" title="Sửa mã nguồn tại đề mục: Số hoàn hảo lẻ"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right; width:30%; margin:0em 1em 1em 1em; background:#eee; border:#ccc solid"> <tbody><tr> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/20px-Question%2C_Web_Fundamentals.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/30px-Question%2C_Web_Fundamentals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/40px-Question%2C_Web_Fundamentals.svg.png 2x" data-file-width="44" data-file-height="44" /></span></span></td> <td><b>Vấn đề mở trong toán học</b>:<br /><div style="padding:5px; font-style:italic;">Liệu có tồn tại hay không số hoàn hảo lẻ?</div><small>(<a href="/wiki/Danh_s%C3%A1ch_v%E1%BA%A5n_%C4%91%E1%BB%81_m%E1%BB%9F_trong_to%C3%A1n_h%E1%BB%8Dc" title="Danh sách vấn đề mở trong toán học">các vấn đề mở khác trong toán học</a>)</small> </td></tr></tbody></table> <p>Hiện tại người ta vẫn chưa biết được liệu số hoàn hảo lẻ nào không mặc dù đã có nhiều kết quả nghiên cứu. Trong 1946, <a href="/w/index.php?title=Jacques_Lef%C3%A8vre_d%27%C3%89taples&action=edit&redlink=1" class="new" title="Jacques Lefèvre d'Étaples (trang không tồn tại)">Jacques Lefèvre</a> phát biểu rằng luật của Euclid cho mọi số hoàn hảo<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>, nghĩa là cho rằng không có số hoàn hảo lẻ nào tồn tại cả. Euler thì nói rằng: "Liệu ... có số hoàn hảo lẻ nào là câu hỏi rất khó có thể giải đáp".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Gần đây hơn, <a href="/w/index.php?title=Carl_Pomerance&action=edit&redlink=1" class="new" title="Carl Pomerance (trang không tồn tại)">Carl Pomerance</a> đã đưa ra <a href="/w/index.php?title=Tranh_lu%E1%BA%ADn_heuristic&action=edit&redlink=1" class="new" title="Tranh luận heuristic (trang không tồn tại)">tranh luận bằng heuristic</a> rằng quả thật không số hoàn hảo lẻ nào nên tồn tại <sup id="cite_ref-oddperfect_5-0" class="reference"><a href="#cite_note-oddperfect-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Tất cả các số hoàn hảo đều là <a href="/w/index.php?title=S%E1%BB%91_%C4%91i%E1%BB%81u_h%C3%B2a_%C6%B0%E1%BB%9Bc&action=edit&redlink=1" class="new" title="Số điều hòa ước (trang không tồn tại)">số điều hòa của Ore</a> và hiện tại người ta vẫn đang <a href="/wiki/Gi%E1%BA%A3_thuy%E1%BA%BFt_kh%C3%B4ng" title="Giả thuyết không">giả thuyết không</a> có số điều hòa lẻ nào ngoại trừ số 1. </p><p>Bất cứ số hoàn hảo lẻ <i>N</i> phải thỏa mãn các điều kiện sau: </p> <ul><li><i>N</i> > 10<sup>1500</sup>.<sup id="cite_ref-Ochem_and_Rao_(2012)_6-0" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i> không chia hết bởi 105.<sup id="cite_ref-Kühnel_U_7-0" class="reference"><a href="#cite_note-Kühnel_U-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i> dưới dạng <i>N</i> ≡ 1 (mod 12) hoặc <i>N</i> ≡ 117 (mod 468) hoặc <i>N</i> ≡ 81 (mod 324).<sup id="cite_ref-Roberts_T_(2008)_8-0" class="reference"><a href="#cite_note-Roberts_T_(2008)-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i> dưới dạng</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98db26132d3d93255a17f6459fdd7d1ee65f5c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.341ex; height:3.509ex;" alt="{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}"></span></dd></dl></dd> <dd>trong đó: <ul><li><i>q</i>, <i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>k</i></sub> là các số nguyên tố lẻ phân biệt (Euler).</li> <li><i>q</i> ≡ α ≡ 1 (<a href="/w/index.php?title=Modulo&action=edit&redlink=1" class="new" title="Modulo (trang không tồn tại)">mod</a> 4) (Euler).</li> <li>Ước nguyên tố lẻ nhỏ nhất của <i>N</i> nằm dưới <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {k-1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k-1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dbfc93b200572b06d94cfee046d68b1ab7e062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.697ex; height:5.343ex;" alt="{\displaystyle {\frac {k-1}{2}}.}"></span><sup id="cite_ref-Zelinsky_2021_9-0" class="reference"><a href="#cite_note-Zelinsky_2021-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><i>q</i><sup>α</sup> > 10<sup>62</sup>, hoặc <i>p</i><sub><i>j</i></sub><sup>2<i>e</i><sub><i>j</i></sub></sup>  > 10<sup>62</sup> với một vài giá trị <i>j</i>.<sup id="cite_ref-Ochem_and_Rao_(2012)_6-1" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N<2^{(4^{k+1}-2^{k+1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N<2^{(4^{k+1}-2^{k+1})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1058d91f37bc389a35ef129108a75159624d90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.888ex; height:3.009ex;" alt="{\displaystyle N<2^{(4^{k+1}-2^{k+1})}}"></span><sup id="cite_ref-Chen_and_Tang_10-0" class="reference"><a href="#cite_note-Chen_and_Tang-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nielsen_(2003)_11-0" class="reference"><a href="#cite_note-Nielsen_(2003)-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {66k-191}{25}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>66</mn> <mi>k</mi> <mo>−</mo> <mn>191</mn> </mrow> <mn>25</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {66k-191}{25}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a888def72f235b29ba5abfef13556dc34f1c446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.447ex; height:5.343ex;" alt="{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {66k-191}{25}}}"></span>.<sup id="cite_ref-Zelinsky_2021_9-1" class="reference"><a href="#cite_note-Zelinsky_2021-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ochem_and_Rao_(2014)_12-0" class="reference"><a href="#cite_note-Ochem_and_Rao_(2014)-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo><</mo> <mn>2</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>26</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27ccd1f9756752a6deb57c68a1f8b16fd2cb3bf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.283ex; height:4.009ex;" alt="{\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}"></span>.<sup id="cite_ref-LucaPomerance_13-0" class="reference"><a href="#cite_note-LucaPomerance-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <ul><li>Ước nguyên tố lớn nhất của <i>N</i> lớn hơn 10<sup>8</sup><sup id="cite_ref-Goto_and_Ohno_(2008)_14-0" class="reference"><a href="#cite_note-Goto_and_Ohno_(2008)-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> và nhỏ hơn <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 (3N)^{1/3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3N)^{1/3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8092c9897dbe6209909ea43606f02eb5d3217e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.38ex; height:3.343ex;" alt="{\displaystyle (3N)^{1/3}.}"></span> <sup id="cite_ref-AK_2012_15-0" class="reference"><a href="#cite_note-AK_2012-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li>Ước nguyên tố lớn thứ hai của <i>N</i> lớn hơn 10<sup>4</sup>,<sup id="cite_ref-Ianucci_DE_(1999)_16-0" class="reference"><a href="#cite_note-Ianucci_DE_(1999)-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> và nhỏ hơn <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 (2N)^{1/5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2N)^{1/5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b8933a16eb3dc62720ef691be189282c6cca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.734ex; height:3.343ex;" alt="{\displaystyle (2N)^{1/5}}"></span>.<sup id="cite_ref-Zelinsky_2019_17-0" class="reference"><a href="#cite_note-Zelinsky_2019-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li>Ước nguyên tố thứ ba lớn hơn 100,<sup id="cite_ref-Ianucci_DE_(2000)_18-0" class="reference"><a href="#cite_note-Ianucci_DE_(2000)-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> và nhỏ hơn <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 (2N)^{\frac {1}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2N)^{\frac {1}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c56baad8c9ca9645ebcd7ce9898a03450efb8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.418ex; height:4.009ex;" alt="{\displaystyle (2N)^{\frac {1}{6}}.}"></span><sup id="cite_ref-Zelinsky_2021a_19-0" class="reference"><a href="#cite_note-Zelinsky_2021a-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i> có ít nhất 101 ước nguyên tố và ít nhất 10 ước nguyên tố phân biệt.<sup id="cite_ref-Ochem_and_Rao_(2012)_6-2" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nielsen_Pace_P._(2015)_20-0" class="reference"><a href="#cite_note-Nielsen_Pace_P._(2015)-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Nếu 3 không phải là ước của <i>N</i>, thì <i>N</i> có ít nhất 12 ước nguyên tố phân biệt.<sup id="cite_ref-Nielsen_Pace_P._(2007)_21-0" class="reference"><a href="#cite_note-Nielsen_Pace_P._(2007)-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Xem_thêm"><span id="Xem_th.C3.AAm"></span>Xem thêm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=4" title="Sửa đổi phần “Xem thêm”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&action=edit&section=4" title="Sửa mã nguồn tại đề mục: Xem thêm"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Danh_s%C3%A1ch_s%E1%BB%91_nguy%C3%AAn_t%E1%BB%91_Mersenne_v%C3%A0_s%E1%BB%91_ho%C3%A0n_h%E1%BA%A3o" title="Danh sách số nguyên tố Mersenne và số hoàn hảo">Danh sách số nguyên tố Mersenne và số hoàn hảo</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ghi_chú"><span id="Ghi_ch.C3.BA"></span>Ghi chú</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=5" title="Sửa đổi phần “Ghi 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a{background:linear-gradient(transparent,transparent),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:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite class="citation cs2"><a href="/w/index.php?title=John_J._O%27Connor_(nh%C3%A0_to%C3%A1n_h%E1%BB%8Dc)&action=edit&redlink=1" class="new" title="John J. O'Connor (nhà toán học) (trang không tồn tại)">O'Connor, John J.</a>; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Haytham.html">“Abu Ali al-Hasan ibn al-Haytham”</a>, <i><a href="/wiki/B%E1%BB%99_l%C6%B0u_tr%E1%BB%AF_l%E1%BB%8Bch_s%E1%BB%AD_to%C3%A1n_h%E1%BB%8Dc_MacTutor" title="Bộ lưu trữ lịch sử toán học MacTutor">Bộ lưu trữ lịch sử toán học MacTutor</a></i>, <a href="/wiki/%C4%90%E1%BA%A1i_h%E1%BB%8Dc_St._Andrews" title="Đại học St. Andrews">Đại học St. Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abu+Ali+al-Hasan+ibn+al-Haytham&rft.btitle=B%E1%BB%99+l%C6%B0u+tr%E1%BB%AF+l%E1%BB%8Bch+s%E1%BB%AD+to%C3%A1n+h%E1%BB%8Dc+MacTutor&rft.pub=%C4%90%E1%BA%A1i+h%E1%BB%8Dc+St.+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FAl-Haytham.html&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mersenne.org/">“Great Internet Mersenne Prime Search”</a><span class="reference-accessdate">. Truy cập 7 tháng 10 năm 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Great+Internet+Mersenne+Prime+Search&rft_id=http%3A%2F%2Fwww.mersenne.org%2F&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFDickson1919" class="citation book cs1"><a href="/w/index.php?title=L._E._Dickson&action=edit&redlink=1" class="new" title="L. E. Dickson (trang không tồn tại)">Dickson, L. E.</a> (1919). <a rel="nofollow" class="external text" href="https://archive.org/stream/historyoftheoryo01dick#page/6/"><i>History of the Theory of Numbers, Vol. I</i></a>. Washington: Carnegie Institution of Washington. tr. 6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+the+Theory+of+Numbers%2C+Vol.+I&rft.place=Washington&rft.pages=6&rft.pub=Carnegie+Institution+of+Washington&rft.date=1919&rft.aulast=Dickson&rft.aufirst=L.+E.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fhistoryoftheoryo01dick%23page%2F6%2F&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf">http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/w/index.php?title=Wikipedia:Bare_URLs&action=edit&redlink=1" class="new" title="Wikipedia:Bare URLs (trang không tồn tại)"><span title="A full citation of this PDF document is required to prevent link rot. (March 2022)">liên kết URL chỉ có mỗi PDF</span></a></i>]</sup></span> </li> <li id="cite_note-oddperfect-5"><b><a href="#cite_ref-oddperfect_5-0">^</a></b> <span class="reference-text"><a rel="nofollow" class="external text" href="http://oddperfect.org/pomerance.html">Oddperfect.org</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061229094011/http://oddperfect.org/pomerance.html">Lưu trữ</a> 2006-12-29 tại <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-Ochem_and_Rao_(2012)-6">^ <a href="#cite_ref-Ochem_and_Rao_(2012)_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ochem_and_Rao_(2012)_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Ochem_and_Rao_(2012)_6-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFOchemRao2012" class="citation journal cs1">Ochem, Pascal; Rao, Michaël (2012). <a rel="nofollow" class="external text" href="http://www.lirmm.fr/~ochem/opn/opn.pdf">“Odd perfect numbers are greater than 10<sup>1500</sup>”</a> <span class="cs1-format">(PDF)</span>. <i><a href="/w/index.php?title=Mathematics_of_Computation&action=edit&redlink=1" class="new" title="Mathematics of Computation (trang không tồn tại)">Mathematics of Computation</a></i>. <b>81</b> (279): 1869–1877. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2012-02563-4">10.1090/S0025-5718-2012-02563-4</a></span>. <a href="/wiki/M%C3%A3_s%E1%BB%91_ti%C3%AAu_chu%E1%BA%A9n_qu%E1%BB%91c_t%E1%BA%BF_cho_xu%E1%BA%A5t_b%E1%BA%A3n_ph%E1%BA%A9m_nhi%E1%BB%81u_k%E1%BB%B3" class="mw-redirect" title="Mã số tiêu chuẩn quốc tế cho xuất bản phẩm nhiều kỳ">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0025-5718">0025-5718</a>. <a href="/w/index.php?title=Zbl_(%C4%91%E1%BB%8Bnh_danh)&action=edit&redlink=1" class="new" title="Zbl (định danh) (trang không tồn tại)">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1263.11005">1263.11005</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Odd+perfect+numbers+are+greater+than+10%3Csup%3E1500%3C%2Fsup%3E&rft.volume=81&rft.issue=279&rft.pages=1869-1877&rft.date=2012&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1263.11005%23id-name%3DZbl&rft.issn=0025-5718&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-2012-02563-4&rft.aulast=Ochem&rft.aufirst=Pascal&rft.au=Rao%2C+Micha%C3%ABl&rft_id=http%3A%2F%2Fwww.lirmm.fr%2F~ochem%2Fopn%2Fopn.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Kühnel_U-7"><b><a href="#cite_ref-Kühnel_U_7-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFKühnel1950" class="citation journal cs1">Kühnel, Ullrich (1950). “Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen”. <i>Mathematische Zeitschrift</i> (bằng tiếng Đức). <b>52</b>: 202–211. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02230691">10.1007/BF02230691</a>. <a href="/wiki/S2CID_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="S2CID (định danh)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120754476">120754476</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Versch%C3%A4rfung+der+notwendigen+Bedingungen+f%C3%BCr+die+Existenz+von+ungeraden+vollkommenen+Zahlen&rft.volume=52&rft.pages=202-211&rft.date=1950&rft_id=info%3Adoi%2F10.1007%2FBF02230691&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120754476%23id-name%3DS2CID&rft.aulast=K%C3%BChnel&rft.aufirst=Ullrich&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Roberts_T_(2008)-8"><b><a href="#cite_ref-Roberts_T_(2008)_8-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFRoberts2008" class="citation journal cs1">Roberts, T (2008). <a rel="nofollow" class="external text" href="http://www.austms.org.au/Publ/Gazette/2008/Sep08/CommsRoberts.pdf">“On the Form of an Odd Perfect Number”</a> <span class="cs1-format">(PDF)</span>. <i>Australian Mathematical Gazette</i>. <b>35</b> (4): 244.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Australian+Mathematical+Gazette&rft.atitle=On+the+Form+of+an+Odd+Perfect+Number&rft.volume=35&rft.issue=4&rft.pages=244&rft.date=2008&rft.aulast=Roberts&rft.aufirst=T&rft_id=http%3A%2F%2Fwww.austms.org.au%2FPubl%2FGazette%2F2008%2FSep08%2FCommsRoberts.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Zelinsky_2021-9">^ <a href="#cite_ref-Zelinsky_2021_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zelinsky_2021_9-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFZelinsky2021" class="citation journal cs1">Zelinsky, Joshua (3 tháng 8 năm 2021). <a rel="nofollow" class="external text" href="http://math.colgate.edu/~integers/v76/v76.pdf">“On the Total Number of Prime Factors of an Odd Perfect Number”</a> <span class="cs1-format">(PDF)</span>. <i>Integers</i>. <b>21</b><span class="reference-accessdate">. Truy cập ngày 7 tháng 8 năm 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=On+the+Total+Number+of+Prime+Factors+of+an+Odd+Perfect+Number&rft.volume=21&rft.date=2021-08-03&rft.aulast=Zelinsky&rft.aufirst=Joshua&rft_id=http%3A%2F%2Fmath.colgate.edu%2F~integers%2Fv76%2Fv76.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Chen_and_Tang-10"><b><a href="#cite_ref-Chen_and_Tang_10-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFChenTang2014" class="citation journal cs1">Chen, Yong-Gao; Tang, Cui-E (2014). “Improved upper bounds for odd multiperfect numbers”. <i>Bulletin of the Australian Mathematical Society</i>. <b>89</b> (3): 353–359. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0004972713000488">10.1017/S0004972713000488</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Australian+Mathematical+Society&rft.atitle=Improved+upper+bounds+for+odd+multiperfect+numbers.&rft.volume=89&rft.issue=3&rft.pages=353-359&rft.date=2014&rft_id=info%3Adoi%2F10.1017%2FS0004972713000488&rft.aulast=Chen&rft.aufirst=Yong-Gao&rft.au=Tang%2C+Cui-E&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Nielsen_(2003)-11"><b><a href="#cite_ref-Nielsen_(2003)_11-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFNielsen2003" class="citation journal cs1">Nielsen, Pace P. (2003). <a rel="nofollow" class="external text" href="http://www.westga.edu/~integers/vol3.html">“An upper bound for odd perfect numbers”</a>. <i>Integers</i>. <b>3</b>: A14–A22<span class="reference-accessdate">. Truy cập ngày 23 tháng 3 năm 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=An+upper+bound+for+odd+perfect+numbers&rft.volume=3&rft.pages=A14-A22&rft.date=2003&rft.aulast=Nielsen&rft.aufirst=Pace+P.&rft_id=http%3A%2F%2Fwww.westga.edu%2F~integers%2Fvol3.html&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Ochem_and_Rao_(2014)-12"><b><a href="#cite_ref-Ochem_and_Rao_(2014)_12-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFOchemRao2014" class="citation journal cs1">Ochem, Pascal; Rao, Michaël (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2013-02776-7">“On the number of prime factors of an odd perfect number”</a>. <i><a href="/w/index.php?title=Mathematics_of_Computation&action=edit&redlink=1" class="new" title="Mathematics of Computation (trang không tồn tại)">Mathematics of Computation</a></i>. <b>83</b> (289): 2435–2439. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2013-02776-7">10.1090/S0025-5718-2013-02776-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=On+the+number+of+prime+factors+of+an+odd+perfect+number.&rft.volume=83&rft.issue=289&rft.pages=2435-2439&rft.date=2014&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-2013-02776-7&rft.aulast=Ochem&rft.aufirst=Pascal&rft.au=Rao%2C+Micha%C3%ABl&rft_id=%2F%2Fdoi.org%2F10.1090%252FS0025-5718-2013-02776-7&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-LucaPomerance-13"><b><a href="#cite_ref-LucaPomerance_13-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFPomeranceLuca2010" class="citation journal cs1">Pomerance, Carl; Luca, Florian (2010). <a rel="nofollow" class="external text" href="http://nyjm.albany.edu/j/2010/16-3.html">“On the radical of a perfect number”</a>. <i>New York Journal of Mathematics</i>. <b>16</b>: 23–30<span class="reference-accessdate">. Truy cập ngày 7 tháng 12 năm 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+York+Journal+of+Mathematics&rft.atitle=On+the+radical+of+a+perfect+number&rft.volume=16&rft.pages=23-30&rft.date=2010&rft.aulast=Pomerance&rft.aufirst=Carl&rft.au=Luca%2C+Florian&rft_id=http%3A%2F%2Fnyjm.albany.edu%2Fj%2F2010%2F16-3.html&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Goto_and_Ohno_(2008)-14"><b><a href="#cite_ref-Goto_and_Ohno_(2008)_14-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFGotoOhno,_Y2008" class="citation journal cs1">Goto, T; Ohno, Y (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110807101906/http://www.ma.noda.tus.ac.jp/u/tg/perfect/perfect.pdf">“Odd perfect numbers have a prime factor exceeding 10<sup>8</sup>”</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>77</b> (263): 1859–1868. <a href="/wiki/Bibcode_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="Bibcode (định danh)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008MaCom..77.1859G">2008MaCom..77.1859G</a>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-08-02050-9">10.1090/S0025-5718-08-02050-9</a></span>. <a rel="nofollow" class="external text" href="http://www.ma.noda.tus.ac.jp/u/tg/perfect/perfect.pdf">Bản gốc</a> <span class="cs1-format">(PDF)</span> lưu trữ ngày 7 tháng 8 năm 2011<span class="reference-accessdate">. Truy cập ngày 30 tháng 3 năm 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Odd+perfect+numbers+have+a+prime+factor+exceeding+10%3Csup%3E8%3C%2Fsup%3E&rft.volume=77&rft.issue=263&rft.pages=1859-1868&rft.date=2008&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-08-02050-9&rft_id=info%3Abibcode%2F2008MaCom..77.1859G&rft.aulast=Goto&rft.aufirst=T&rft.au=Ohno%2C+Y&rft_id=http%3A%2F%2Fwww.ma.noda.tus.ac.jp%2Fu%2Ftg%2Fperfect%2Fperfect.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-AK_2012-15"><b><a href="#cite_ref-AK_2012_15-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFKonyaginAcquaah2012" class="citation journal cs1">Konyagin, Sergei; Acquaah, Peter (2012). “On Prime Factors of Odd Perfect Numbers”. <i>International Journal of Number Theory</i>. <b>8</b> (6): 1537–1540. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS1793042112500935">10.1142/S1793042112500935</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Number+Theory&rft.atitle=On+Prime+Factors+of+Odd+Perfect+Numbers&rft.volume=8&rft.issue=6&rft.pages=1537-1540&rft.date=2012&rft_id=info%3Adoi%2F10.1142%2FS1793042112500935&rft.aulast=Konyagin&rft.aufirst=Sergei&rft.au=Acquaah%2C+Peter&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Ianucci_DE_(1999)-16"><b><a href="#cite_ref-Ianucci_DE_(1999)_16-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFIannucci1999" class="citation journal cs1">Iannucci, DE (1999). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1999-68-228/S0025-5718-99-01126-6/S0025-5718-99-01126-6.pdf">“The second largest prime divisor of an odd perfect number exceeds ten thousand”</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>68</b> (228): 1749–1760. <a href="/wiki/Bibcode_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="Bibcode (định danh)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999MaCom..68.1749I">1999MaCom..68.1749I</a>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-99-01126-6">10.1090/S0025-5718-99-01126-6</a></span><span class="reference-accessdate">. Truy cập ngày 30 tháng 3 năm 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=The+second+largest+prime+divisor+of+an+odd+perfect+number+exceeds+ten+thousand&rft.volume=68&rft.issue=228&rft.pages=1749-1760&rft.date=1999&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-99-01126-6&rft_id=info%3Abibcode%2F1999MaCom..68.1749I&rft.aulast=Iannucci&rft.aufirst=DE&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1999-68-228%2FS0025-5718-99-01126-6%2FS0025-5718-99-01126-6.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Zelinsky_2019-17"><b><a href="#cite_ref-Zelinsky_2019_17-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFZelinsky2019" class="citation journal cs1">Zelinsky, Joshua (tháng 7 năm 2019). “Upper bounds on the second largest prime factor of an odd perfect number”. <i>International Journal of Number Theory</i>. <b>15</b> (6): 1183–1189. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1810.11734">1810.11734</a></span>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS1793042119500659">10.1142/S1793042119500659</a>. <a href="/wiki/S2CID_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="S2CID (định danh)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62885986">62885986</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Number+Theory&rft.atitle=Upper+bounds+on+the+second+largest+prime+factor+of+an+odd+perfect+number&rft.volume=15&rft.issue=6&rft.pages=1183-1189&rft.date=2019-07&rft_id=info%3Aarxiv%2F1810.11734&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62885986%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS1793042119500659&rft.aulast=Zelinsky&rft.aufirst=Joshua&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span>.</span> </li> <li id="cite_note-Ianucci_DE_(2000)-18"><b><a href="#cite_ref-Ianucci_DE_(2000)_18-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFIannucci2000" class="citation journal cs1">Iannucci, DE (2000). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01127-8/S0025-5718-99-01127-8.pdf">“The third largest prime divisor of an odd perfect number exceeds one hundred”</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>69</b> (230): 867–879. <a href="/wiki/Bibcode_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="Bibcode (định danh)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000MaCom..69..867I">2000MaCom..69..867I</a>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-99-01127-8">10.1090/S0025-5718-99-01127-8</a></span><span class="reference-accessdate">. Truy cập ngày 30 tháng 3 năm 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=The+third+largest+prime+divisor+of+an+odd+perfect+number+exceeds+one+hundred&rft.volume=69&rft.issue=230&rft.pages=867-879&rft.date=2000&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-99-01127-8&rft_id=info%3Abibcode%2F2000MaCom..69..867I&rft.aulast=Iannucci&rft.aufirst=DE&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F2000-69-230%2FS0025-5718-99-01127-8%2FS0025-5718-99-01127-8.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Zelinsky_2021a-19"><b><a href="#cite_ref-Zelinsky_2021a_19-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFBibbyVynckeZelinsky2021" class="citation journal cs1">Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23 tháng 11 năm 2021). <a rel="nofollow" class="external text" href="http://math.colgate.edu/~integers/v115/v115.pdf">“On the Third Largest Prime Divisor of an Odd Perfect Number”</a> <span class="cs1-format">(PDF)</span>. <i>Integers</i>. <b>21</b><span class="reference-accessdate">. Truy cập ngày 6 tháng 12 năm 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=On+the+Third+Largest+Prime+Divisor+of+an+Odd+Perfect+Number&rft.volume=21&rft.date=2021-11-23&rft.aulast=Bibby&rft.aufirst=Sean&rft.au=Vyncke%2C+Pieter&rft.au=Zelinsky%2C+Joshua&rft_id=http%3A%2F%2Fmath.colgate.edu%2F~integers%2Fv115%2Fv115.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Nielsen_Pace_P._(2015)-20"><b><a href="#cite_ref-Nielsen_Pace_P._(2015)_20-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFNielsen2015" class="citation journal cs1">Nielsen, Pace P. (2015). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150708185554/https://math.byu.edu/~pace/BestBound_web.pdf">“Odd perfect numbers, Diophantine equations, and upper bounds”</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>84</b> (295): 2549–2567. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2015-02941-X">10.1090/S0025-5718-2015-02941-X</a></span>. <a rel="nofollow" class="external text" href="https://math.byu.edu/~pace/BestBound_web.pdf">Bản gốc</a> <span class="cs1-format">(PDF)</span> lưu trữ ngày 8 tháng 7 năm 2015<span class="reference-accessdate">. Truy cập ngày 13 tháng 8 năm 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Odd+perfect+numbers%2C+Diophantine+equations%2C+and+upper+bounds&rft.volume=84&rft.issue=295&rft.pages=2549-2567&rft.date=2015&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-2015-02941-X&rft.aulast=Nielsen&rft.aufirst=Pace+P.&rft_id=https%3A%2F%2Fmath.byu.edu%2F~pace%2FBestBound_web.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> <li id="cite_note-Nielsen_Pace_P._(2007)-21"><b><a href="#cite_ref-Nielsen_Pace_P._(2007)_21-0">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFNielsen2007" class="citation journal cs1">Nielsen, Pace P. (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211103100630/https://math.byu.edu/~pace/NotEight_web.pdf">“Odd perfect numbers have at least nine distinct prime factors”</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>76</b> (260): 2109–2126. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="//arxiv.org/abs/math/0602485">math/0602485</a></span>. <a href="/wiki/Bibcode_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="Bibcode (định danh)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007MaCom..76.2109N">2007MaCom..76.2109N</a>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-07-01990-4">10.1090/S0025-5718-07-01990-4</a>. <a href="/wiki/S2CID_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="S2CID (định danh)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2767519">2767519</a>. <a rel="nofollow" class="external text" href="https://math.byu.edu/~pace/NotEight_web.pdf">Bản gốc</a> <span class="cs1-format">(PDF)</span> lưu trữ ngày 3 tháng 11 năm 2021<span class="reference-accessdate">. Truy cập ngày 30 tháng 3 năm 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Odd+perfect+numbers+have+at+least+nine+distinct+prime+factors&rft.volume=76&rft.issue=260&rft.pages=2109-2126&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0602485&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2767519%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-07-01990-4&rft_id=info%3Abibcode%2F2007MaCom..76.2109N&rft.aulast=Nielsen&rft.aufirst=Pace+P.&rft_id=https%3A%2F%2Fmath.byu.edu%2F~pace%2FNotEight_web.pdf&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+ho%C3%A0n+thi%E1%BB%87n" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Liên_kết_ngoài"><span id="Li.C3.AAn_k.E1.BA.BFt_ngo.C3.A0i"></span>Liên kết ngoài</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&veaction=edit&section=6" title="Sửa đổi phần “Liên kết ngoài”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n&action=edit&section=6" title="Sửa mã nguồn tại đề mục: Liên kết ngoài"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>David Moews: <a rel="nofollow" class="external text" href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a></li> <li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html">Perfect numbers - History and Theory</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-perfect_number"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PerfectNumber.html">"perfect number"</a>, <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</span></li> <li><a rel="nofollow" class="external text" href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000396">List of Perfect Numbers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20010715150601/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000396">Lưu trữ</a> 2001-07-15 tại <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at the On-Line Encyclopedia of Integer Sequences</li> <li><a rel="nofollow" class="external text" href="http://amicable.homepage.dk/perfect.htm">List of known Perfect Numbers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090503154707/http://amicable.homepage.dk/perfect.htm">Lưu trữ</a> 2009-05-03 tại <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> All known perfect numbers are here.</li> <li><a rel="nofollow" class="external text" href="http://www.oddperfect.org">OddPerfect.org</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181106015226/http://oddperfect.org/">Lưu trữ</a> 2018-11-06 tại <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> A projected distributed computing project to search for odd perfect numbers.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Lấy từ “<a dir="ltr" href="https://vi.wikipedia.org/w/index.php?title=Số_hoàn_thiện&oldid=71527840">https://vi.wikipedia.org/w/index.php?title=Số_hoàn_thiện&oldid=71527840</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BB%83_lo%E1%BA%A1i" title="Đặc biệt:Thể loại">Thể loại</a>: <ul><li><a href="/w/index.php?title=Th%E1%BB%83_lo%E1%BA%A1i:Chu%E1%BB%97i_s%E1%BB%91_t%E1%BB%B1_nhi%C3%AAn&action=edit&redlink=1" class="new" title="Thể loại:Chuỗi số tự nhiên (trang không tồn tại)">Chuỗi số tự nhiên</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:H%C3%A0m_%C6%B0%E1%BB%9Bc_s%E1%BB%91" title="Thể loại:Hàm ước số">Hàm ước số</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:V%E1%BA%A5n_%C4%91%E1%BB%81_ch%C6%B0a_%C4%91%C6%B0%E1%BB%A3c_gi%E1%BA%A3i_quy%E1%BA%BFt_trong_to%C3%A1n_h%E1%BB%8Dc" title="Thể loại:Vấn đề chưa được giải quyết trong toán học">Vấn đề chưa được giải quyết trong toán học</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:V%E1%BA%A5n_%C4%91%E1%BB%81_m%E1%BB%9F_trong_l%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91" title="Thể loại:Vấn đề mở trong lý thuyết số">Vấn đề mở trong lý thuyết số</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:Chu%E1%BB%97i_s%E1%BB%91_nguy%C3%AAn" title="Thể loại:Chuỗi số nguyên">Chuỗi số nguyên</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Thể loại ẩn: <ul><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:Articles_with_bare_URLs_for_citations" title="Thể loại:Articles with bare URLs for citations">Articles with bare URLs for citations</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:B%E1%BA%A3n_m%E1%BA%ABu_webarchive_d%C3%B9ng_li%C3%AAn_k%E1%BA%BFt_wayback" title="Thể loại:Bản mẫu webarchive dùng liên kết wayback">Bản mẫu webarchive dùng liên kết wayback</a></li><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:Ngu%E1%BB%93n_CS1_ti%E1%BA%BFng_%C4%90%E1%BB%A9c_(de)" title="Thể loại:Nguồn CS1 tiếng Đức (de)">Nguồn CS1 tiếng Đức (de)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Trang này được sửa đổi lần cuối vào ngày 15 tháng 7 năm 2024, 10:49.</li> <li id="footer-info-copyright">Văn bản được phát hành theo <a href="/wiki/Wikipedia:Nguy%C3%AAn_v%C4%83n_Gi%E1%BA%A5y_ph%C3%A9p_Creative_Commons_Ghi_c%C3%B4ng%E2%80%93Chia_s%E1%BA%BB_t%C6%B0%C6%A1ng_t%E1%BB%B1_phi%C3%AAn_b%E1%BA%A3n_4.0_Qu%E1%BB%91c_t%E1%BA%BF" title="Wikipedia:Nguyên văn Giấy phép Creative Commons Ghi công–Chia sẻ tương tự phiên bản 4.0 Quốc tế">Giấy phép Creative Commons Ghi công–Chia sẻ tương tự</a>; có thể áp dụng điều khoản bổ sung. 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