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class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko" class=""><span>기부</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%99%84%EC%A0%84%EC%88%98" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%99%84%EC%A0%84%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%99%84%EC%A0%84%EC%88%98" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%99%84%EC%A0%84%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-짝수_완전수" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#짝수_완전수"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>짝수 완전수</span> </div> </a> <ul id="toc-짝수_완전수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-홀수_완전수" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#홀수_완전수"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>홀수 완전수</span> </div> </a> <ul id="toc-홀수_완전수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>각주</span> </div> </a> <ul id="toc-각주-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="목차" 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="목차 토글" > <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">목차 토글</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">완전수</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="다른 언어로 문서를 방문합니다. 64개 언어로 읽을 수 있습니다" > <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">64개 언어</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 – 아랍어" lang="ar" hreflang="ar" data-title="عدد تام" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D1%81%D0%BA%D0%B0%D0%BD%D0%B0%D0%BB%D1%8B_%D0%BB%D1%96%D0%BA" title="Дасканалы лік – 벨라루스어" lang="be" hreflang="be" data-title="Дасканалы лік" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</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 – 불가리아어" lang="bg" hreflang="bg" data-title="Съвършено число" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%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 – 벵골어" lang="bn" hreflang="bn" data-title="নিখুঁত সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_peurvat" title="Niver peurvat – 브르타뉴어" lang="br" hreflang="br" data-title="Niver peurvat" data-language-autonym="Brezhoneg" data-language-local-name="브르타뉴어" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_perfecte" title="Nombre perfecte – 카탈로니아어" lang="ca" hreflang="ca" data-title="Nombre perfecte" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%A9%D8%A7%D9%85%DA%B5" title="ژمارەی کامڵ – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="ژمارەی کامڵ" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dokonal%C3%A9_%C4%8D%C3%ADslo" title="Dokonalé číslo – 체코어" lang="cs" hreflang="cs" data-title="Dokonalé číslo" data-language-autonym="Čeština" data-language-local-name="체코어" 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 – 덴마크어" lang="da" hreflang="da" data-title="Fuldkomne tal" data-language-autonym="Dansk" data-language-local-name="덴마크어" 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 – 독일어" lang="de" hreflang="de" data-title="Vollkommene Zahl" data-language-autonym="Deutsch" data-language-local-name="독일어" 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 – 그리스어" lang="el" hreflang="el" data-title="Τέλειος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" 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 – 영어" lang="en" hreflang="en" data-title="Perfect number" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Perfekta_nombro" title="Perfekta nombro – 에스페란토어" lang="eo" hreflang="eo" data-title="Perfekta nombro" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_perfecto" title="Número perfecto – 스페인어" lang="es" hreflang="es" data-title="Número perfecto" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_perfektu" title="Zenbaki perfektu – 바스크어" lang="eu" hreflang="eu" data-title="Zenbaki perfektu" data-language-autonym="Euskara" data-language-local-name="바스크어" 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 – 페르시아어" lang="fa" hreflang="fa" data-title="عدد تام" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/T%C3%A4ydellinen_luku" title="Täydellinen luku – 핀란드어" lang="fi" hreflang="fi" data-title="Täydellinen luku" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/T%C3%A4vvelidseq_arvoq" title="Tävvelidseq arvoq – Võro" lang="vro" hreflang="vro" data-title="Tävvelidseq arvoq" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_parfait" title="Nombre parfait – 프랑스어" lang="fr" hreflang="fr" data-title="Nombre parfait" data-language-autonym="Français" data-language-local-name="프랑스어" 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 – 아일랜드어" lang="ga" hreflang="ga" data-title="Uimhir fhoirfe" data-language-autonym="Gaeilge" data-language-local-name="아일랜드어" 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 – 갈리시아어" lang="gl" hreflang="gl" data-title="Número perfecto" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מספר משוכלל – 히브리어" lang="he" hreflang="he" data-title="מספר משוכלל" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6k%C3%A9letes_sz%C3%A1mok" title="Tökéletes számok – 헝가리어" lang="hu" hreflang="hu" data-title="Tökéletes számok" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BF%D5%A1%D6%80%D5%B5%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Կատարյալ թիվ – 아르메니아어" lang="hy" hreflang="hy" data-title="Կատարյալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" 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 – 인터링구아" lang="ia" hreflang="ia" data-title="Numero perfecte" data-language-autonym="Interlingua" data-language-local-name="인터링구아" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_sempurna" title="Bilangan sempurna – 인도네시아어" lang="id" hreflang="id" data-title="Bilangan sempurna" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fullkomin_tala" title="Fullkomin tala – 아이슬란드어" lang="is" hreflang="is" data-title="Fullkomin tala" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" 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 – 이탈리아어" lang="it" hreflang="it" data-title="Numero perfetto" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – 일본어" lang="ja" hreflang="ja" data-title="完全数" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</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 – 쿠르드어" lang="ku" hreflang="ku" data-title="Hejmarên nuwaze" data-language-autonym="Kurdî" data-language-local-name="쿠르드어" 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 – 라틴어" lang="la" hreflang="la" data-title="Numerus perfectus" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_parfett" title="Nümar parfett – Lombard" lang="lmo" hreflang="lmo" data-title="Nümar parfett" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tobulasis_skai%C4%8Dius" title="Tobulasis skaičius – 리투아니아어" lang="lt" hreflang="lt" data-title="Tobulasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BE%D0%B2%D1%80%D1%88%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Совршен број – 마케도니아어" lang="mk" hreflang="mk" data-title="Совршен број" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Nummero_perfetto" title="Nummero perfetto – 나폴리어" lang="nap" hreflang="nap" data-title="Nummero perfetto" data-language-autonym="Napulitano" data-language-local-name="나폴리어" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Perfect_getal" title="Perfect getal – 네덜란드어" lang="nl" hreflang="nl" data-title="Perfect getal" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fulkomne_tal" title="Fulkomne tal – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Fulkomne tal" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Perfekt_tall" title="Perfekt tall – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Perfekt tall" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_doskona%C5%82e" title="Liczby doskonałe – 폴란드어" lang="pl" hreflang="pl" data-title="Liczby doskonałe" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_p%C3%ABrfet" title="Nùmer përfet – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer përfet" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_perfeito" title="Número perfeito – 포르투갈어" lang="pt" hreflang="pt" data-title="Número perfeito" data-language-autonym="Português" data-language-local-name="포르투갈어" 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 – 루마니아어" lang="ro" hreflang="ro" data-title="Număr perfect" data-language-autonym="Română" data-language-local-name="루마니아어" 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 – 러시아어" lang="ru" hreflang="ru" data-title="Совершенное число" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_pirfettu" title="Nùmmuru pirfettu – 시칠리아어" lang="scn" hreflang="scn" data-title="Nùmmuru pirfettu" data-language-autonym="Sicilianu" data-language-local-name="시칠리아어" 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 – 슬로바키아어" lang="sk" hreflang="sk" data-title="Dokonalé číslo" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" 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 – 슬로베니아어" lang="sl" hreflang="sl" data-title="Popolno število" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_p%C3%ABrsosur" title="Numrat e përsosur – 알바니아어" lang="sq" hreflang="sq" data-title="Numrat e përsosur" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B0%D0%B2%D1%80%D1%88%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Савршен број – 세르비아어" lang="sr" hreflang="sr" data-title="Савршен број" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Perfekt_tal" title="Perfekt tal – 스웨덴어" lang="sv" hreflang="sv" data-title="Perfekt tal" data-language-autonym="Svenska" data-language-local-name="스웨덴어" 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) – 타밀어" lang="ta" hreflang="ta" data-title="நிறைவெண் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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 – 텔루구어" lang="te" hreflang="te" data-title="పరిపూర్ణసంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="텔루구어" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%BC%D1%83%D0%BA%D0%B0%D0%BC%D0%BC%D0%B0%D0%BB" title="Адади мукаммал – 타지크어" lang="tg" hreflang="tg" data-title="Адади мукаммал" data-language-autonym="Тоҷикӣ" data-language-local-name="타지크어" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AA%E0%B8%A1%E0%B8%9A%E0%B8%B9%E0%B8%A3%E0%B8%93%E0%B9%8C" title="จำนวนสมบูรณ์ – 태국어" lang="th" hreflang="th" data-title="จำนวนสมบูรณ์" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/M%C3%BCkemmel_say%C4%B1" title="Mükemmel sayı – 터키어" lang="tr" hreflang="tr" data-title="Mükemmel sayı" data-language-autonym="Türkçe" data-language-local-name="터키어" 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 – 우크라이나어" lang="uk" hreflang="uk" data-title="Досконале число" data-language-autonym="Українська" data-language-local-name="우크라이나어" 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 – 우르두어" lang="ur" hreflang="ur" data-title="کامل عدد" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mukammal_son" title="Mukammal son – 우즈베크어" lang="uz" hreflang="uz" data-title="Mukammal son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ho%C3%A0n_thi%E1%BB%87n" title="Số hoàn thiện – 베트남어" lang="vi" hreflang="vi" data-title="Số hoàn thiện" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – 우어" lang="wuu" hreflang="wuu" data-title="完全数" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数 – 중국어" lang="zh" hreflang="zh" data-title="完全数" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</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" 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</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">위키백과, 우리 모두의 백과사전.</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="ko" dir="ltr"><p><span class="nowrap"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Perfect_number_Cuisenaire_rods_6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Perfect_number_Cuisenaire_rods_6.png/220px-Perfect_number_Cuisenaire_rods_6.png" decoding="async" width="220" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Perfect_number_Cuisenaire_rods_6.png/330px-Perfect_number_Cuisenaire_rods_6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/66/Perfect_number_Cuisenaire_rods_6.png 2x" data-file-width="437" data-file-height="345" /></a><figcaption></figcaption></figure> <p><a href="/wiki/%EC%88%98%EB%A1%A0" title="수론">수론</a>에서 <b>완전수</b>(完全數)는 자기 자신을 제외한 양의 <a href="/wiki/%EC%95%BD%EC%88%98" title="약수">약수</a>(진약수)를 더했을 때 자기 자신이 되는 <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">양의 정수</a>를 말한다. 또는 모든 양의 약수를 더했을 때 자기 자신의 2배가 되는 수를 말하기도 한다. </p><p>최초 6개의 완전수는 <a href="/wiki/0" title="0">0</a>, <a href="/wiki/6" title="6">6</a>, <a href="/wiki/28" title="28">28</a>, <a href="/wiki/496" title="496">496</a>, <a href="/wiki/8128" title="8128">8128</a>, 33550336이다. </p> <pre> 0 = 0 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 </pre> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="짝수_완전수"><span id=".EC.A7.9D.EC.88.98_.EC.99.84.EC.A0.84.EC.88.98"></span>짝수 완전수</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EC%A0%84%EC%88%98&action=edit&section=1" title="부분 편집: 짝수 완전수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>고대 그리스인들은 이들 네 개의 완전수밖에는 알지 못했다. <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C" class="mw-redirect" title="유클리드">유클리드</a>는 이들을 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n-1}\cdot (2^{n}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}\cdot (2^{n}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56add7d8caed550996c464f9ea844aed14c4b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.354ex; height:3.176ex;" alt="{\displaystyle 2^{n-1}\cdot (2^{n}-1)}"></span> 에 알맞은 수를 대입해 구할 수 있다는 것을 발견했다. </p> <dl><dd><i>n</i> = 1 일 때:   2<sup>0</sup> · (2<sup>1</sup> − 1) = 0</dd> <dd><i>n</i> = 2 일 때:   2<sup>1</sup> · (2<sup>2</sup> − 1) = 6</dd> <dd><i>n</i> = 3 일 때:   2<sup>2</sup> · (2<sup>3</sup> − 1) = 28</dd> <dd><i>n</i> = 5 일 때:   2<sup>4</sup> · (2<sup>5</sup> − 1) = 496</dd> <dd><i>n</i> = 7 일 때:   2<sup>6</sup> · (2<sup>7</sup> − 1) = 8128</dd></dl> <p>이때 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>은 언제나 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>이지만 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>이 소수라고 2<sup><i>n</i></sup> − 1도 꼭 소수가 되지는 않는다. 2<sup><i>n</i></sup> − 1이 소수일 때는 이를 <a href="/wiki/%EB%A9%94%EB%A5%B4%EC%84%BC_%EC%86%8C%EC%88%98" title="메르센 소수">메르센 소수</a>라고 부른다. <a href="/wiki/%EB%A7%88%EB%9E%AD_%EB%A9%94%EB%A5%B4%EC%84%BC" title="마랭 메르센">마랭 메르센</a>은 <a href="/wiki/17%EC%84%B8%EA%B8%B0" title="17세기">17세기</a>에 <a href="/wiki/%EC%A0%95%EC%88%98%EB%A1%A0" class="mw-redirect" title="정수론">정수론</a>과 완전수를 연구한 수도승이었다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n-1}\cdot (2^{n}-1)={\frac {M_{n}(M_{n}+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}\cdot (2^{n}-1)={\frac {M_{n}(M_{n}+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c7b3bf80e07b517796ff3c30e9e3f1764debb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.045ex; height:5.676ex;" alt="{\displaystyle 2^{n-1}\cdot (2^{n}-1)={\frac {M_{n}(M_{n}+1)}{2}}}"></span> 즉, 짝수 완전수와 메르센 소수 사이에는 일대일 대응이 있다는 것이 밝혀졌다.</dd></dl> <p>모든 짝수 완전수가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n-1}\cdot (2^{n}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}\cdot (2^{n}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56add7d8caed550996c464f9ea844aed14c4b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.354ex; height:3.176ex;" alt="{\displaystyle 2^{n-1}\cdot (2^{n}-1)}"></span> 꼴이므로, 모든 짝수 완전수는 연속된 자연수의 합으로 표현할 수 있다. 그러나 <a href="/wiki/%EB%A9%94%EB%A5%B4%EC%84%BC_%EC%88%98" class="mw-redirect" title="메르센 수">메르센 수</a>가 소수가 아닌 경우에는 해당 숫자는 <a href="/wiki/%EA%B3%BC%EC%9E%89%EC%88%98" title="과잉수">과잉수</a>가 된다. 그와 동시에 모두 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>이기도 하다. 그러한 예는 120, 2016, 32640, 130816 등이 있다. 15, 63, 255, 511 등은 모두 메르센 수들 중에서 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>가 아닌 <a href="/wiki/%ED%95%A9%EC%84%B1%EC%88%98" title="합성수">합성수</a>이기 때문이다. </p> <pre> 0 = 0 6 = 1 + 2 + 3 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + . . . + 30 + 31 </pre> <p><br /> 메르센 소수의 수가 유한한지 무한한지는 알려져 있지 않다. 그러므로 짝수 완전수의 수가 무한한지도 알려져 있지 않다. </p> <div class="mw-heading mw-heading2"><h2 id="홀수_완전수"><span id=".ED.99.80.EC.88.98_.EC.99.84.EC.A0.84.EC.88.98"></span>홀수 완전수</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EC%A0%84%EC%88%98&action=edit&section=2" title="부분 편집: 홀수 완전수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right; width:330px; 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/e/e4/Fluent_Emoji_flat_2754.svg/50px-Fluent_Emoji_flat_2754.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Fluent_Emoji_flat_2754.svg/75px-Fluent_Emoji_flat_2754.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Fluent_Emoji_flat_2754.svg/100px-Fluent_Emoji_flat_2754.svg.png 2x" data-file-width="320" data-file-height="320" /></span></span></td> <td><b>수학의 미해결 문제</b><br /><div style="padding:5px;">홀수 완전수는 존재하는가?<br /><small><a href="/wiki/%EC%88%98%ED%95%99%EC%9D%98_%EB%AF%B8%ED%95%B4%EA%B2%B0_%EB%AC%B8%EC%A0%9C_%EB%AA%A9%EB%A1%9D" title="수학의 미해결 문제 목록">(더 많은 수학의 미해결 문제 보기)</a></small></div> </td></tr></tbody></table> <p>만약 홀수 완전수가 존재한다면 그 수는 다음 조건을 만족한다. </p> <ul><li>10<sup>1500</sup>보다 크다.<sup id="cite_ref-Ochem_and_Rao_(2012)_1-0" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>105로 나누어떨어지지 않으며<sup id="cite_ref-Kühnel_U_2-0" class="reference"><a href="#cite_note-Kühnel_U-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <i>N</i> ≡ 1 (mod 12) 또는 <i>N</i> ≡ 81 (mod 324) 또는 <i>N</i> ≡ 117 (mod 468)꼴이다.<sup id="cite_ref-Roberts_T_(2008)_3-0" class="reference"><a href="#cite_note-Roberts_T_(2008)-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>가장 큰 소인수는 10<sup>8</sup>보다 크고<sup id="cite_ref-Goto_and_Ohno_(2008)_4-0" class="reference"><a href="#cite_note-Goto_and_Ohno_(2008)-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <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 {\sqrt[{3}]{3N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>3</mn> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{3N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af60d573cccbdd77584f482029a46e04db3590e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.162ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{3N}}}"></span>보다 작다.<sup id="cite_ref-AK_2012_5-0" class="reference"><a href="#cite_note-AK_2012-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>두 번째로 가장 큰 소인수는 10000보다 크고,<sup id="cite_ref-Ianucci_DE_(1999)_6-0" class="reference"><a href="#cite_note-Ianucci_DE_(1999)-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <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 {\sqrt[{5}]{2N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{5}]{2N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5871ef69c5781b0267353a19bfa37d91b90e0c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.162ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{5}]{2N}}}"></span>보다 작다.<sup id="cite_ref-Zelinsky_2019_7-0" class="reference"><a href="#cite_note-Zelinsky_2019-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>세 번째로 가장 큰 소인수는 100보다 크고,<sup id="cite_ref-Ianucci_DE_(2000)_8-0" class="reference"><a href="#cite_note-Ianucci_DE_(2000)-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <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 {\sqrt[{6}]{2N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{2N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2765a66355e477513834b928a6c1c7fd5731ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.162ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{6}]{2N}}}"></span>보다 작다.<sup id="cite_ref-Zelinsky_2021a_9-0" class="reference"><a href="#cite_note-Zelinsky_2021a-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>소인수는 중복을 포함하여 적어도 101개이고 서로 다른 소인수는 10개 이상이다.<sup id="cite_ref-Ochem_and_Rao_(2012)_1-1" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nielsen_Pace_P._(2015)_10-0" class="reference"><a href="#cite_note-Nielsen_Pace_P._(2015)-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> 만약 3이 인수가 아니면, 서로 다른 소인수는 적어도 12개이다.<sup id="cite_ref-Nielsen_Pace_P._(2007)_11-0" class="reference"><a href="#cite_note-Nielsen_Pace_P._(2007)-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i>은 <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> </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/d34fa52a2089388efc3644d0dd9c9126738c9f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.694ex; height:3.509ex;" alt="{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}}}"></span>의 형태이며 다음을 만족시킨다.</li></ul> <dl><dd><ul><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 q,p_{1},\cdots ,p_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q,p_{1},\cdots ,p_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecbc29cfdcc1b69eec7e981a5eda67c290fd1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.764ex; height:2.009ex;" alt="{\displaystyle q,p_{1},\cdots ,p_{k}}"></span>는 서로 다른 소수이다. (오일러)</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 q\equiv \alpha \equiv 1{\pmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>≡</mo> <mi>α</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\equiv \alpha \equiv 1{\pmod {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270c7f34d6b7ba653e0948a53d63d78546d95118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.763ex; height:2.843ex;" alt="{\displaystyle q\equiv \alpha \equiv 1{\pmod {4}}}"></span> (오일러)</li> <li><i>N</i>의 가장 작은 소인수는 <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> </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/0c37a7ebb4d274bf85b38a999004a4f7fb81382a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.05ex; height:5.343ex;" alt="{\displaystyle {\frac {k-1}{2}}}"></span> 이하이다.<sup id="cite_ref-Zelinsky_2021_12-0" class="reference"><a href="#cite_note-Zelinsky_2021-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><i>N</i>을 나누는, 10<sup>62</sup>보다 큰 소수의 거듭제곱수가 적어도 하나 있다.<sup id="cite_ref-Ochem_and_Rao_(2012)_1-2" class="reference"><a href="#cite_note-Ochem_and_Rao_(2012)-1"><span class="cite-bracket">[</span>1<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_13-0" class="reference"><a href="#cite_note-Chen_and_Tang-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nielsen_(2003)_14-0" class="reference"><a href="#cite_note-Nielsen_(2003)-14"><span class="cite-bracket">[</span>14<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 {99k-224}{37}}}"> <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>99</mn> <mi>k</mi> <mo>−</mo> <mn>224</mn> </mrow> <mn>37</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {99k-224}{37}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5a7e3126bffe13c79b0ba6ead683ae3884684d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:45.447ex; height:5.509ex;" alt="{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {99k-224}{37}}}"></span>.<sup id="cite_ref-Zelinsky_2021_12-1" class="reference"><a href="#cite_note-Zelinsky_2021-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ochem_and_Rao_(2014)_15-0" class="reference"><a href="#cite_note-Ochem_and_Rao_(2014)-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ClayotonHansen_16-0" class="reference"><a href="#cite_note-ClayotonHansen-16"><span class="cite-bracket">[</span>16<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_17-0" class="reference"><a href="#cite_note-LucaPomerance-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl> <p>더 나아가서 지수 <i>e</i><sub>1</sub>, ..., <i>e</i><sub><i>k</i></sub>에 대해서는 다음과 같은 결과가 알려져 있다. </p> <ul><li>모든 <i>e</i><sub><i>i</i></sub>가 <i>e</i><sub><i>i</i></sub> ≡ 1 (<a href="/w/index.php?title=Modular_arithmetic&action=edit&redlink=1" class="new" title="Modular arithmetic (없는 문서)">mod</a> 3)인 것은 아니다.<sup id="cite_ref-McDaniel_(1970)_18-0" class="reference"><a href="#cite_note-McDaniel_(1970)-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li>(<i>e</i><sub>1</sub>, ..., <i>e</i><sub><i>k</i></sub>) ≠ (1, ..., 1, 3),<sup id="cite_ref-Kanold_(1950)_19-0" class="reference"><a href="#cite_note-Kanold_(1950)-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> (1, ..., 1, 5), (1, ..., 1, 6).<sup id="cite_ref-Cohen_and_Williams_(1985)_20-0" class="reference"><a href="#cite_note-Cohen_and_Williams_(1985)-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li><style data-mw-deduplicate="TemplateStyles:r25030363">.mw-parser-output .texhtml{font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;white-space:nowrap;line-height:1;font-size:118%}</style><span class="texhtml"><i>e</i><sub>1</sub> = ... = <i>e</i><sub><i>k</i></sub> = <i>e</i></span>인 경우, <ul><li><i>e</i>는 3,<sup id="cite_ref-Hagis_and_McDaniel_(1972)_21-0" class="reference"><a href="#cite_note-Hagis_and_McDaniel_(1972)-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> 5, 6, 8, 11, 14, 18,<sup id="cite_ref-Cohen_and_Williams_(1985)_20-1" class="reference"><a href="#cite_note-Cohen_and_Williams_(1985)-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> 24<sup id="cite_ref-McDaniel_and_Hagis_(1975)_22-0" class="reference"><a href="#cite_note-McDaniel_and_Hagis_(1975)-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>가 아니다.</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 k\leq 2e^{2}+8e+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≤</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>e</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\leq 2e^{2}+8e+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f262dc1f08d926f8e38335a4221432e01110858b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.699ex; height:2.843ex;" alt="{\displaystyle k\leq 2e^{2}+8e+2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N<2^{4^{2e^{2}+8e+3}}}"> <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"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>e</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N<2^{4^{2e^{2}+8e+3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352c47ec4267fb122ccc355b38e6ee713520ac90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.666ex; height:3.509ex;" alt="{\displaystyle N<2^{4^{2e^{2}+8e+3}}}"></span>.<sup id="cite_ref-Yamada_(2019)_23-0" class="reference"><a href="#cite_note-Yamada_(2019)-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EC%A0%84%EC%88%98&action=edit&section=3" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%B4%88%EC%99%84%EC%A0%84%EC%88%98" title="초완전수">초완전수</a></li> <li><a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a></li> <li><a href="/wiki/%EC%A4%80%EC%99%84%EC%A0%84%EC%88%98" title="준완전수">준완전수</a></li> <li><a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a></li> <li><a href="/wiki/%EA%B3%BC%EC%9E%89%EC%88%98" title="과잉수">과잉수</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EC%A0%84%EC%88%98&action=edit&section=4" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Ochem_and_Rao_(2012)-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Ochem_and_Rao_(2012)_1-0">가</a></sup> <sup><a href="#cite_ref-Ochem_and_Rao_(2012)_1-1">나</a></sup> <sup><a href="#cite_ref-Ochem_and_Rao_(2012)_1-2">다</a></sup></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《<a href="/w/index.php?title=Mathematics_of_Computation&action=edit&redlink=1" class="new" title="Mathematics of Computation (없는 문서)">Mathematics of Computation</a>》 <b>81</b> (279): 1869–1877. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-2012-02563-4">10.1090/S0025-5718-2012-02563-4</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0025-5718">0025-5718</a>. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">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&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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Kühnel_U-2"><span class="mw-cite-backlink"><a href="#cite_ref-Kühnel_U_2-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Kühnel, Ullrich (1950). “Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen”. 《Mathematische Zeitschrift》 (독일어) <b>52</b>: 202–211. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF02230691">10.1007/BF02230691</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">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&rft.aulast=K%C3%BChnel&rft.aufirst=Ullrich&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Roberts_T_(2008)-3"><span class="mw-cite-backlink"><a href="#cite_ref-Roberts_T_(2008)_3-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Australian Mathematical Gazette》 <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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Goto_and_Ohno_(2008)-4"><span class="mw-cite-backlink"><a href="#cite_ref-Goto_and_Ohno_(2008)_4-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Mathematics of Computation》 <b>77</b> (263): 1859–1868. <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2008MaCom..77.1859G">2008MaCom..77.1859G</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-08-02050-9">10.1090/S0025-5718-08-02050-9</a>. 2011년 8월 7일에 <a rel="nofollow" class="external text" href="http://www.ma.noda.tus.ac.jp/u/tg/perfect/perfect.pdf">원본 문서</a> <span style="font-size:85%;">(PDF)</span>에서 보존된 문서<span class="reference-accessdate">. 2011년 3월 30일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-AK_2012-5"><span class="mw-cite-backlink"><a href="#cite_ref-AK_2012_5-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Konyagin, Sergei; Acquaah, Peter (2012). “On Prime Factors of Odd Perfect Numbers”. 《International Journal of Number Theory》 <b>8</b> (6): 1537–1540. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Ianucci_DE_(1999)-6"><span class="mw-cite-backlink"><a href="#cite_ref-Ianucci_DE_(1999)_6-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Mathematics of Computation》 <b>68</b> (228): 1749–1760. <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1999MaCom..68.1749I">1999MaCom..68.1749I</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-99-01126-6">10.1090/S0025-5718-99-01126-6</a><span class="reference-accessdate">. 2011년 3월 30일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Zelinsky_2019-7"><span class="mw-cite-backlink"><a href="#cite_ref-Zelinsky_2019_7-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Zelinsky, Joshua (July 2019). “Upper bounds on the second largest prime factor of an odd perfect number”. 《International Journal of Number Theory》 <b>15</b> (6): 1183–1189. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/1810.11734">1810.11734</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1142%2FS1793042119500659">10.1142/S1793042119500659</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">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&rft_id=info%3Adoi%2F10.1142%2FS1793042119500659&rft.aulast=Zelinsky&rft.aufirst=Joshua&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span>.</span> </li> <li id="cite_note-Ianucci_DE_(2000)-8"><span class="mw-cite-backlink"><a href="#cite_ref-Ianucci_DE_(2000)_8-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Mathematics of Computation》 <b>69</b> (230): 867–879. <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2000MaCom..69..867I">2000MaCom..69..867I</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-99-01127-8">10.1090/S0025-5718-99-01127-8</a><span class="reference-accessdate">. 2011년 3월 30일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Zelinsky_2021a-9"><span class="mw-cite-backlink"><a href="#cite_ref-Zelinsky_2021a_9-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (2021년 11월 23일). <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 style="font-size:85%;">(PDF)</span>. 《Integers》 <b>21</b><span class="reference-accessdate">. 2021년 12월 6일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Nielsen_Pace_P._(2015)-10"><span class="mw-cite-backlink"><a href="#cite_ref-Nielsen_Pace_P._(2015)_10-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Mathematics of Computation》 <b>84</b> (295): 2549–2567. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-2015-02941-X">10.1090/S0025-5718-2015-02941-X</a>. 2015년 7월 8일에 <a rel="nofollow" class="external text" href="https://math.byu.edu/~pace/BestBound_web.pdf">원본 문서</a> <span style="font-size:85%;">(PDF)</span>에서 보존된 문서<span class="reference-accessdate">. 2015년 8월 13일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Nielsen_Pace_P._(2007)-11"><span class="mw-cite-backlink"><a href="#cite_ref-Nielsen_Pace_P._(2007)_11-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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 style="font-size:85%;">(PDF)</span>. 《Mathematics of Computation》 <b>76</b> (260): 2109–2126. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/math/0602485">math/0602485</a>. <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2007MaCom..76.2109N">2007MaCom..76.2109N</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-07-01990-4">10.1090/S0025-5718-07-01990-4</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2767519">2767519</a>. 2021년 11월 3일에 <a rel="nofollow" class="external text" href="https://math.byu.edu/~pace/NotEight_web.pdf">원본 문서</a> <span style="font-size:85%;">(PDF)</span>에서 보존된 문서<span class="reference-accessdate">. 2011년 3월 30일에 확인함</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&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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Zelinsky_2021-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Zelinsky_2021_12-0">가</a></sup> <sup><a href="#cite_ref-Zelinsky_2021_12-1">나</a></sup></span> <span class="reference-text"><cite class="citation journal">Zelinsky, Joshua (2021년 8월 3일). <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 style="font-size:85%;">(PDF)</span>. 《Integers》 <b>21</b><span class="reference-accessdate">. 2021년 8월 7일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Chen_and_Tang-13"><span class="mw-cite-backlink"><a href="#cite_ref-Chen_and_Tang_13-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Chen, Yong-Gao; Tang, Cui-E (2014). “Improved upper bounds for odd multiperfect numbers.”. 《Bulletin of the Australian Mathematical Society》 <b>89</b> (3): 353–359. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Nielsen_(2003)-14"><span class="mw-cite-backlink"><a href="#cite_ref-Nielsen_(2003)_14-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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>. 《Integers》 <b>3</b>: A14–A22<span class="reference-accessdate">. 2021년 3월 23일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Ochem_and_Rao_(2014)-15"><span class="mw-cite-backlink"><a href="#cite_ref-Ochem_and_Rao_(2014)_15-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Ochem, Pascal; Rao, Michaël (2014). “On the number of prime factors of an odd perfect number.”. 《<a href="/w/index.php?title=Mathematics_of_Computation&action=edit&redlink=1" class="new" title="Mathematics of Computation (없는 문서)">Mathematics of Computation</a>》 <b>83</b> (289): 2435–2439. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-2013-02776-7">10.1090/S0025-5718-2013-02776-7</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=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&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-ClayotonHansen-16"><span class="mw-cite-backlink"><a href="#cite_ref-ClayotonHansen_16-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Graeme Clayton, Cody Hansen (2023). <a rel="nofollow" class="external text" href="http://math.colgate.edu/~integers/x79/x79.pdf">“On inequalities involving counts of the prime factors of an odd perfect number”</a> <span style="font-size:85%;">(PDF)</span>. 《Integers》 <b>23</b>. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/2303.11974">2303.11974</a><span class="reference-accessdate">. 2023년 11월 29일에 확인함</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+inequalities+involving+counts+of+the+prime+factors+of+an+odd+perfect+number&rft.volume=23&rft.date=2023&rft_id=info%3Aarxiv%2F2303.11974&rft.au=Graeme+Clayton%2C+Cody+Hansen&rft_id=http%3A%2F%2Fmath.colgate.edu%2F~integers%2Fx79%2Fx79.pdf&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-LucaPomerance-17"><span class="mw-cite-backlink"><a href="#cite_ref-LucaPomerance_17-0">↑</a></span> <span class="reference-text"><cite class="citation journal">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>. 《New York Journal of Mathematics》 <b>16</b>: 23–30<span class="reference-accessdate">. 2018년 12월 7일에 확인함</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%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-McDaniel_(1970)-18"><span class="mw-cite-backlink"><a href="#cite_ref-McDaniel_(1970)_18-0">↑</a></span> <span class="reference-text"><cite class="citation journal">McDaniel, Wayne L. (1970). “The non-existence of odd perfect numbers of a certain form”. 《Archiv der Mathematik》 <b>21</b> (1): 52–53. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF01220877">10.1007/BF01220877</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1420-8938">1420-8938</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0258723">0258723</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121251041">121251041</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+der+Mathematik&rft.atitle=The+non-existence+of+odd+perfect+numbers+of+a+certain+form&rft.volume=21&rft.issue=1&rft.pages=52-53&rft.date=1970&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121251041&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0258723&rft.issn=1420-8938&rft_id=info%3Adoi%2F10.1007%2FBF01220877&rft.aulast=McDaniel&rft.aufirst=Wayne+L.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Kanold_(1950)-19"><span class="mw-cite-backlink"><a href="#cite_ref-Kanold_(1950)_19-0">↑</a></span> <span class="reference-text"><cite class="citation journal"><a href="https://de.wikipedia.org/wiki/Hans-Joachim_Kanold" class="extiw" title="de:Hans-Joachim Kanold">Kanold, Hans-Joachim</a> (1950). “Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II”. 《<a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>》 <b>188</b> (1): 129–146. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1515%2Fcrll.1950.188.129">10.1515/crll.1950.188.129</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1435-5345">1435-5345</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0044579">0044579</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122452828">122452828</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Satze+uber+Kreisteilungspolynome+und+ihre+Anwendungen+auf+einige+zahlentheoretisehe+Probleme.+II&rft.volume=188&rft.issue=1&rft.pages=129-146&rft.date=1950&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122452828&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0044579&rft.issn=1435-5345&rft_id=info%3Adoi%2F10.1515%2Fcrll.1950.188.129&rft.aulast=Kanold&rft.aufirst=Hans-Joachim&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Cohen_and_Williams_(1985)-20"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Cohen_and_Williams_(1985)_20-0">가</a></sup> <sup><a href="#cite_ref-Cohen_and_Williams_(1985)_20-1">나</a></sup></span> <span class="reference-text"><cite class="citation journal">Cohen, G. L.; Williams, R. J. (1985). <a rel="nofollow" class="external text" href="https://www.fq.math.ca/Scanned/23-1/cohen.pdf">“Extensions of some results concerning odd perfect numbers”</a> <span style="font-size:85%;">(PDF)</span>. 《<a href="/w/index.php?title=Fibonacci_Quarterly&action=edit&redlink=1" class="new" title="Fibonacci Quarterly (없는 문서)">Fibonacci Quarterly</a>》 <b>23</b> (1): 70–76. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0015-0517">0015-0517</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0786364">0786364</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fibonacci+Quarterly&rft.atitle=Extensions+of+some+results+concerning+odd+perfect+numbers&rft.volume=23&rft.issue=1&rft.pages=70-76&rft.date=1985&rft.issn=0015-0517&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0786364&rft.aulast=Cohen&rft.aufirst=G.+L.&rft.au=Williams%2C+R.+J.&rft_id=https%3A%2F%2Fwww.fq.math.ca%2FScanned%2F23-1%2Fcohen.pdf&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Hagis_and_McDaniel_(1972)-21"><span class="mw-cite-backlink"><a href="#cite_ref-Hagis_and_McDaniel_(1972)_21-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Hagis, Peter Jr.; McDaniel, Wayne L. (1972). “A new result concerning the structure of odd perfect numbers”. 《Proceedings of the American Mathematical Society》 <b>32</b> (1): 13–15. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0002-9939-1972-0292740-5">10.1090/S0002-9939-1972-0292740-5</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1088-6826">1088-6826</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0292740">0292740</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=A+new+result+concerning+the+structure+of+odd+perfect+numbers&rft.volume=32&rft.issue=1&rft.pages=13-15&rft.date=1972&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0292740&rft.issn=1088-6826&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1972-0292740-5&rft.aulast=Hagis&rft.aufirst=Peter+Jr.&rft.au=McDaniel%2C+Wayne+L.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-McDaniel_and_Hagis_(1975)-22"><span class="mw-cite-backlink"><a href="#cite_ref-McDaniel_and_Hagis_(1975)_22-0">↑</a></span> <span class="reference-text"><cite class="citation journal">McDaniel, Wayne L.; Hagis, Peter Jr. (1975). <a rel="nofollow" class="external text" href="https://www.fq.math.ca/Scanned/13-1/mcdaniel.pdf">“Some results concerning the non-existence of odd perfect numbers of the form <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^{\alpha }M^{2\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{\alpha }M^{2\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8c90a06b90826fac41894d97aee2551761e8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.038ex; height:3.009ex;" alt="{\displaystyle p^{\alpha }M^{2\beta }}"></span>”</a> <span style="font-size:85%;">(PDF)</span>. 《<a href="/w/index.php?title=Fibonacci_Quarterly&action=edit&redlink=1" class="new" title="Fibonacci Quarterly (없는 문서)">Fibonacci Quarterly</a>》 <b>13</b> (1): 25–28. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0015-0517">0015-0517</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0354538">0354538</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fibonacci+Quarterly&rft.atitle=Some+results+concerning+the+non-existence+of+odd+perfect+numbers+of+the+form+MATH+RENDER+ERROR&rft.volume=13&rft.issue=1&rft.pages=25-28&rft.date=1975&rft.issn=0015-0517&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0354538&rft.aulast=McDaniel&rft.aufirst=Wayne+L.&rft.au=Hagis%2C+Peter+Jr.&rft_id=https%3A%2F%2Fwww.fq.math.ca%2FScanned%2F13-1%2Fmcdaniel.pdf&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span> <span style="font-size:100%" class="error citation-comment"><code style="color:inherit; border:inherit; padding:inherit;">|title=</code>에 지움 문자가 있음(위치 78) (<a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%9D%B8%EC%9A%A9_%EC%98%A4%EB%A5%98_%EB%8F%84%EC%9B%80%EB%A7%90#invisible_char" title="위키백과:인용 오류 도움말">도움말</a>)</span></span> </li> <li id="cite_note-Yamada_(2019)-23"><span class="mw-cite-backlink"><a href="#cite_ref-Yamada_(2019)_23-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Yamada, Tomohiro (2019). “A new upper bound for odd perfect numbers of a special form”. 《Colloquium Mathematicum》 <b>156</b> (1): 15–21. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/1706.09341">1706.09341</a>. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.4064%2Fcm7339-3-2018">10.4064/cm7339-3-2018</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1730-6302">1730-6302</a>. <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119175632">119175632</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloquium+Mathematicum&rft.atitle=A+new+upper+bound+for+odd+perfect+numbers+of+a+special+form&rft.volume=156&rft.issue=1&rft.pages=15-21&rft.date=2019&rft_id=info%3Aarxiv%2F1706.09341&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119175632&rft.issn=1730-6302&rft_id=info%3Adoi%2F10.4064%2Fcm7339-3-2018&rft.aulast=Yamada&rft.aufirst=Tomohiro&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EC%A0%84%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output 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href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%A0%84%EA%B1%B0_%ED%86%B5%EC%A0%9C" title="위키백과:전거 통제">전거 통제</a>: 국가 <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q170043#identifiers" title="위키데이터에서 편집하기"><img alt="위키데이터에서 편집하기" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a 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