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type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="เมนูหลัก" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">เมนูหลัก</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">เมนูหลัก</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ซ่อน</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> การนำทาง </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%AB%E0%B8%A5%E0%B8%B1%E0%B8%81" title="เยี่ยมชมหน้าหลัก [z]" accesskey="z"><span>หน้าหลัก</span></a></li><li id="n-ask" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%81%E0%B8%B4%E0%B8%9E%E0%B8%B5%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2:%E0%B8%96%E0%B8%B2%E0%B8%A1%E0%B8%84%E0%B8%B3%E0%B8%96%E0%B8%B2%E0%B8%A1"><span>ถามคำถาม</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%E0%B8%AA%E0%B8%96%E0%B8%B2%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B9%88%E0%B8%AD%E0%B8%A2:%E0%B9%80%E0%B8%AB%E0%B8%95%E0%B8%B8%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%93%E0%B9%8C%E0%B8%9B%E0%B8%B1%E0%B8%88%E0%B8%88%E0%B8%B8%E0%B8%9A%E0%B8%B1%E0%B8%99" title="ค้นหาข้อมูลเบื้องหลังในเหตุการณ์ปัจจุบัน"><span>เหตุการณ์ปัจจุบัน</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%B8%E0%B9%88%E0%B8%A1" title="โหลดหน้าแบบสุ่ม [x]" accesskey="x"><span>สุ่มบทความ</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%81%E0%B8%B4%E0%B8%9E%E0%B8%B5%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2:%E0%B9%80%E0%B8%81%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A7%E0%B8%81%E0%B8%B1%E0%B8%9A" title="ทำความรู้จักวิกิพีเดีย"><span>เกี่ยวกับวิกิพีเดีย</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%81%E0%B8%B4%E0%B8%9E%E0%B8%B5%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2:%E0%B8%95%E0%B8%B4%E0%B8%94%E0%B8%95%E0%B9%88%E0%B8%AD" title="วิธีการติดต่อวิกิพีเดีย"><span>ติดต่อเรา</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> มีส่วนร่วม </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B9%83%E0%B8%8A%E0%B9%89:%E0%B8%AA%E0%B8%B2%E0%B8%A3%E0%B8%9A%E0%B8%B1%E0%B8%8D" title="ข้อแนะนำการใช้และแก้ไขวิกิพีเดีย"><span>คำอธิบาย</span></a></li><li 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placeholder="ค้นหาใน วิกิพีเดีย" aria-label="ค้นหาใน วิกิพีเดีย" autocapitalize="sentences" title="ค้นหาวิกิ [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="พิเศษ:ค้นหา"> </div> <button class="cdx-button cdx-search-input__end-button">ค้นหา</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="เครื่องมือส่วนตัว"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="หน้าตา"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="หน้าตา" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">หน้าตา</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_th.wikipedia.org&uselang=th" class=""><span>บริจาคให้วิกิพีเดีย</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=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%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="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ" class=""><span>สร้างบัญชี</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=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%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="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o" class=""><span>เข้าสู่ระบบ</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="ตัวเลือกเพิ่มเติม" > <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="เครื่องมือส่วนตัว" > <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">เครื่องมือส่วนตัว</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="เมนูผู้ใช้" > <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_th.wikipedia.org&uselang=th"><span>บริจาคให้วิกิพีเดีย</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%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="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>สร้างบัญชี</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%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="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>เข้าสู่ระบบ</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"> หน้าสำหรับผู้แก้ไขที่ออกจากระบบ <a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B9%83%E0%B8%8A%E0%B9%89:%E0%B8%9A%E0%B8%97%E0%B8%99%E0%B8%B3" aria-label="เรียนรู้เพิ่มเกี่ยวกับการแก้ไข"><span>เรียนรู้เพิ่มเติม</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/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%80%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%89%E0%B8%B1%E0%B8%99%E0%B9%80%E0%B8%82%E0%B8%B5%E0%B8%A2%E0%B8%99" title="รายการการแก้ไขจากเลขที่อยู่ไอพีนี้ [y]" accesskey="y"><span>ส่วนร่วม</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%9E%E0%B8%B9%E0%B8%94%E0%B8%84%E0%B8%B8%E0%B8%A2%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%89%E0%B8%B1%E0%B8%99" title="อภิปรายเกี่ยวกับการแก้ไขจากเลขที่อยู่ไอพีนี้ [n]" accesskey="n"><span>คุย</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="ไซต์"> <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="สารบัญ" 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">สารบัญ</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ซ่อน</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">บทนำ</div> </a> </li> <li id="toc-จำนวนสมบูรณ์คู่" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#จำนวนสมบูรณ์คู่"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>จำนวนสมบูรณ์คู่</span> </div> </a> <ul id="toc-จำนวนสมบูรณ์คู่-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-จำนวนสมบูรณ์คี่" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#จำนวนสมบูรณ์คี่"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>จำนวนสมบูรณ์คี่</span> </div> </a> <ul id="toc-จำนวนสมบูรณ์คี่-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-อ้างอิง" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#อ้างอิง"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>อ้างอิง</span> </div> </a> <ul id="toc-อ้างอิง-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="สารบัญ" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">จำนวนสมบูรณ์</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="ไปที่บทความในภาษาอื่น ซึ่งมีใน 64 ภาษา" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">64 ภาษา</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 – อาหรับ" lang="ar" hreflang="ar" data-title="عدد تام" data-language-autonym="العربية" data-language-local-name="อาหรับ" 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 – เบลารุส" lang="be" hreflang="be" data-title="Дасканалы лік" data-language-autonym="Беларуская" data-language-local-name="เบลารุส" 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 – บัลแกเรีย" lang="bg" hreflang="bg" data-title="Съвършено число" data-language-autonym="Български" data-language-local-name="บัลแกเรีย" 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 – บังกลา" lang="bn" hreflang="bn" data-title="নিখুঁত সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="บังกลา" 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 – เบรตัน" lang="br" hreflang="br" data-title="Niver peurvat" data-language-autonym="Brezhoneg" data-language-local-name="เบรตัน" 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 – คาตาลัน" lang="ca" hreflang="ca" data-title="Nombre perfecte" data-language-autonym="Català" data-language-local-name="คาตาลัน" 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 – เคิร์ดตอนกลาง" lang="ckb" hreflang="ckb" data-title="ژمارەی کامڵ" data-language-autonym="کوردی" data-language-local-name="เคิร์ดตอนกลาง" 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 – เช็ก" lang="cs" hreflang="cs" data-title="Dokonalé číslo" data-language-autonym="Čeština" data-language-local-name="เช็ก" 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 – เดนมาร์ก" lang="da" hreflang="da" data-title="Fuldkomne tal" data-language-autonym="Dansk" data-language-local-name="เดนมาร์ก" 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 – เยอรมัน" lang="de" hreflang="de" data-title="Vollkommene Zahl" data-language-autonym="Deutsch" data-language-local-name="เยอรมัน" 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 – กรีก" lang="el" hreflang="el" data-title="Τέλειος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="กรีก" 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 – อังกฤษ" lang="en" hreflang="en" data-title="Perfect number" data-language-autonym="English" data-language-local-name="อังกฤษ" 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 – เอสเปรันโต" lang="eo" hreflang="eo" data-title="Perfekta nombro" data-language-autonym="Esperanto" data-language-local-name="เอสเปรันโต" 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 – สเปน" lang="es" hreflang="es" data-title="Número perfecto" data-language-autonym="Español" data-language-local-name="สเปน" 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 – บาสก์" lang="eu" hreflang="eu" data-title="Zenbaki perfektu" data-language-autonym="Euskara" data-language-local-name="บาสก์" 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 – เปอร์เซีย" lang="fa" hreflang="fa" data-title="عدد تام" data-language-autonym="فارسی" data-language-local-name="เปอร์เซีย" 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 – ฟินแลนด์" lang="fi" hreflang="fi" data-title="Täydellinen luku" data-language-autonym="Suomi" data-language-local-name="ฟินแลนด์" 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 – โวโร" lang="vro" hreflang="vro" data-title="Tävvelidseq arvoq" data-language-autonym="Võro" data-language-local-name="โวโร" 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 – ฝรั่งเศส" lang="fr" hreflang="fr" data-title="Nombre parfait" data-language-autonym="Français" data-language-local-name="ฝรั่งเศส" 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 – ไอริช" lang="ga" hreflang="ga" data-title="Uimhir fhoirfe" data-language-autonym="Gaeilge" data-language-local-name="ไอริช" 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 – กาลิเซีย" lang="gl" hreflang="gl" data-title="Número perfecto" data-language-autonym="Galego" data-language-local-name="กาลิเซีย" 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 – ฮิบรู" lang="he" hreflang="he" data-title="מספר משוכלל" data-language-autonym="עברית" data-language-local-name="ฮิบรู" 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 – ฮังการี" lang="hu" hreflang="hu" data-title="Tökéletes számok" data-language-autonym="Magyar" data-language-local-name="ฮังการี" 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 – อาร์เมเนีย" lang="hy" hreflang="hy" data-title="Կատարյալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="อาร์เมเนีย" 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 – อินเตอร์ลิงกัว" lang="ia" hreflang="ia" data-title="Numero perfecte" data-language-autonym="Interlingua" data-language-local-name="อินเตอร์ลิงกัว" 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 – อินโดนีเซีย" lang="id" hreflang="id" data-title="Bilangan sempurna" data-language-autonym="Bahasa Indonesia" data-language-local-name="อินโดนีเซีย" 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 – ไอซ์แลนด์" lang="is" hreflang="is" data-title="Fullkomin tala" data-language-autonym="Íslenska" data-language-local-name="ไอซ์แลนด์" 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 – อิตาลี" lang="it" hreflang="it" data-title="Numero perfetto" data-language-autonym="Italiano" data-language-local-name="อิตาลี" 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 – ญี่ปุ่น" lang="ja" hreflang="ja" data-title="完全数" data-language-autonym="日本語" data-language-local-name="ญี่ปุ่น" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수 – เกาหลี" lang="ko" hreflang="ko" data-title="완전수" data-language-autonym="한국어" data-language-local-name="เกาหลี" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_nuwaze" title="Hejmarên nuwaze – เคิร์ด" lang="ku" hreflang="ku" data-title="Hejmarên nuwaze" data-language-autonym="Kurdî" data-language-local-name="เคิร์ด" 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 – ละติน" lang="la" hreflang="la" data-title="Numerus perfectus" data-language-autonym="Latina" data-language-local-name="ละติน" 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 – ลอมบาร์ด" lang="lmo" hreflang="lmo" data-title="Nümar parfett" data-language-autonym="Lombard" data-language-local-name="ลอมบาร์ด" 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 – ลิทัวเนีย" lang="lt" hreflang="lt" data-title="Tobulasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="ลิทัวเนีย" 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 – มาซิโดเนีย" lang="mk" hreflang="mk" data-title="Совршен број" data-language-autonym="Македонски" data-language-local-name="มาซิโดเนีย" 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 – นาโปลี" lang="nap" hreflang="nap" data-title="Nummero perfetto" data-language-autonym="Napulitano" data-language-local-name="นาโปลี" 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 – ดัตช์" lang="nl" hreflang="nl" data-title="Perfect getal" data-language-autonym="Nederlands" data-language-local-name="ดัตช์" 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 – นอร์เวย์นีนอสก์" lang="nn" hreflang="nn" data-title="Fulkomne tal" data-language-autonym="Norsk nynorsk" data-language-local-name="นอร์เวย์นีนอสก์" 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 – นอร์เวย์บุคมอล" lang="nb" hreflang="nb" data-title="Perfekt tall" data-language-autonym="Norsk bokmål" data-language-local-name="นอร์เวย์บุคมอล" 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 – โปแลนด์" lang="pl" hreflang="pl" data-title="Liczby doskonałe" data-language-autonym="Polski" data-language-local-name="โปแลนด์" 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 – พีดมอนต์" lang="pms" hreflang="pms" data-title="Nùmer përfet" data-language-autonym="Piemontèis" data-language-local-name="พีดมอนต์" 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 – โปรตุเกส" lang="pt" hreflang="pt" data-title="Número perfeito" data-language-autonym="Português" data-language-local-name="โปรตุเกส" 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 – โรมาเนีย" lang="ro" hreflang="ro" data-title="Număr perfect" data-language-autonym="Română" data-language-local-name="โรมาเนีย" 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 – รัสเซีย" lang="ru" hreflang="ru" data-title="Совершенное число" data-language-autonym="Русский" data-language-local-name="รัสเซีย" 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 – ซิซิลี" lang="scn" hreflang="scn" data-title="Nùmmuru pirfettu" data-language-autonym="Sicilianu" data-language-local-name="ซิซิลี" 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 – สโลวัก" lang="sk" hreflang="sk" data-title="Dokonalé číslo" data-language-autonym="Slovenčina" data-language-local-name="สโลวัก" 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 – สโลวีเนีย" lang="sl" hreflang="sl" data-title="Popolno število" data-language-autonym="Slovenščina" data-language-local-name="สโลวีเนีย" 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 – แอลเบเนีย" lang="sq" hreflang="sq" data-title="Numrat e përsosur" data-language-autonym="Shqip" data-language-local-name="แอลเบเนีย" 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 – เซอร์เบีย" lang="sr" hreflang="sr" data-title="Савршен број" data-language-autonym="Српски / srpski" data-language-local-name="เซอร์เบีย" 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 – สวีเดน" lang="sv" hreflang="sv" data-title="Perfekt tal" data-language-autonym="Svenska" data-language-local-name="สวีเดน" 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) – ทมิฬ" lang="ta" hreflang="ta" data-title="நிறைவெண் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="ทมิฬ" 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 – เตลูกู" lang="te" hreflang="te" data-title="పరిపూర్ణసంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="เตลูกู" 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 – ทาจิก" lang="tg" hreflang="tg" data-title="Адади мукаммал" data-language-autonym="Тоҷикӣ" data-language-local-name="ทาจิก" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/M%C3%BCkemmel_say%C4%B1" title="Mükemmel sayı – ตุรกี" lang="tr" hreflang="tr" data-title="Mükemmel sayı" data-language-autonym="Türkçe" data-language-local-name="ตุรกี" 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 – ยูเครน" lang="uk" hreflang="uk" data-title="Досконале число" data-language-autonym="Українська" data-language-local-name="ยูเครน" 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 – อูรดู" lang="ur" hreflang="ur" data-title="کامل عدد" data-language-autonym="اردو" data-language-local-name="อูรดู" 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 – อุซเบก" lang="uz" hreflang="uz" data-title="Mukammal son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="อุซเบก" 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 – เวียดนาม" lang="vi" hreflang="vi" data-title="Số hoàn thiện" data-language-autonym="Tiếng Việt" data-language-local-name="เวียดนาม" 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 – จีนอู๋" lang="wuu" hreflang="wuu" data-title="完全数" data-language-autonym="吴语" data-language-local-name="จีนอู๋" 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 – จีน" lang="zh" hreflang="zh" data-title="完全数" data-language-autonym="中文" data-language-local-name="จีน" 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 – จีนคลาสสิก" lang="lzh" hreflang="lzh" data-title="完全數" data-language-autonym="文言" data-language-local-name="จีนคลาสสิก" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E6%95%B8" title="完全數 – กวางตุ้ง" lang="yue" hreflang="yue" data-title="完全數" data-language-autonym="粵語" data-language-local-name="กวางตุ้ง" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q170043#sitelinks-wikipedia" title="แก้ไขลิงก์ข้ามภาษา" class="wbc-editpage">แก้ไขลิงก์</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="เนมสเปซ"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%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="ดูหน้าเนื้อหา [c]" accesskey="c"><span>บทความ</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B9%E0%B8%94%E0%B8%84%E0%B8%B8%E0%B8%A2:%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" rel="discussion" title="อภิปรายเกี่ยวกับหน้าเนื้อหา [t]" 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data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">เครื่องมือ</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ซ่อน</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="ตัวเลือกเพิ่มเติม" > <div class="vector-menu-heading"> การกระทำ </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/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"><span>อ่าน</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%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&action=edit" title="แก้ไขรหัสต้นฉบับของหน้านี้ [e]" accesskey="e"><span>แก้ไข</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%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&action=history"><span>ดูประวัติ</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> ทั่วไป </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%9A%E0%B8%97%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%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="รายการหน้าวิกิทุกหน้าที่ลิงก์มาที่นี่ [j]" accesskey="j"><span>หน้าที่ลิงก์มา</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9B%E0%B8%A3%E0%B8%B1%E0%B8%9A%E0%B8%9B%E0%B8%A3%E0%B8%B8%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%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" rel="nofollow" title="รายการเปลี่ยนแปลงล่าสุดในหน้าที่ลิงก์จากหน้านี้ [k]" accesskey="k"><span>การเปลี่ยนแปลงที่เกี่ยวโยง</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%81%E0%B8%B4%E0%B8%9E%E0%B8%B5%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2:%E0%B8%AD%E0%B8%B1%E0%B8%9B%E0%B9%82%E0%B8%AB%E0%B8%A5%E0%B8%94" title="อัปโหลดไฟล์ [u]" accesskey="u"><span>อัปโหลดไฟล์</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9" title="รายการหน้าพิเศษทั้งหมด [q]" accesskey="q"><span>หน้าพิเศษ</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%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&oldid=11910670" title="ลิงก์ถาวรมารุ่นนี้ของหน้านี้"><span>ลิงก์ถาวร</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%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&action=info" title="ข้อมูลเพิ่มเติมเกี่ยวกับหน้านี้"><span>สารสนเทศหน้า</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AD%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%AD%E0%B8%B4%E0%B8%87&page=%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&id=11910670&wpFormIdentifier=titleform" title="สารสนเทศเกี่ยวกับวิธีการอ้างอิงหน้านี้"><span>อ้างอิงบทความนี้</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:UrlShortener&url=https%3A%2F%2Fth.wikipedia.org%2Fwiki%2F%25E0%25B8%2588%25E0%25B8%25B3%25E0%25B8%2599%25E0%25B8%25A7%25E0%25B8%2599%25E0%25B8%25AA%25E0%25B8%25A1%25E0%25B8%259A%25E0%25B8%25B9%25E0%25B8%25A3%25E0%25B8%2593%25E0%25B9%258C"><span>รับยูอาร์แอลแบบสั้น</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:QrCode&url=https%3A%2F%2Fth.wikipedia.org%2Fwiki%2F%25E0%25B8%2588%25E0%25B8%25B3%25E0%25B8%2599%25E0%25B8%25A7%25E0%25B8%2599%25E0%25B8%25AA%25E0%25B8%25A1%25E0%25B8%259A%25E0%25B8%25B9%25E0%25B8%25A3%25E0%25B8%2593%25E0%25B9%258C"><span>ดาวน์โหลดคิวอาร์โค้ด</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> พิมพ์/ส่งออก </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B7%E0%B8%AD&bookcmd=book_creator&referer=%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"><span>สร้างหนังสือ</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:DownloadAsPdf&page=%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&action=show-download-screen"><span>ดาวน์โหลดเป็น PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%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&printable=yes" title="รุ่นที่พร้อมพิมพ์ของหน้านี้ [p]" accesskey="p"><span>รุ่นพร้อมพิมพ์</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> ในโครงการอื่น </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Perfect_number" hreflang="en"><span>วิกิมีเดียคอมมอนส์</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q170043" title="ลิงก์ไปยังสิ่งนี้ในคลังซึ่งเชื่อมโยงข้อมูลต่าง ๆ เข้าด้วยกัน [g]" accesskey="g"><span>สิ่งนี้ในวิกิสนเทศ</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="หน้าเครื่องมือ"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="หน้าตา"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">หน้าตา</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ซ่อน</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">จากวิกิพีเดีย สารานุกรมเสรี</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="th" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/%E0%B9%84%E0%B8%9F%E0%B8%A5%E0%B9%8C: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/280px-Perfect_number_Cuisenaire_rods_6.png" decoding="async" width="280" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Perfect_number_Cuisenaire_rods_6.png/420px-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>ภาพแสดงสถานะจำนวนสมบูรณ์ของเลข 6</figcaption></figure> <p><b>จำนวนสมบูรณ์</b> (<a href="/wiki/%E0%B8%A0%E0%B8%B2%E0%B8%A9%E0%B8%B2%E0%B8%AD%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B8%A4%E0%B8%A9" title="ภาษาอังกฤษ">อังกฤษ</a>: <span lang="en">perfect number</span>) คือ <a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="จำนวนเต็ม">จำนวนเต็ม</a>ที่มีผลบวกของ<a href="/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%AB%E0%B8%B2%E0%B8%A3" title="ตัวหาร">ตัวหาร</a>แท้ เท่ากับตัวมันเอง </p><p>หก (<a href="/wiki/6_(%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99)" class="mw-redirect" title="6 (จำนวน)">6</a>) เป็นจำนวนสมบูรณ์ตัวแรก เพราะว่า 6 มีตัวหารแท้คือ 1, 2, 3 และ 1 + 2 + 3 = 6 จำนวนสมบูรณ์ตัวถัดไปคือ 28 = 1 + 2 + 4 + 7 + 14 จำนวนสมบูรณ์ตัวถัดไปอีกคือ 496 และ 8128 (ลำดับ <span class="nowrap"><span typeof="mw:File"><a href="/wiki/%E0%B8%AA%E0%B8%B2%E0%B8%A3%E0%B8%B2%E0%B8%99%E0%B8%B8%E0%B8%81%E0%B8%A3%E0%B8%A1%E0%B8%AD%E0%B8%AD%E0%B8%99%E0%B9%84%E0%B8%A5%E0%B8%99%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="OEIS"><img alt="OEIS" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" decoding="async" width="11" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/17px-OEISicon_light.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/22px-OEISicon_light.svg.png 2x" data-file-width="409" data-file-height="556" /></a></span> <a href="//oeis.org/A000396" class="extiw" title="oeis:A000396">A000396</a>)</span> </p><p>จำนวนสมบูรณ์สี่ตัวแรกนั้นถูกค้นพบโดยชาว<a href="/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B5%E0%B8%81%E0%B9%82%E0%B8%9A%E0%B8%A3%E0%B8%B2%E0%B8%93" class="mw-redirect" title="กรีกโบราณ">กรีกโบราณ</a> </p> <div class="mw-heading mw-heading2"><h2 id="จำนวนสมบูรณ์คู่"><span id=".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.E0.B8.84.E0.B8.B9.E0.B9.88"></span>จำนวนสมบูรณ์คู่</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&action=edit&section=1" title="แก้ไขส่วน: จำนวนสมบูรณ์คู่"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E0%B8%A2%E0%B8%B8%E0%B8%84%E0%B8%A5%E0%B8%B4%E0%B8%94" title="ยุคลิด">ยุคลิด</a>ได้ค้นพบว่า จำนวนสมบูรณ์สี่ตัวแรกนั้นสามารถหาโดยใช้สูตร 2<sup><i>n</i>−1</sup>(2<sup><i>n</i></sup> − 1) ได้ </p> <dl><dd>สำหรับ <i>n</i> = 2:   2<sup>1</sup>(2<sup>2</sup> − 1) = 6</dd> <dd>สำหรับ <i>n</i> = 3:   2<sup>2</sup>(2<sup>3</sup> − 1) = 28</dd> <dd>สำหรับ <i>n</i> = 5:   2<sup>4</sup>(2<sup>5</sup> − 1) = 496</dd> <dd>สำหรับ <i>n</i> = 7:   2<sup>6</sup>(2<sup>7</sup> − 1) = 8128</dd></dl> <p>สังเกตว่าในแต่ละตัวอย่างที่ยกมา 2<sup><i>n</i></sup> − 1 จะเป็น<a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="จำนวนเฉพาะ">จำนวนเฉพาะ</a>เท่านั้น ยุคลิดได้พิสูจน์ว่า ถ้า 2<sup><i>n</i></sup> − 1 เป็นจำนวนเฉพาะแล้ว สูตร 2<sup><i>n</i>−1</sup>(2<sup><i>n</i></sup> − 1) จะให้ผลลัพธ์เป็นจำนวนสมบูรณ์คู่เสมอ </p><p>นักคณิตศาสตร์สมัยก่อน ได้ตั้งสมมติฐานเกี่ยวกับจำนวนสมบูรณ์จากจำนวนสมบูรณ์ที่เขารู้เพียง 4 ตัว ซึ่งสมมติฐานที่เขาได้ตั้งส่วนใหญ่จะผิด เช่น สมมติฐานที่ว่า เพราะว่า <i>n</i> = 2, 3, 5, 7 เป็นจำนวนเฉพาะ 4 ตัวแรก ที่นำไปแทนในสูตรแล้วได้ผลลัพธ์เป็นจำนวนสมบูรณ์ ดังนั้น <i>n</i> = 11 ซึ่งเป็นจำนวนเฉพาะตัวที่ 5 จะทำให้ได้ผลลัพธ์เป็นจำนวนสมบูรณ์เช่นกัน อย่างไรก็ตาม 2<sup>11</sup> − 1 = 2047 = 23 · 89 ซึ่งไม่เป็นจำนวนเฉพาะ ดังนั้น สมมติฐานนี้จึงผิด <b>สมมติฐานที่ผิด</b>อีกสองข้อ ได้แก่ </p> <ul><li>จำนวนสมบูรณ์ตัวที่ 5 จะต้องมี 5 หลัก เพราะว่าจำนวนสมบูรณ์ 4 ตัวแรกมี 1, 2, 3, 4 หลัก ตามลำดับ</li> <li>จำนวนสมบูรณ์จะลงท้ายด้วยเลข 6 หรือ 8 สลับกันเสมอ</li></ul> <p>จำนวนสมบูรณ์ตัวที่ห้า (<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 33550336=2^{12}(2^{13}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>33550336</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 33550336=2^{12}(2^{13}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723be6b5b68b7b183bfb6d19bcdb4dc681271dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.288ex; height:3.176ex;" alt="{\displaystyle 33550336=2^{12}(2^{13}-1)}"></span>) มี 8 หลัก ดังนั้นสมมติฐานข้อแรกจึงผิด. สำหรับสมมติฐานข้อสองนั้น แม้ว่าจำนวนสมบูรณ์ตัวที่ห้า จะลงท้ายด้วยเลข 6 แต่จำนวนสมบูรณ์ตัวที่หก (8 589 869 056) ไม่ได้ลงท้ายด้วยเลข 8 สมมติฐานข้อสองจึงผิด. เราสามารถพิสูจน์ได้ว่าจำนวนสมบูรณ์จะมีเลขหลักสุดท้ายเป็น 6 หรือ 8 เสมอ (แต่ไม่จำเป็นต้องสลับกัน) </p><p>คุณสมบัติของจำนวนสมบูรณ์อันหนึ่งที่น่าสนใจก็คือ <a href="/wiki/%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%9A" class="mw-redirect" title="ส่วนกลับ">ส่วนกลับ</a>ของตัวประกอบของจำนวนสมบูรณ์ จะรวมกันได้ 2 เสมอ เช่น </p> <ul><li>สำหรับ 6, จะได้ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/6+1/3+1/2+1/1=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/6+1/3+1/2+1/1=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30570fedd4b66a0bac48f8ff314fa46406d93e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.732ex; height:2.843ex;" alt="{\displaystyle 1/6+1/3+1/2+1/1=2}"></span></li> <li>สำหรับ 28, จะได้ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>28</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>14</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64b0bc6752f0147725ac8060fc2ed814c886ecfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.712ex; height:2.843ex;" alt="{\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="จำนวนสมบูรณ์คี่"><span id=".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.E0.B8.84.E0.B8.B5.E0.B9.88"></span>จำนวนสมบูรณ์คี่</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&action=edit&section=2" title="แก้ไขส่วน: จำนวนสมบูรณ์คี่"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>ยังไม่มีใครรู้ว่าจำนวนสมบูรณ์คี่นั้นมีอยู่จริงหรือไม่ เมื่อไม่นานมานี้ <a href="/w/index.php?title=Carl_Pomerance&action=edit&redlink=1" class="new" title="Carl Pomerance (ไม่มีหน้านี้)">Carl Pomerance</a> และ <a href="/w/index.php?title=Joshua_Zelinsky&action=edit&redlink=1" class="new" title="Joshua Zelinsky (ไม่มีหน้านี้)">Joshua Zelinsky</a> ได้แสดง<a href="/wiki/%E0%B8%AE%E0%B8%B4%E0%B8%A7%E0%B8%A3%E0%B8%B4%E0%B8%AA%E0%B8%95%E0%B8%B4%E0%B8%81" class="mw-redirect" title="ฮิวริสติก">ฮิวริสติก</a>ว่าไม่มีจำนวนสมบูรณ์คี่อยู่จริง </p><p>สมมติว่ามีจำนวนสมบูรณ์คี่ <i>N</i> อยู่จริงแล้ว มันจะต้องมีคุณสมบัติต่อไปนี้ </p> <ul><li><i>N</i> จะอยู่ในรูป</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\ldots p_{k}^{2e_{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\ldots p_{k}^{2e_{k}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b1a66c5b6bea7d6054798825a3306797084b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.341ex; height:3.509ex;" alt="{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\ldots p_{k}^{2e_{k}},}"></span></dd></dl></dd> <dd>เมื่อ <i>q</i>, <i>p</i><sub>1</sub>, …, <i>p</i><sub><i>k</i></sub> คือ จำนวนเฉพาะที่แตกต่างกัน และ <i>q</i> ≡ α ≡ 1 (mod 4) (<a href="/wiki/%E0%B8%AD%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C" class="mw-redirect" title="ออยเลอร์">ออยเลอร์</a>)</dd></dl> <ul><li><i>N</i> จะมากกว่า 10<sup>300</sup></li> <li><i>N</i> จะอยู่ในรูป 4<i>j</i> + 1 (A. Stern, 1896)</li> <li><i>N</i> จะมีตัวประกอบเฉพาะที่แตกต่างกันอย่างน้อย 8 ตัว (และอย่างน้อย 11 ตัว ถ้ามันหาร 3 ไม่ลงตัว) (<a href="/w/index.php?title=Peter_Hagis&action=edit&redlink=1" class="new" title="Peter Hagis (ไม่มีหน้านี้)">Peter Hagis</a>)</li> <li><i>N</i> จะมีตัวประกอบเฉพาะทั้งหมดอย่างน้อย 75 ตัว, รวมที่นับซ้ำด้วย (<a href="/w/index.php?title=Kevin_Hare&action=edit&redlink=1" class="new" title="Kevin Hare (ไม่มีหน้านี้)">Kevin Hare</a>, 2005)</li> <li><i>N</i> จะมีตัวประกอบเฉพาะอย่างน้อย 1 ตัวที่มากกว่า 10<sup>7</sup>, ตัวประกอบเฉพาะอย่างน้อย 2 ตัวที่มากกว่า 10<sup>4</sup>, ตัวประกอบเฉพาะอย่างน้อย 3 ตัวที่มากกว่า 100</li> <li><i>N</i> จะน้อยกว่า <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f0fa63c02614549b40f2bc329f4880fad12904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{4^{n}}}"></span> เมื่อ <i>n</i>=<i>k</i>+1 คือ จำนวนตัวประกอบเฉพาะที่แตกต่างกัน</li> <li><i>N</i> จะอยู่ในรูป 12<i>j</i> + 1 หริอ 36<i>j</i> + 9 (<a href="/w/index.php?title=Jacques_Touchard&action=edit&redlink=1" class="new" title="Jacques Touchard (ไม่มีหน้านี้)">Jacques Touchard</a>) (วิธีพิสูจน์พื้นฐานถูกค้นพบโดย <a href="/w/index.php?title=Judy_A._Holdener&action=edit&redlink=1" class="new" title="Judy A. Holdener (ไม่มีหน้านี้)">Judy A. Holdener</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="อ้างอิง"><span id=".E0.B8.AD.E0.B9.89.E0.B8.B2.E0.B8.87.E0.B8.AD.E0.B8.B4.E0.B8.87"></span>อ้างอิง</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&action=edit&section=3" title="แก้ไขส่วน: อ้างอิง"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> </div> <style data-mw-deduplicate="TemplateStyles:r8938631">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin" style=""> <ul><li>Kevin Hare, <i>New techniques for bounds on the total number of prime factors of an odd perfect number.</i> Preprint, 2005. อ่านได้จาก<a rel="nofollow" class="external text" href="http://www.math.uwaterloo.ca/~kghare/Preprints/">เว็บเพจของเขา</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051127040850/http://www.math.uwaterloo.ca/~kghare/Preprints/">เก็บถาวร</a> 2005-11-27 ที่ <a href="/wiki/%E0%B9%80%E0%B8%A7%E0%B8%A2%E0%B9%8C%E0%B9%81%E0%B8%9A%E0%B9%87%E0%B8%81%E0%B9%81%E0%B8%A1%E0%B8%8A%E0%B8%8A%E0%B8%B5%E0%B8%99" title="เวย์แบ็กแมชชีน">เวย์แบ็กแมชชีน</a></li></ul> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r10552737">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output 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id="เซตจำนวนเต็มตามคุณสมบัติหารลงตัว" style="font-size:114%;margin:0 4em">เซตจำนวนเต็มตามคุณสมบัติหารลงตัว</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">รูปแบบ<a href="/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%A2%E0%B8%81%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%81%E0%B8%AD%E0%B8%9A" title="การแยกตัวประกอบ">การแยกตัวประกอบ</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="จำนวนเฉพาะ">จำนวนเฉพาะ</a></li> <li><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%81%E0%B8%AD%E0%B8%9A" title="จำนวนประกอบ">จำนวนประกอบ</a></li> <li><a href="/w/index.php?title=Powerful_number&action=edit&redlink=1" class="new" title="Powerful number (ไม่มีหน้านี้)">Powerful number</a></li> <li><a href="/w/index.php?title=Square-free_number&action=edit&redlink=1" class="new" title="Square-free number (ไม่มีหน้านี้)">Square-free number</a></li> <li><a href="/w/index.php?title=Achilles_number&action=edit&redlink=1" class="new" title="Achilles number (ไม่มีหน้านี้)">Achilles number</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/%E0%B9%84%E0%B8%9F%E0%B8%A5%E0%B9%8C:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E0%B8%9C%E0%B8%A5%E0%B8%9A%E0%B8%A7%E0%B8%81" class="mw-redirect" title="ผลบวก">ผลบวก</a>ตัวหารตามเงื่อนไข</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">จำนวนสมบูรณ์</a></li> <li><a href="/w/index.php?title=Almost_perfect_number&action=edit&redlink=1" class="new" title="Almost perfect number (ไม่มีหน้านี้)">Almost perfect number</a></li> <li><a href="/w/index.php?title=Quasiperfect_number&action=edit&redlink=1" class="new" title="Quasiperfect number (ไม่มีหน้านี้)">Quasiperfect number</a></li> <li><a href="/w/index.php?title=Multiply_perfect_number&action=edit&redlink=1" class="new" title="Multiply perfect number (ไม่มีหน้านี้)">Multiply perfect number</a></li> <li><a href="/w/index.php?title=Hyperperfect_number&action=edit&redlink=1" class="new" title="Hyperperfect number (ไม่มีหน้านี้)">Hyperperfect number</a></li> <li><a href="/w/index.php?title=Unitary_perfect_number&action=edit&redlink=1" class="new" title="Unitary perfect number (ไม่มีหน้านี้)">Unitary perfect number</a></li> <li><a href="/w/index.php?title=Semiperfect_number&action=edit&redlink=1" class="new" title="Semiperfect number (ไม่มีหน้านี้)">Semiperfect number</a></li> <li><a href="/w/index.php?title=Primitive_semiperfect_number&action=edit&redlink=1" class="new" title="Primitive semiperfect number (ไม่มีหน้านี้)">Primitive semiperfect number</a></li> <li><a href="/w/index.php?title=Practical_number&action=edit&redlink=1" class="new" title="Practical number (ไม่มีหน้านี้)">Practical number</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">เปรียบเทียบผลบวกตัวหาร</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%81%E0%B8%B4%E0%B8%99&action=edit&redlink=1" class="new" title="จำนวนเกิน (ไม่มีหน้านี้)">จำนวนเกิน</a></li> <li><a href="/w/index.php?title=Highly_abundant_number&action=edit&redlink=1" class="new" title="Highly abundant number (ไม่มีหน้านี้)">Highly abundant number</a></li> <li><a href="/w/index.php?title=Superabundant_number&action=edit&redlink=1" class="new" title="Superabundant number (ไม่มีหน้านี้)">Superabundant number</a></li> <li><a href="/w/index.php?title=Colossally_abundant_number&action=edit&redlink=1" class="new" title="Colossally abundant number (ไม่มีหน้านี้)">Colossally abundant number</a></li> <li><a href="/w/index.php?title=Highly_composite_number&action=edit&redlink=1" class="new" title="Highly composite number (ไม่มีหน้านี้)">Highly composite number</a></li> <li><a href="/w/index.php?title=Superior_highly_composite_number&action=edit&redlink=1" class="new" title="Superior highly composite number (ไม่มีหน้านี้)">Superior highly composite number</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">อื่น ๆ</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9E%E0%B8%A3%E0%B9%88%E0%B8%AD%E0%B8%87&action=edit&redlink=1" class="new" title="จำนวนพร่อง (ไม่มีหน้านี้)">จำนวนพร่อง</a></li> <li><a href="/w/index.php?title=Weird_number&action=edit&redlink=1" class="new" title="Weird number (ไม่มีหน้านี้)">Weird number</a></li> <li><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%A3" title="จำนวนเชิงมิตร">จำนวนเชิงมิตร</a></li> <li><a href="/w/index.php?title=Friendly_number&action=edit&redlink=1" class="new" title="Friendly number (ไม่มีหน้านี้)">Friendly number</a></li> <li><a href="/w/index.php?title=Sociable_number&action=edit&redlink=1" class="new" title="Sociable number (ไม่มีหน้านี้)">Sociable number</a></li> <li><a href="/w/index.php?title=Solitary_number&action=edit&redlink=1" class="new" title="Solitary number (ไม่มีหน้านี้)">Solitary number</a></li> <li><a href="/w/index.php?title=Sublime_number&action=edit&redlink=1" class="new" title="Sublime number (ไม่มีหน้านี้)">Sublime number</a></li> <li><a href="/w/index.php?title=Harmonic_divisor_number&action=edit&redlink=1" class="new" title="Harmonic divisor number (ไม่มีหน้านี้)">Harmonic divisor number</a></li> <li><a href="/w/index.php?title=Frugal_number&action=edit&redlink=1" class="new" title="Frugal number (ไม่มีหน้านี้)">Frugal number</a></li> <li><a href="/w/index.php?title=Equidigital_number&action=edit&redlink=1" class="new" title="Equidigital number (ไม่มีหน้านี้)">Equidigital number</a></li> <li><a href="/w/index.php?title=Extravagant_number&action=edit&redlink=1" class="new" title="Extravagant number (ไม่มีหน้านี้)">Extravagant number</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <dl><dt>ดูเพิ่ม</dt> <dd><a href="/w/index.php?title=%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%AB%E0%B8%B2%E0%B8%A3&action=edit&redlink=1" class="new" title="ฟังก์ชันตัวหาร (ไม่มีหน้านี้)">ฟังก์ชันตัวหาร</a></dd> <dd><a href="/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%AB%E0%B8%B2%E0%B8%A3" title="ตัวหาร">ตัวหาร</a></dd> <dd><a href="/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%81%E0%B8%AD%E0%B8%9A%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="ตัวประกอบเฉพาะ">ตัวประกอบเฉพาะ</a></dd></dl> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r9895811">.mw-parser-output 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href="/wiki/%E0%B8%AB%E0%B8%A1%E0%B8%A7%E0%B8%94%E0%B8%AB%E0%B8%A1%E0%B8%B9%E0%B9%88:%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="หมวดหมู่:ลำดับจำนวนเต็ม">ลำดับจำนวนเต็ม</a></li><li><a href="/wiki/%E0%B8%AB%E0%B8%A1%E0%B8%A7%E0%B8%94%E0%B8%AB%E0%B8%A1%E0%B8%B9%E0%B9%88:%E0%B8%9B%E0%B8%B1%E0%B8%8D%E0%B8%AB%E0%B8%B2%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%B1%E0%B8%87%E0%B9%81%E0%B8%81%E0%B9%89%E0%B9%84%E0%B8%A1%E0%B9%88%E0%B9%84%E0%B8%94%E0%B9%89" title="หมวดหมู่:ปัญหาทางคณิตศาสตร์ที่ยังแก้ไม่ได้">ปัญหาทางคณิตศาสตร์ที่ยังแก้ไม่ได้</a></li><li><a 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