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href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E5%B9%B3%E6%96%B9%E6%95%B0" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%B9%B3%E6%96%B9%E6%95%B0" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</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"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</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/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-表达式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#表达式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>表达式</span> </div> </a> <ul id="toc-表达式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>性质</span> </div> </a> <ul id="toc-性质-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-註釋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註釋"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>註釋</span> </div> </a> <ul id="toc-註釋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考資料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>參考資料</span> </div> </a> <ul id="toc-參考資料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參看" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參看"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>參看</span> </div> </a> <ul id="toc-參看-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="目录" 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="开关目录" > <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">开关目录</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">平方数</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="前往另一种语言写成的文章。55种语言可用" > <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-55" 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">55种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9_%D9%83%D8%A7%D9%85%D9%84" title="مربع كامل – 阿拉伯语" lang="ar" hreflang="ar" data-title="مربع كامل" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99din_kvadrat%C4%B1" title="Ədədin kvadratı – 阿塞拜疆语" lang="az" hreflang="az" data-title="Ədədin kvadratı" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Квадратно число – 保加利亚语" lang="bg" hreflang="bg" data-title="Квадратно число" data-language-autonym="Български" data-language-local-name="保加利亚语" 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%AC%E0%A6%B0%E0%A7%8D%E0%A6%97_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বর্গ সংখ্যা – 孟加拉语" lang="bn" hreflang="bn" data-title="বর্গ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quadrat_perfecte" title="Quadrat perfecte – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Quadrat perfecte" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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/%D8%AF%D9%88%D9%88%D8%AC%D8%A7%DB%8C_%D8%AA%DB%95%D9%88%D8%A7%D9%88" title="دووجای تەواو – 中库尔德语" lang="ckb" hreflang="ckb" data-title="دووجای تەواو" data-language-autonym="کوردی" data-language-local-name="中库尔德语" 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/%C4%8Ctvercov%C3%A9_%C4%8D%C3%ADslo" title="Čtvercové číslo – 捷克语" lang="cs" hreflang="cs" data-title="Čtvercové číslo" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%83%D0%BB%D0%BB%D0%B8_%D1%82%C4%83%D0%B2%D0%B0%D1%82%D0%BA%D0%B0%D0%BB" title="Тулли тăваткал – 楚瓦什语" lang="cv" hreflang="cv" data-title="Тулли тăваткал" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_sgw%C3%A2r" title="Rhif sgwâr – 威尔士语" lang="cy" hreflang="cy" data-title="Rhif sgwâr" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvadrattal" title="Kvadrattal – 丹麦语" lang="da" hreflang="da" data-title="Kvadrattal" data-language-autonym="Dansk" data-language-local-name="丹麦语" 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/Quadratzahl" title="Quadratzahl – 德语" lang="de" hreflang="de" data-title="Quadratzahl" data-language-autonym="Deutsch" data-language-local-name="德语" 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%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Τετραγωνικός αριθμός – 希腊语" lang="el" hreflang="el" data-title="Τετραγωνικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" 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_quadr%C3%AA" title="Nùmer quadrê – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer quadrê" 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/Square_number" title="Square number – 英语" lang="en" hreflang="en" data-title="Square number" data-language-autonym="English" data-language-local-name="英语" 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/Kvadrata_nombro" title="Kvadrata nombro – 世界语" lang="eo" hreflang="eo" data-title="Kvadrata nombro" data-language-autonym="Esperanto" data-language-local-name="世界语" 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/Cuadrado_perfecto" title="Cuadrado perfecto – 西班牙语" lang="es" hreflang="es" data-title="Cuadrado perfecto" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B1%D8%A8%D8%B9_%DA%A9%D8%A7%D9%85%D9%84" title="مربع کامل – 波斯语" lang="fa" hreflang="fa" data-title="مربع کامل" data-language-autonym="فارسی" data-language-local-name="波斯语" 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/Neli%C3%B6luku" title="Neliöluku – 芬兰语" lang="fi" hreflang="fi" data-title="Neliöluku" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Carr%C3%A9_parfait" title="Carré parfait – 法语" lang="fr" hreflang="fr" data-title="Carré parfait" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kwadroottaal" title="Kwadroottaal – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Kwadroottaal" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cadrado_perfecto" title="Cadrado perfecto – 加利西亚语" lang="gl" hreflang="gl" data-title="Cadrado perfecto" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%97" title="વર્ગ – 古吉拉特语" lang="gu" hreflang="gu" data-title="વર્ગ" data-language-autonym="ગુજરાતી" data-language-local-name="古吉拉特语" class="interlanguage-link-target"><span>ગુજરાતી</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%A8%D7%99%D7%91%D7%95%D7%A2%D7%99" title="מספר ריבועי – 希伯来语" lang="he" hreflang="he" data-title="מספר ריבועי" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Potpuni_kvadrat" title="Potpuni kvadrat – 克罗地亚语" lang="hr" hreflang="hr" data-title="Potpuni kvadrat" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/%C5%A0tw%C3%B3rcowa_li%C4%8Dba" title="Štwórcowa ličba – 上索布语" lang="hsb" hreflang="hsb" data-title="Štwórcowa ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="上索布语" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzetsz%C3%A1mok" title="Négyzetszámok – 匈牙利语" lang="hu" hreflang="hu" data-title="Négyzetszámok" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_persegi" title="Bilangan persegi – 印度尼西亚语" lang="id" hreflang="id" data-title="Bilangan persegi" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/Ferningstala" title="Ferningstala – 冰岛语" lang="is" hreflang="is" data-title="Ferningstala" data-language-autonym="Íslenska" data-language-local-name="冰岛语" 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/Quadrato_perfetto" title="Quadrato perfetto – 意大利语" lang="it" hreflang="it" data-title="Quadrato perfetto" data-language-autonym="Italiano" data-language-local-name="意大利语" 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%B9%B3%E6%96%B9%E6%95%B0" title="平方数 – 日语" lang="ja" hreflang="ja" data-title="平方数" data-language-autonym="日本語" data-language-local-name="日语" 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%A0%95%EC%82%AC%EA%B0%81%EC%88%98" title="정사각수 – 韩语" lang="ko" hreflang="ko" data-title="정사각수" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_quadratus" title="Numerus quadratus – 拉丁语" lang="la" hreflang="la" data-title="Numerus quadratus" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skait%C4%BCa_kvadr%C4%81ts" title="Skaitļa kvadrāts – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Skaitļa kvadrāts" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="वर्ग (गणित) – 马拉地语" lang="mr" hreflang="mr" data-title="वर्ग (गणित)" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kuasa_dua_sempurna" title="Kuasa dua sempurna – 马来语" lang="ms" hreflang="ms" data-title="Kuasa dua sempurna" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwadraatgetal" title="Kwadraatgetal – 荷兰语" lang="nl" hreflang="nl" data-title="Kwadraatgetal" data-language-autonym="Nederlands" data-language-local-name="荷兰语" 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/Kvadrattal" title="Kvadrattal – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Kvadrattal" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" 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/Kvadrattall" title="Kvadrattall – 书面挪威语" lang="nb" hreflang="nb" data-title="Kvadrattall" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kaaslamee(square)" title="Kaaslamee(square) – 奥罗莫语" lang="om" hreflang="om" data-title="Kaaslamee(square)" data-language-autonym="Oromoo" data-language-local-name="奥罗莫语" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_kwadratowe" title="Liczby kwadratowe – 波兰语" lang="pl" hreflang="pl" data-title="Liczby kwadratowe" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_quadrado" title="Número quadrado – 葡萄牙语" lang="pt" hreflang="pt" data-title="Número quadrado" data-language-autonym="Português" data-language-local-name="葡萄牙语" 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/P%C4%83trat_perfect" title="Pătrat perfect – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Pătrat perfect" data-language-autonym="Română" data-language-local-name="罗马尼亚语" 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%9F%D0%BE%D0%BB%D0%BD%D1%8B%D0%B9_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82" title="Полный квадрат – 俄语" lang="ru" hreflang="ru" data-title="Полный квадрат" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvadrat_(algebra)" title="Kvadrat (algebra) – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Kvadrat (algebra)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Square_number" title="Square number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Square 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/%C5%A0tvorcov%C3%A9_%C4%8D%C3%ADslo" title="Štvorcové číslo – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Štvorcové číslo" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" 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/Kvadratno_%C5%A1tevilo" title="Kvadratno število – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Kvadratno število" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வர்க்கம் (கணிதம்) – 泰米尔语" lang="ta" hreflang="ta" data-title="வர்க்கம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" 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%B5%E0%B0%B0%E0%B1%8D%E0%B0%97_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="వర్గ సంఖ్యలు – 泰卢固语" lang="te" hreflang="te" data-title="వర్గ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tam_kare" title="Tam kare – 土耳其语" lang="tr" hreflang="tr" data-title="Tam kare" data-language-autonym="Türkçe" data-language-local-name="土耳其语" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Квадратне число – 乌克兰语" lang="uk" hreflang="uk" data-title="Квадратне число" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ch%C3%ADnh_ph%C6%B0%C6%A1ng" title="Số chính phương – 越南语" lang="vi" hreflang="vi" data-title="Số chính phương" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%95%D7%95%D7%90%D7%93%D7%A8%D7%90%D7%98%D7%A6%D7%90%D7%9C" title="קוואדראטצאל – 意第绪语" lang="yi" hreflang="yi" data-title="קוואדראטצאל" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%95%B8" title="平方數 – 文言文" lang="lzh" hreflang="lzh" data-title="平方數" data-language-autonym="文言" data-language-local-name="文言文" 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%B9%B3%E6%96%B9%E6%95%B8" title="平方數 – 粤语" lang="yue" hreflang="yue" data-title="平方數" data-language-autonym="粵語" data-language-local-name="粤语" 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/Q50705#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div 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class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Square_numbers" hreflang="en"><span>维基共享资源</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/Q50705" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</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="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <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">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</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">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">（重定向自<a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B8&redirect=no" class="mw-redirect" title="平方數">平方數</a>）</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Refimprove plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>需要补充更多<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2019年4月21日</span>)</i></small><span class="hide-when-compact"><br /><small>请协助補充多方面<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit">改善这篇条目</a>，<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AF%81" class="mw-redirect" title="Wikipedia:可供查证">无法查证</a>的内容可能會因為<a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">异议提出</a>而被移除。<br />致使用者：请搜索一下条目的标题（来源搜索：<span class="plainlinks"><a rel="nofollow" class="external text" href="//www.google.com/search?&as_eq=wikipedia&q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">"平方数"</a> — <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">网页</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">新闻</a>、<a rel="nofollow" class="external text" href="//books.google.com/books?&as_brr=0&as_pub=-icon&q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">书籍</a>、<a rel="nofollow" class="external text" href="//scholar.google.com/scholar?&q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">学术</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=isch&safe=off&q=%22%E5%B9%B3%E6%96%B9%E6%95%B0%22">图像</a></span>），以检查网络上是否存在该主题的更多可靠来源（<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">判定指引</a>）。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:X_square.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/X_square.png/350px-X_square.png" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/27/X_square.png 1.5x" data-file-width="500" data-file-height="500" /></a><figcaption><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 y=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1108c4c9ee8ac7de90b77f9bd27415b13b6bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.638ex; height:3.009ex;" alt="{\displaystyle y=x^{2}}"></span> 的<a href="/wiki/%E5%87%BD%E6%95%B8%E5%9C%96%E5%BD%A2" class="mw-redirect" title="函數圖形">函數圖形</a>。</figcaption></figure> <p><a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a>上，<b>平方数</b>，或称<b>完全平方数</b>，是指可以写成某个<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>的<a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a>的数，即其<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根">平方根</a>为<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>的数。例如，9 = 3 × 3，它是一个平方数。 </p><p>平方数也称<b>正方形数</b>，若 <i>n</i> 为平方数，将 <i>n</i> 个点排成<a href="/wiki/%E7%9F%A9%E5%BD%A2" title="矩形">矩形</a>，可以排成一个<a href="/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形">正方形</a>。 </p><p>若将平方数概念扩展到<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>，则两个平方数的比仍然是平方数，例如， (2 × 2) / (3 × 3) = 4/9 = 2/3 × 2/3。 </p><p>若一个整数没有除了 1 之外的平方数为其<a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a>，则称其为<a href="/wiki/%E6%97%A0%E5%B9%B3%E6%96%B9%E6%95%B0%E5%9B%A0%E6%95%B0%E7%9A%84%E6%95%B0" class="mw-redirect" title="无平方数因数的数">无平方数因数的数</a>。 </p><p>前n個平方數 </p><p>（<a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">OEIS</a>數列<a href="//oeis.org/A000290" class="extiw" title="oeis:A000290">A000290</a>）： </p> <div style="float:left; padding: 1em;"> <dl><dd>0<sup>2</sup> = <a href="/wiki/0" title="0">0</a></dd></dl> </div> <div style="float:left; padding: 1em;"> <dl><dd>1<sup>2</sup> = <a href="/wiki/1" title="1">1</a></dd> <dd>2<sup>2</sup> = <a href="/wiki/4" title="4">4</a></dd> <dd>3<sup>2</sup> = <a href="/wiki/9" title="9">9</a></dd> <dd>4<sup>2</sup> = <a href="/wiki/16" title="16">16</a></dd> <dd>5<sup>2</sup> = <a href="/wiki/25" title="25">25</a></dd> <dd>6<sup>2</sup> = <a href="/wiki/36" title="36">36</a></dd> <dd>7<sup>2</sup> = <a href="/wiki/49" title="49">49</a></dd> <dd>8<sup>2</sup> = <a href="/wiki/64" title="64">64</a></dd> <dd>9<sup>2</sup> = <a href="/wiki/81" title="81">81</a></dd> <dd>10<sup>2</sup> = <a href="/wiki/100" title="100">100</a></dd></dl> </div> <div style="float:left; padding: 1em;"> <dl><dd>11<sup>2</sup> = <a href="/wiki/121" title="121">121</a></dd> <dd>12<sup>2</sup> = <a href="/wiki/144" title="144">144</a></dd> <dd>13<sup>2</sup> = <a href="/wiki/169" title="169">169</a></dd> <dd>14<sup>2</sup> = <a href="/wiki/196" title="196">196</a></dd> <dd>15<sup>2</sup> = <a href="/wiki/225" title="225">225</a></dd> <dd>16<sup>2</sup> = <a href="/wiki/256" title="256">256</a></dd> <dd>17<sup>2</sup> = <a href="/wiki/289" title="289">289</a></dd> <dd>18<sup>2</sup> = <a href="/wiki/324" title="324">324</a></dd> <dd>19<sup>2</sup> = <a href="/wiki/361" class="mw-redirect" title="361">361</a></dd> <dd>20<sup>2</sup> = <a href="/wiki/400" title="400">400</a></dd></dl> </div> <div style="float:left; padding: 1em;"> <dl><dd>21<sup>2</sup> = <a href="/wiki/441" class="mw-redirect" title="441">441</a></dd> <dd>22<sup>2</sup> = <a href="/wiki/484" class="mw-redirect" title="484">484</a></dd> <dd>23<sup>2</sup> = 529</dd> <dd>24<sup>2</sup> = <a href="/wiki/576" class="mw-redirect" title="576">576</a></dd> <dd>25<sup>2</sup> = 625</dd> <dd>26<sup>2</sup> = 676</dd> <dd>27<sup>2</sup> = 729</dd> <dd>28<sup>2</sup> = 784</dd> <dd>29<sup>2</sup> = 841</dd> <dd>30<sup>2</sup> = <a href="/wiki/900" title="900">900</a></dd></dl> </div> <div style="float:left; padding: 1em;"> <dl><dd>31<sup>2</sup> = 961</dd> <dd>32<sup>2</sup> = <a href="/wiki/1024" title="1024">1024</a></dd> <dd>33<sup>2</sup> = <a href="/wiki/1089" title="1089">1089</a></dd> <dd>34<sup>2</sup> = <a href="/wiki/1156" class="mw-redirect" title="1156">1156</a></dd> <dd>35<sup>2</sup> = 1225</dd> <dd>36<sup>2</sup> = 1296</dd> <dd>37<sup>2</sup> = 1369</dd> <dd>38<sup>2</sup> = 1444</dd> <dd>39<sup>2</sup> = 1521</dd> <dd>40<sup>2</sup> = <a href="/wiki/1600" class="mw-redirect" title="1600">1600</a></dd></dl> </div> <div style="float:left; padding: 1em;"> <dl><dd>41<sup>2</sup> = 1681</dd> <dd>42<sup>2</sup> = 1764</dd> <dd>43<sup>2</sup> = 1849</dd> <dd>44<sup>2</sup> = 1936</dd> <dd>45<sup>2</sup> = <a href="/wiki/2025" class="mw-redirect" title="2025">2025</a></dd> <dd>46<sup>2</sup> = 2116</dd> <dd>47<sup>2</sup> = 2209</dd> <dd>48<sup>2</sup> = 2304</dd> <dd>49<sup>2</sup> = 2401</dd> <dd>50<sup>2</sup> = 2500</dd></dl> </div> <div style="clear:both;"></div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="表达式"><span id=".E8.A1.A8.E8.BE.BE.E5.BC.8F"></span>表达式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit&section=1" title="编辑章节：表达式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个整数是完全平方数当且仅当相同数目的点能够在平面上排成一个正方形的点阵，使得每行每列的点都一样多。 </p> <table cellpadding="8"> <tbody><tr> <td><a href="/wiki/1" title="1">1</a><sup>2</sup> = 1 </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Square_number_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/78/Square_number_1.png" decoding="async" width="136" height="16" class="mw-file-element" data-file-width="136" data-file-height="16" /></a></span> </td></tr> <tr> <td><a href="/wiki/2" title="2">2</a><sup>2</sup> = 4 </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Square_number_4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e1/Square_number_4.png" decoding="async" width="152" height="32" class="mw-file-element" data-file-width="152" data-file-height="32" /></a></span> </td></tr> <tr> <td><a href="/wiki/3" title="3">3</a><sup>2</sup> = 9 </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Square_number_9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/4c/Square_number_9.png" decoding="async" width="168" height="48" class="mw-file-element" data-file-width="168" data-file-height="48" /></a></span> </td></tr> <tr> <td><a href="/wiki/4" title="4">4</a><sup>2</sup> = 16 </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Square_number_16.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f8/Square_number_16.png" decoding="async" width="184" height="64" class="mw-file-element" data-file-width="184" data-file-height="64" /></a></span> </td></tr> <tr> <td><a href="/wiki/5" title="5">5</a><sup>2</sup> = 25 </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Square_number_25.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0a/Square_number_25.png" decoding="async" width="200" height="80" class="mw-file-element" data-file-width="200" data-file-height="80" /></a></span> </td></tr></tbody></table> <dl><dt><style data-mw-deduplicate="TemplateStyles:r79700083">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}</style><span class="vanchor"><span class="anchor" id="通项公式"></span><span class="anchor" id="通项公式"></span><span>通项公式</span></span></dt></dl> <p>对于一个整数 <i>n</i>，它的<a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a>写成 <i>n</i><sup>2</sup>。<i>n</i><sup>2</sup>等于头 <i>n</i> 个正<a href="/wiki/%E5%A5%87%E6%95%B0" class="mw-redirect" title="奇数">奇数</a>的和（<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}=\sum _{k=1}^{n}(2k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04871a4c1ba6b61eebf21135580311ff72382e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.088ex; height:6.843ex;" alt="{\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1)}"></span>）。在上图中，从1开始，第 <i>n</i> 个平方数表示为前一个平方数加上第 <i>n</i> 个正奇数，如 5<sup>2</sup> = 25 = 1 + 3 + 5 + 7 + 9 = 16 + 9。即第五个平方数25等于第四个平方数16加上第五个正奇数：9。 </p> <dl><dt><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r79700083"><span class="vanchor"><span class="anchor" id="递归公式"></span><span class="anchor" id="递归公式"></span><span>递归公式</span></span></dt></dl> <p>每个平方数可以从之前的两个平方数计算得到，递推公式为 <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}=2(n-1)^{2}-(n-2)^{2}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}=2(n-1)^{2}-(n-2)^{2}+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e71b2c4bd9b64d130500943af67070bf12f1198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.075ex; height:3.176ex;" alt="{\displaystyle n^{2}=2(n-1)^{2}-(n-2)^{2}+2}"></span>。例如，2×5<sup>2</sup> − 4<sup>2</sup> + 2 = 2×25 − 16 + 2 = 50 − 16 + 2 = 36 = 6<sup>2</sup>。 </p> <dl><dt><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r79700083"><span class="vanchor"><span class="anchor" id="连续整数的和"></span><span class="anchor" id="连续整数的和"></span><span>连续整数的和</span></span></dt></dl> <p>平方数还可以表示成 <i>n</i><sup>2</sup> = 1 + 1 + 2 + 2 + ... + <i>n</i> − 1 + <i>n</i> − 1 + <i>n</i>。例如，4<sup>2</sup> = 16 = 1 + 1 + 2 + 2 + 3 + 3 + 4。可以将其解释为在边长为 3 的矩形上添加宽度为 1 的一行和一列，即得到边长为 4 的矩形。这对于计算较大的数的平方数非常有用。例如， 52<sup>2</sup> = 50<sup>2</sup> + 50 + 51 + 51 + 52 = 2500 + 204 = 2704. </p> <div class="mw-heading mw-heading2"><h2 id="性质"><span id=".E6.80.A7.E8.B4.A8"></span>性质</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit&section=2" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>一个平方数是两个相邻<a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="三角形數">三角形數</a>之和。两个相邻平方数之和为一个<a href="/wiki/%E4%B8%AD%E5%BF%83%E6%AD%A3%E6%96%B9%E5%BD%A2%E6%95%B8" title="中心正方形數">中心正方形數</a>。所有的奇数平方数同时也是中心八边形数。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li></ul> <ul><li><a href="/wiki/%E5%9B%9B%E5%B9%B3%E6%96%B9%E5%92%8C%E5%AE%9A%E7%90%86" title="四平方和定理">四平方和定理</a>說明所有正整数均可表示为最多四个平方数的和。特别的，三个平方数之和不能表示形如 4<sup><i>k</i></sup>(8<i>m</i> + 7) 的数。<a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>一个正整数可以表示<a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a>中没有形如 4<i>k</i> + 3 的素数的奇次方，则它可以表示成两个平方数之和。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li></ul> <ul><li>在<a href="/wiki/%E5%8D%81%E8%BF%9B%E5%88%B6" title="十进制">十进制</a>中，平方数只能以 1，4，6，9 或 00 25 结尾。</li></ul> <ol><li>若一个数以 0 结尾，它的平方数以 0 结尾（除 0 外，其他數字的個位和十位數字都是 0 ），且00前面的數也是平方数（例如：0x0=0、10x10=100）</li> <li>若一个数以 1 或 9 结尾，它的平方数以 1 结尾，且前面的兩位數字构成的兩位数能被 4 整除（例如：1x1=1、11x11=121；9x9=81、19x19=361）</li> <li>若一个数以 2 或 8 结尾，它的平方数以 4 结尾，且前面的一位數字為偶数（例如：2x2=4、12x12=144；8x8=64、18x18=324）</li> <li>若一个数以 3 或 7 结尾，它的平方数以 9 结尾，且前面的兩位數字构成的兩位数能被 4 整除（例如：3x3=9、13x13=169；7x7=49、17x17=289）</li> <li>若一个数以 4 或 6 结尾，它的平方数以 6 结尾，且前面的一位數字為奇数（例如：4x4=16、14x14=196；6x6=36、16x16=256）</li> <li>若一个数以 5 结尾，它的平方数以 25 结尾，且前面的一位或两位数字必定为 0，2，06，56 之一，25前面的數是<a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B8" class="mw-redirect" title="普洛尼克數">普洛尼克數</a>（例如：5x5=25、15x15=225）</li></ol> <p>至於為什麼祇能以00、25结尾，可以將該數字除以100。可以發現，n.5若寫成分數形式，則為(2n+1)/2。設2n+1=p，則p與n互質。根據<a href="/wiki/%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E5%85%AC%E5%BC%8F" class="mw-redirect" title="完全平方公式">完全平方公式</a>可得，( 2n/2 + 1/2 )^2=n^2 + 1 + 0.25。由於前面均為整數，所以最終結果小數部分必為.25。乘以100后，則最後兩位必為25。 </p> <ul><li>在<a href="/wiki/%E5%8D%81%E4%BA%8C%E8%BF%9B%E5%88%B6" title="十二进制">十二进制</a>中，平方數的末位數必定是平方數（0, 1, 4或9）：<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li></ul> <ol><li>若一個數同時是2和3的倍數（也就是為6的倍數），它的平方数以 0 结尾，且前面的一位數字為0或3。</li> <li>若一個數既不是2的倍數也不是3的倍數（也就是與12互質），它的平方数以 1 结尾，且前面的一位數字為偶数。</li> <li>若一個數是2的倍數但不是3的倍數，它的平方数以 4 结尾，且前面的一位數字除以4的餘數為0或1（也就是說，前一位數為0,1,4,5,8,9）。</li> <li>若一個數不是2的倍數而是3的倍數，它的平方数以 9 结尾，且前面的一位數字為0或6。</li></ol> <ul><li>每4个连续的<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</a>相乘加 1，必定会等於一个平方数，即<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(n+1)(n+2)(n+3)+1=(n^{2}+3n+1)^{2}=[n+(n+1)^{2}]^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(n+1)(n+2)(n+3)+1=(n^{2}+3n+1)^{2}=[n+(n+1)^{2}]^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aec4b311a9c21dfc77ebac3f703ce22a35aa127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.772ex; height:3.176ex;" alt="{\displaystyle n(n+1)(n+2)(n+3)+1=(n^{2}+3n+1)^{2}=[n+(n+1)^{2}]^{2}}"></span>。<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span id="noteTag-cite_ref-sup"><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>註 1<span class="cite-bracket">]</span></a></sup></span></li></ul> <ul><li>平方数必定不是<a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a>。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>註 2<span class="cite-bracket">]</span></a></sup></span></li> <li>平方數必定是3的倍數或者3的倍數+1。</li> <li>平方數必定是4的倍數或者4的倍數+1。<br />（以上兩者均包括 0 （ 0 倍））</li></ul> <ul><li>0以外的平方數每一位數數字相加之和，不停重複地相加到剩一位數時必定是 1, 4, 9, 7 。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>註 3<span class="cite-bracket">]</span></a></sup></span></li></ul> <ul><li>是否在相继正方形数之间存在一个素数这一命题，对9000000以内的数目是正确的。<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>除了00以外，平方數末2位數若相同，必為44：如12<sup>2</sup>=144，38<sup>2</sup>=1444，62<sup>2</sup>=3844。</li> <li>除了000以外，平方數末3位數若相同，必為444：如38<sup>2</sup>=1444，462<sup>2</sup>=213444。<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>除了0000以外，平方數末4位數不可能相同。</li> <li>除了0以外，平方數不可能是<a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B8" class="mw-redirect" title="普洛尼克數">普洛尼克數</a>。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>註 4<span class="cite-bracket">]</span></a></sup></span>。</li> <li>除了0以外，平方數也不可能是連續若干個（至少兩個）數的積。</li> <li>除了0，1，<a href="/wiki/144" title="144">144</a>以外，平方數不可能是<a href="/wiki/%E8%B2%BB%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B8" class="mw-redirect" title="費波那契數">費波那契數</a>。<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>除了1跟4以外，平方數也<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><mark class="template-facttext" title="需要提供文献来源">不可能</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup>是<a href="/wiki/%E5%8D%A2%E5%8D%A1%E6%96%AF%E6%95%B0" title="卢卡斯数">盧卡斯數</a>。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>除了0，1，<a href="/wiki/169" title="169">169</a>以外，平方數<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><mark class="template-facttext" title="需要提供文献来源">不可能</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup>是<a href="/wiki/%E4%BD%A9%E5%B0%94%E6%95%B0" title="佩尔数">佩爾數</a>。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>除了0，1，<a href="/wiki/4" title="4">4</a>，19600以外，平方數不可能是<a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">四面體數</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>。</li> <li>除了1跟4900以外，平方數不可能是<a href="/wiki/%E5%9B%9B%E8%A7%92%E9%8C%90%E6%95%B8" title="四角錐數">四角錐數</a>。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>平方數不可能是<a href="/wiki/%E6%A5%94%E5%BD%A2%E6%95%B8" class="mw-redirect" title="楔形數">楔形數</a>。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>平方數是模任何整數的<a href="/wiki/%E4%BA%8C%E6%AC%A1%E5%89%A9%E4%BD%99" title="二次剩余">二次剩餘</a>；另外，如果某個整數是模任何整數的二次剩餘，那麼她一定是平方數。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>平方數的正<a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a>總和(含自己)一定是<a href="/wiki/%E5%A5%87%E6%95%B8" class="mw-redirect" title="奇數">奇數</a>。<sup class="noprint Inline-Template" style="white-space:nowrap;">[<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證"><span title="该标签附近的材料需要与引用的来源进行查证。">查证请求</span></a>]</sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83946278"><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup><sup class="noprint"><a href="/wiki/Wikipedia:%E9%9D%9E%E5%8E%9F%E5%88%9B%E7%A0%94%E7%A9%B6" title="Wikipedia:非原创研究"><span title="原创研究验证请求"><span style="white-space: nowrap;">[原創研究？]</span></span></a></sup></li> <li>平方數的正<a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a>個數是<a href="/wiki/%E5%A5%87%E6%95%B8" class="mw-redirect" title="奇數">奇數</a>。<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>當<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 m\leq 300000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≤</mo> <mn>300000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\leq 300000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2150abe87c31b94820c3639ef7ea7660a6c581b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.114ex; height:2.343ex;" alt="{\displaystyle m\leq 300000}"></span>時，<a href="/wiki/%E4%B8%8D%E5%AE%9A%E6%96%B9%E7%A8%8B" class="mw-redirect" title="不定方程">不定方程</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}+2^{2}+3^{2}+......+m^{2}=\sum _{k=1}^{m}k^{2}={\color {Red}{\frac {m(m+1)(2m+1)}{6}}=n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}+2^{2}+3^{2}+......+m^{2}=\sum _{k=1}^{m}k^{2}={\color {Red}{\frac {m(m+1)(2m+1)}{6}}=n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1274b98535b06211d1dc27984c656304fba0d184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.741ex; height:6.843ex;" alt="{\displaystyle 1^{2}+2^{2}+3^{2}+......+m^{2}=\sum _{k=1}^{m}k^{2}={\color {Red}{\frac {m(m+1)(2m+1)}{6}}=n^{2}}}"></span>的正整數解(m , n)只有(1 , 1)與(24 , 70)。</li></ul> <div class="mw-heading mw-heading2"><h2 id="註釋"><span id=".E8.A8.BB.E9.87.8B"></span>註釋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit&section=3" title="编辑章节：註釋"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="references-NoteFoot"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">更一般地，任何整數等差數列連續4項之乘積加上公差的4次方必為平方數，亦即a(a+d)(a+2d)(a+3d)+d<sup>4</sup>=(a<sup>2</sup>+3ad+d<sup>2</sup>)<sup>2</sup>。當公差d=1時，即為前述性質。</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">因為<a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a>的正因數總和(含自己)必為偶數，但平方數的正因數總和必為奇數。</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">亦即0以外的平方數必為9的倍數+1, 9的倍數+4, 9的倍數+9, 9的倍數+7 。</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">因為n與(n+1)差1，所以兩數互質，故若n×(n+1)為平方數，則n與(n+1)也皆為平方數，2個平方數差1，則必為0與1，因此唯一的普洛尼克數兼平方數為0=0×1。</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="參考資料"><span id=".E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit&section=4" title="编辑章节：參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web"><a href="/wiki/Neil_Sloane" class="mw-redirect" title="Neil Sloane">Sloane, N.J.A.</a> (编). <a rel="nofollow" class="external text" href="https://oeis.org/A062938">Sequence A062938 (a(n)= n*(n+1)*(n+2)*(n+3)+1 = (n^2 +3*n + 1)^2.)</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" class="mw-redirect" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%95%B0&rft.atitle=Sequence%26%23x20%3BA062938%26%23x20%3B%28a%28n%29%3D+n%2A%28n%2B1%29%2A%28n%2B2%29%2A%28n%2B3%29%2B1+%3D+%28n%5E2+%2B3%2An+%2B+1%29%5E2.%29&rft.aufirst=N.J.A.&rft.aulast=Sloane&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft_id=https%3A%2F%2Foeis.org%2FA062938&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web"><a href="/wiki/Neil_Sloane" class="mw-redirect" title="Neil Sloane">Sloane, N.J.A.</a> (编). <a rel="nofollow" class="external text" href="https://oeis.org/A028387">Sequence A028387</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" class="mw-redirect" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%95%B0&rft.atitle=Sequence%26%23x20%3BA028387&rft.aufirst=N.J.A.&rft.aulast=Sloane&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft_id=https%3A%2F%2Foeis.org%2FA028387&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">《数论妙趣》267页[美国]阿尔伯特-贝勒著谈祥柏译，上海教育出版社，<a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787532054732" class="internal mw-magiclink-isbn">ISBN 9787532054732</a>。</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web">Bernard Schott. <a rel="nofollow" class="external text" href="https://oeis.org/A039685">Numbers m such that m^2 ends in 444.</a>. <a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">整數數列線上大全</a>. 2019-10-31 <span class="reference-accessdate"> [<span class="nowrap">2023-05-27</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20230527104625/https://oeis.org/A039685">存档</a>于2023-05-27）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%95%B0&rft.au=Bernard+Schott&rft.btitle=Numbers+m+such+that+m%5E2+ends+in+444.&rft.date=2019-10-31&rft.genre=unknown&rft.pub=%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8&rft_id=https%3A%2F%2Foeis.org%2FA039685&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation web">JOHN H. E. COHN. <a rel="nofollow" class="external text" href="https://archive.today/20120630214035/http://math.la.asu.edu/~checkman/SquareFibonacci.html">〈Square Fibonacci Numbers, Etc.〉</a>. Bedford College, University of London, London, N.W.1. <span class="reference-accessdate"> [<span class="nowrap">2019-05-12</span>]</span>. （<a rel="nofollow" class="external text" href="https://math.la.asu.edu/~checkman/SquareFibonacci.html">原始内容</a>存档于2012-06-30）. <q><u>Theorem 3.</u> If F<sub>n</sub> = x<sup>2</sup>, then n = 0, ±1, 2 or 12.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%95%B0&rft.au=JOHN+H.+E.+COHN&rft.btitle=%E3%80%88Square+Fibonacci+Numbers%2C+Etc.%E3%80%89&rft.genre=unknown&rft.pub=Bedford+College%2C+University+of+London%2C+London%2C+N.W.1.&rft_id=https%3A%2F%2Fmath.la.asu.edu%2F~checkman%2FSquareFibonacci.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 600.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 165 (Rev. ed. 1997). </span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite class="citation web">郭耀元. <a rel="nofollow" class="external text" href="https://archive.today/20180106113331/http://210.60.110.11/reading/wp-content/uploads/2015/05/10303003.pdf">探討完全平方數在數論領域中之研究</a> <span style="font-size:85%;">(PDF)</span>. 私立高英高級工商職業學校. （<a rel="nofollow" class="external text" href="http://210.60.110.11/reading/wp-content/uploads/2015/05/10303003.pdf">原始内容</a> <span style="font-size:85%;">(PDF)</span>存档于2018年1月6日）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%95%B0&rft.au=%E9%83%AD%E8%80%80%E5%85%83&rft.btitle=%E6%8E%A2%E8%A8%8E%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B8%E5%9C%A8%E6%95%B8%E8%AB%96%E9%A0%98%E5%9F%9F%E4%B8%AD%E4%B9%8B%E7%A0%94%E7%A9%B6&rft.genre=unknown&rft.pub=%E7%A7%81%E7%AB%8B%E9%AB%98%E8%8B%B1%E9%AB%98%E7%B4%9A%E5%B7%A5%E5%95%86%E8%81%B7%E6%A5%AD%E5%AD%B8%E6%A0%A1&rft_id=http%3A%2F%2F210.60.110.11%2Freading%2Fwp-content%2Fuploads%2F2015%2F05%2F10303003.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="參看"><span id=".E5.8F.83.E7.9C.8B"></span>參看</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%95%B0&action=edit&section=5" title="编辑章节：參看"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">立方數</a>：平方數在立體的推廣</li> <li><a href="/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E6%95%B8" title="四次方數">四次方數</a>：平方數在<a href="/wiki/%E5%9B%9B%E7%BB%B4%E7%A9%BA%E9%97%B4" title="四维空间">四維空間</a>的推廣</li> <li><a href="/wiki/%E4%BA%94%E6%AC%A1%E6%96%B9%E6%95%B8" title="五次方數">五次方數</a>：平方數在<a href="/wiki/%E4%BA%94%E7%BB%B4%E7%A9%BA%E9%97%B4" title="五维空间">五維空間</a>的推廣</li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B0" class="mw-redirect" title="三角形数">三角形数</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B9%B3%E6%96%B9%E6%95%B8" title="三角平方數">三角平方數</a>：同時為三角形數和平方數的數</li> <li><a href="/wiki/%E5%A4%9A%E9%82%8A%E5%BD%A2%E6%95%B8" title="多邊形數">多邊形數</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/%E7%BB%B4%E5%9F%BA%E5%85%B1%E4%BA%AB%E8%B5%84%E6%BA%90" title="维基共享资源">维基共享资源</a>上的相关多媒体资源：<a href="https://commons.wikimedia.org/wiki/Category:Square_numbers" class="extiw" title="commons:Category:Square numbers"><span style="font-weight:bold;">平方数</span></a></div></div> </div> <div class="navbox-styles"><style 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style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="三角形數">三角形數</a></li> <li><a class="mw-selflink selflink">四邊形數</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%95%B0" title="五角数">五邊形數</a></li> <li><a href="/wiki/%E5%85%AD%E9%82%8A%E5%BD%A2%E6%95%B8" title="六邊形數">六邊形數</a></li> <li><a href="/wiki/%E4%B8%83%E9%82%8A%E5%BD%A2%E6%95%B8" title="七邊形數">七邊形數</a></li> <li><a href="/wiki/%E5%85%AB%E9%82%8A%E5%BD%A2%E6%95%B8" title="八邊形數">八邊形數</a></li> <li><a href="/wiki/%E4%B9%9D%E9%82%8A%E5%BD%A2%E6%95%B8" title="九邊形數">九邊形數</a></li> <li><a href="/wiki/%E5%8D%81%E9%82%8A%E5%BD%A2%E6%95%B8" title="十邊形數">十邊形數</a></li> <li><a href="/wiki/%E5%8D%81%E4%BA%8C%E9%82%8A%E5%BD%A2%E6%95%B8" title="十二邊形數">十二邊形數</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Triangle10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/100px-Triangle10.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/150px-Triangle10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/200px-Triangle10.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">多面體數</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">四面體數</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">六面體數</a></li> <li><a href="/wiki/%E5%85%AB%E9%9D%A2%E9%AB%94%E6%95%B8" title="八面體數">八面體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">錐體數</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">三角錐數</a></li> <li><a href="/wiki/%E5%9B%9B%E8%A7%92%E9%8C%90%E6%95%B8" title="四角錐數">四角錐數</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E9%8C%90%E6%95%B8" title="五角錐數">五角錐數</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E9%8C%90%E6%95%B8" title="六角錐數">六角錐數</a></li> <li><a href="/wiki/%E4%B8%83%E8%A7%92%E9%94%A5%E6%95%B0" title="七角锥数">七角錐數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%A4%9A%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心多邊形數">中心多邊形數</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="中心三角形數">中心三角形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E6%AD%A3%E6%96%B9%E5%BD%A2%E6%95%B8" title="中心正方形數">中心四邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%BA%94%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心五邊形數">中心五邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%85%AD%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心六邊形數">中心六邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%B8%83%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心七邊形數">中心七邊形數</a></li> <li><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">中心十二邊形數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">中心多面體數</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="中心四面體數">中心四面體數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E7%AB%8B%E6%96%B9%E9%AB%94%E6%95%B8" title="中心立方體數">中心六面體數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%85%AB%E9%9D%A2%E9%AB%94%E6%95%B8" title="中心八面體數">中心八面體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">多胞體數</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">1-多胞體數</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="三角形數">2-多胞體數</a></li> <li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">3-多胞體數</a></li> <li><a href="/wiki/%E4%BA%94%E8%83%9E%E9%AB%94%E6%95%B8" title="五胞體數">4-多胞體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">星數</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="五角星數">五角星數</a></li> <li><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">六角星數</a></li> <li><a href="/wiki/%E4%B8%83%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="七角星數">七角星數</a></li> <li><a href="/wiki/%E5%85%AB%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="八角星數">八角星數</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E6%98%9F%E8%B3%AA%E6%95%B8" class="mw-redirect" title="六角星質數">六角星質數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他<a href="/wiki/%E6%9C%89%E5%BD%A2%E6%95%B8" title="有形數">有形數</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="mw-selflink selflink">平方数</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">立方數</a></li> <li><a href="/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E6%95%B8" title="四次方數">四次方數</a></li> <li><a href="/wiki/%E4%BA%94%E6%AC%A1%E6%96%B9%E6%95%B8" title="五次方數">五次方數</a></li> <li><a href="/wiki/%E5%85%AD%E6%AC%A1%E6%96%B9%E6%95%B8" title="六次方數">六次方數</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B9%B3%E6%96%B9%E6%95%B8" title="三角平方數">三角平方數</a></li> <li><a href="/wiki/%E6%A2%AF%E5%BD%A2%E6%95%B8" class="mw-redirect" title="梯形數">梯形數</a></li> <li><a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B0" title="普洛尼克数">普洛尼克数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B4%B9%E9%A9%AC%E5%A4%9A%E8%BE%B9%E5%BD%A2%E6%95%B0%E5%AE%9A%E7%90%86" title="费马多边形数定理">费马多边形数定理</a></li> <li><a href="/wiki/%E5%9B%9B%E5%B9%B3%E6%96%B9%E5%92%8C%E5%AE%9A%E7%90%86" title="四平方和定理">四平方和定理</a></li> <li><a href="/wiki/%E4%BA%94%E9%82%8A%E5%BD%A2%E6%95%B8%E5%AE%9A%E7%90%86" title="五邊形數定理">五邊形數定理</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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