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</div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Innholdsfortegnelse" > <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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Kvadrattall</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="Gå til en artikkel på et annet språk. Tilgjengelig på 55 språk" > <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 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvadrattal" title="Kvadrattal – norsk nynorsk" lang="nn" hreflang="nn" data-title="Kvadrattal" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvadrattal" title="Kvadrattal – dansk" lang="da" hreflang="da" data-title="Kvadrattal" data-language-autonym="Dansk" data-language-local-name="dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferningstala" title="Ferningstala – islandsk" lang="is" hreflang="is" data-title="Ferningstala" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><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 – arabisk" lang="ar" hreflang="ar" data-title="مربع كامل" data-language-autonym="العربية" data-language-local-name="arabisk" 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 – aserbajdsjansk" lang="az" hreflang="az" data-title="Ədədin kvadratı" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</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 – bengali" lang="bn" hreflang="bn" data-title="বর্গ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</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 – bulgarsk" lang="bg" hreflang="bg" data-title="Квадратно число" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quadrat_perfecte" title="Quadrat perfecte – katalansk" lang="ca" hreflang="ca" data-title="Quadrat perfecte" data-language-autonym="Català" data-language-local-name="katalansk" class="interlanguage-link-target"><span>Català</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 – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Тулли тăваткал" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</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 – tsjekkisk" lang="cs" hreflang="cs" data-title="Čtvercové číslo" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</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 – walisisk" lang="cy" hreflang="cy" data-title="Rhif sgwâr" data-language-autonym="Cymraeg" data-language-local-name="walisisk" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quadratzahl" title="Quadratzahl – tysk" lang="de" hreflang="de" data-title="Quadratzahl" data-language-autonym="Deutsch" data-language-local-name="tysk" 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 – gresk" lang="el" hreflang="el" data-title="Τετραγωνικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="gresk" 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 – engelsk" lang="en" hreflang="en" data-title="Square number" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuadrado_perfecto" title="Cuadrado perfecto – spansk" lang="es" hreflang="es" data-title="Cuadrado perfecto" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvadrata_nombro" title="Kvadrata nombro – esperanto" lang="eo" hreflang="eo" data-title="Kvadrata nombro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</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 – persisk" lang="fa" hreflang="fa" data-title="مربع کامل" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Carr%C3%A9_parfait" title="Carré parfait – fransk" lang="fr" hreflang="fr" data-title="Carré parfait" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cadrado_perfecto" title="Cadrado perfecto – galisisk" lang="gl" hreflang="gl" data-title="Cadrado perfecto" data-language-autonym="Galego" data-language-local-name="galisisk" 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 – gujarati" lang="gu" hreflang="gu" data-title="વર્ગ" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</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 – koreansk" lang="ko" hreflang="ko" data-title="정사각수" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</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 – høysorbisk" lang="hsb" hreflang="hsb" data-title="Štwórcowa ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="høysorbisk" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Potpuni_kvadrat" title="Potpuni kvadrat – kroatisk" lang="hr" hreflang="hr" data-title="Potpuni kvadrat" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_persegi" title="Bilangan persegi – indonesisk" lang="id" hreflang="id" data-title="Bilangan persegi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quadrato_perfetto" title="Quadrato perfetto – italiensk" lang="it" hreflang="it" data-title="Quadrato perfetto" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99" title="מספר ריבועי – hebraisk" lang="he" hreflang="he" data-title="מספר ריבועי" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_quadratus" title="Numerus quadratus – latin" lang="la" hreflang="la" data-title="Numerus quadratus" data-language-autonym="Latina" data-language-local-name="latin" 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 – latvisk" lang="lv" hreflang="lv" data-title="Skaitļa kvadrāts" data-language-autonym="Latviešu" data-language-local-name="latvisk" class="interlanguage-link-target"><span>Latviešu</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 – ungarsk" lang="hu" hreflang="hu" data-title="Négyzetszámok" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</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) – marathi" lang="mr" hreflang="mr" data-title="वर्ग (गणित)" data-language-autonym="मराठी" data-language-local-name="marathi" 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 – malayisk" lang="ms" hreflang="ms" data-title="Kuasa dua sempurna" data-language-autonym="Bahasa Melayu" data-language-local-name="malayisk" 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 – nederlandsk" lang="nl" hreflang="nl" data-title="Kwadraatgetal" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%95%B0" title="平方数 – japansk" lang="ja" hreflang="ja" data-title="平方数" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kwadroottaal" title="Kwadroottaal – nordfrisisk" lang="frr" hreflang="frr" data-title="Kwadroottaal" data-language-autonym="Nordfriisk" data-language-local-name="nordfrisisk" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kaaslamee(square)" title="Kaaslamee(square) – oromo" lang="om" hreflang="om" data-title="Kaaslamee(square)" data-language-autonym="Oromoo" data-language-local-name="oromo" 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 – polsk" lang="pl" hreflang="pl" data-title="Liczby kwadratowe" data-language-autonym="Polski" data-language-local-name="polsk" 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 – portugisisk" lang="pt" hreflang="pt" data-title="Número quadrado" data-language-autonym="Português" data-language-local-name="portugisisk" 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 – rumensk" lang="ro" hreflang="ro" data-title="Pătrat perfect" data-language-autonym="Română" data-language-local-name="rumensk" 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 – russisk" lang="ru" hreflang="ru" data-title="Полный квадрат" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Square_number" title="Square number – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Square number" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" 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 – slovakisk" lang="sk" hreflang="sk" data-title="Štvorcové číslo" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" 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 – slovensk" lang="sl" hreflang="sl" data-title="Kvadratno število" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%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 – sentralkurdisk" lang="ckb" hreflang="ckb" data-title="دووجای تەواو" data-language-autonym="کوردی" data-language-local-name="sentralkurdisk" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvadrat_(algebra)" title="Kvadrat (algebra) – serbokroatisk" lang="sh" hreflang="sh" data-title="Kvadrat (algebra)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neli%C3%B6luku" title="Neliöluku – finsk" lang="fi" hreflang="fi" data-title="Neliöluku" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</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) – tamil" lang="ta" hreflang="ta" data-title="வர்க்கம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%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 – telugu" lang="te" hreflang="te" data-title="వర్గ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="telugu" 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 – tyrkisk" lang="tr" hreflang="tr" data-title="Tam kare" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" 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 – ukrainsk" lang="uk" hreflang="uk" data-title="Квадратне число" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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 – vietnamesisk" lang="vi" hreflang="vi" data-title="Số chính phương" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" 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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">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</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">Fra Wikipedia, den frie encyklopedi</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="nb" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Square_number_16_as_sum_of_gnomons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Square_number_16_as_sum_of_gnomons.svg/200px-Square_number_16_as_sum_of_gnomons.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Square_number_16_as_sum_of_gnomons.svg/300px-Square_number_16_as_sum_of_gnomons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Square_number_16_as_sum_of_gnomons.svg/400px-Square_number_16_as_sum_of_gnomons.svg.png 2x" data-file-width="101" data-file-height="101" /></a><figcaption>16 kuler danner et kvadrat hvor hver sidekant har 4 kuler.</figcaption></figure> <p>Et <b>kvadrattall</b> er det positive <a href="/wiki/Heltall" title="Heltall">heltallet</a> som oppstår når et heltall <a href="/wiki/Multiplikasjon" title="Multiplikasjon">multipliseres</a> med seg selv. Eksempelvis er 25 et kvadrattall ettersom 5 · 5 = 25. En annen, vanlig skrivemåte er <a href="/wiki/Potens_(matematikk)" title="Potens (matematikk)">potensformen</a> 5<sup>2</sup> som sies å være «fem opphøyd i to» eller «fem i kvadrat». De første kvadrattallene er: </p><p>1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, ... </p><p>Navnet <i>kvadrattall</i> skyldes at et slikt antall prikker eller kuler kan arrangeres som et geometrisk <a href="/wiki/Kvadrat" title="Kvadrat">kvadrat</a>. Det betyr at det <i>n</i>-te kvadrattallet har <i>n</i> prikker eller kuler i sine sidekanter. Disse tallene danner en klasse av <a href="/wiki/Figurtall" title="Figurtall">figurtall</a> på samme måte som <a href="/wiki/Trekanttall" title="Trekanttall">trekanttallene</a> og andre <a href="/wiki/Polygontall" title="Polygontall">polygontall</a>. </p><p>Kvadratet av et <a href="/wiki/Partall" title="Partall">liketall</a> er alltid et liketall, mens kvadratet av et <a href="/wiki/Oddetall" title="Oddetall">oddetall</a> alltid er et oddetall. </p> <div class="mw-heading mw-heading2"><h2 id="Aritmetiske_egenskaper">Aritmetiske egenskaper</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadrattall&veaction=edit&section=1" title="Rediger avsnitt: Aritmetiske egenskaper" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadrattall&action=edit&section=1" title="Rediger kildekoden til seksjonen Aritmetiske egenskaper"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Det <i>n</i>-te kvadrattallet <i>K<sub>n</sub> </i> kan skrives som en sum av de <i>n</i> første <a href="/wiki/Oddetall" title="Oddetall">oddetallene</a>. For eksempel, så er </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&1&=1&=1^{2}\\&1+3&={4}&=2^{2}\\&1+3+5&={9}&=3^{2}\\&1+3+5+7&=16&=4^{2}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <mn>3</mn> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>7</mn> </mtd> <mtd> <mo>=</mo> <mn>16</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&1&=1&=1^{2}\\&1+3&={4}&=2^{2}\\&1+3+5&={9}&=3^{2}\\&1+3+5+7&=16&=4^{2}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1cda5c8aed04e31e6541f14b4c673479211c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:28.661ex; height:12.676ex;" alt="{\displaystyle {\begin{aligned}&1&=1&=1^{2}\\&1+3&={4}&=2^{2}\\&1+3+5&={9}&=3^{2}\\&1+3+5+7&=16&=4^{2}\\\end{aligned}}}"></span></dd></dl> <p>Da det <i>n</i>-te oddetall er 2<i>n</i> + 1, tilsvarer denne egenskapen ved kvadrattallene den <a href="/wiki/Rekursjon" title="Rekursjon">rekursive</a> sammenhengen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n+1}=K_{n}+(2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n+1}=K_{n}+(2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a4b44d67212b51de7ec2e876fdd09f4499b108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.792ex; height:2.843ex;" alt="{\displaystyle K_{n+1}=K_{n}+(2n+1)}"></span></dd></dl> <p>Ved å benytte at <i>K</i><sub>1</sub> = 1, kan alle kvadrattallene herav beregnes ved <a href="/wiki/Addisjon" title="Addisjon">addisjon</a>. Rekursjonsrelasjonen kan også illustreres geometrisk ved figurene </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <td width="25%"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fil:Square_number_1_with_gnomon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Square_number_1_with_gnomon.svg/24px-Square_number_1_with_gnomon.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Square_number_1_with_gnomon.svg/36px-Square_number_1_with_gnomon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Square_number_1_with_gnomon.svg/48px-Square_number_1_with_gnomon.svg.png 2x" data-file-width="24" data-file-height="24" /></a></span> </td> <td width="25%"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fil:Square_number_4_with_gnomon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_number_4_with_gnomon.svg/48px-Square_number_4_with_gnomon.svg.png" decoding="async" width="48" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_number_4_with_gnomon.svg/72px-Square_number_4_with_gnomon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Square_number_4_with_gnomon.svg/96px-Square_number_4_with_gnomon.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span> </td> <td width="25%"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fil:Square_number_9_with_gnomon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Square_number_9_with_gnomon.svg/70px-Square_number_9_with_gnomon.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Square_number_9_with_gnomon.svg/105px-Square_number_9_with_gnomon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Square_number_9_with_gnomon.svg/140px-Square_number_9_with_gnomon.svg.png 2x" data-file-width="70" data-file-height="70" /></a></span> </td> <td width="25%"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fil:Square_number_16_with_gnomon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Square_number_16_with_gnomon.svg/92px-Square_number_16_with_gnomon.svg.png" decoding="async" width="92" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Square_number_16_with_gnomon.svg/138px-Square_number_16_with_gnomon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Square_number_16_with_gnomon.svg/184px-Square_number_16_with_gnomon.svg.png 2x" data-file-width="92" data-file-height="92" /></a></span> </td></tr> <tr> <td><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 \color {black}1\color {black}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="black"> <mn>1</mn> <mstyle /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {black}1\color {black}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3060b6e346d05eb712d9f5d2c36ecde239191b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \color {black}1\color {black}}"></span> </td> <td><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+\color {blue}3\color {black}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mstyle mathcolor="blue"> <mn>3</mn> <mstyle mathcolor="black"> <mo>=</mo> <mn>4</mn> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\color {blue}3\color {black}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb1ac38f2397f4871bf565c9fa6fe7ef2fc2768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+\color {blue}3\color {black}=4}"></span> </td> <td><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 4+\color {blue}5\color {black}=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>+</mo> <mstyle mathcolor="blue"> <mn>5</mn> <mstyle mathcolor="black"> <mo>=</mo> <mn>9</mn> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4+\color {blue}5\color {black}=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a80a65aaccf020a92a030c493bcbdb148d9234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 4+\color {blue}5\color {black}=9}"></span> </td> <td><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 9+\color {blue}7\color {black}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>+</mo> <mstyle mathcolor="blue"> <mn>7</mn> <mstyle mathcolor="black"> <mo>=</mo> <mn>16</mn> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9+\color {blue}7\color {black}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebb3bcc228201ee16264eb372dd2d6321e7f521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.589ex; height:2.343ex;" alt="{\displaystyle 9+\color {blue}7\color {black}=16}"></span> </td></tr></tbody></table> <p>hvor oddetallet som legges til, er gitt ved de blå kulene. Dens riktighet følger også <a href="/wiki/Algebra" title="Algebra">algebraisk</a> fra definisjonen <span class="nowrap"><i>K<sub>n</sub></i> = <i>n</i><sup> 2</sup> </span> som betyr at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n+1}=n^{2}+2n+1=(n+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n+1}=n^{2}+2n+1=(n+1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4217c9c86a944ac31940bc745dca5342afdc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.599ex; height:3.176ex;" alt="{\displaystyle K_{n+1}=n^{2}+2n+1=(n+1)^{2}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sammenheng_med_trekanttall">Sammenheng med trekanttall</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadrattall&veaction=edit&section=2" title="Rediger avsnitt: Sammenheng med trekanttall" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadrattall&action=edit&section=2" title="Rediger kildekoden til seksjonen Sammenheng med trekanttall"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Square_number_25_as_sum_of_two_triangular_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Square_number_25_as_sum_of_two_triangular_numbers.svg/180px-Square_number_25_as_sum_of_two_triangular_numbers.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Square_number_25_as_sum_of_two_triangular_numbers.svg/270px-Square_number_25_as_sum_of_two_triangular_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Square_number_25_as_sum_of_two_triangular_numbers.svg/360px-Square_number_25_as_sum_of_two_triangular_numbers.svg.png 2x" data-file-width="112" data-file-height="112" /></a><figcaption>Summen av de to påfølgende <a href="/wiki/Trekanttall" title="Trekanttall">trekanttallene</a> 10 og 15 gir kvadrattallet 25.</figcaption></figure> <p>Hvert kvadrattall kan skrives som summen av et <a href="/wiki/Trekanttall" title="Trekanttall">trekanttall</a> og det foregående trekanttallet, det vil si </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}=\Delta _{n}+\Delta _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}=\Delta _{n}+\Delta _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22270258f3a08b8cb541ee6e14397423cb84758c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.54ex; height:2.509ex;" alt="{\displaystyle K_{n}=\Delta _{n}+\Delta _{n-1}}"></span></dd></dl> <p>Det følger lett fra den algebraiske sammenhengen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}={1 \over 2}n(n+1)+{1 \over 2}n(n-1)=n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}={1 \over 2}n(n+1)+{1 \over 2}n(n-1)=n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e697b531d92d09152d91a41bef37e20d8ebf251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.878ex; height:5.176ex;" alt="{\displaystyle K_{n}={1 \over 2}n(n+1)+{1 \over 2}n(n-1)=n^{2}}"></span></dd></dl> <p>og kan geometrisk illustreres ved å arrangere kulene i de to trekantene med sidekanter <i>n</i> og <i>n</i> - 1 til å utgjøre et <a href="/wiki/Kvadrat" title="Kvadrat">kvadrat</a> med sidekant <i>n</i>. </p><p>Trekanttallene kan også benyttes til å utlede formelen for summen av de <i>n</i> første kvadrattallene. Den er </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{2}(n)&=1^{2}+2^{2}+3^{2}+\cdots +n^{2}={1 \over 3}n^{3}+{1 \over 2}n^{2}+{1 \over 6}n\\&={1 \over 6}n(n+1)(2n+1)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>n</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{2}(n)&=1^{2}+2^{2}+3^{2}+\cdots +n^{2}={1 \over 3}n^{3}+{1 \over 2}n^{2}+{1 \over 6}n\\&={1 \over 6}n(n+1)(2n+1)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8433d19488c2c6556663f6152b1214ef1deb6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.536ex; margin-bottom: -0.302ex; width:53.785ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}S_{2}(n)&=1^{2}+2^{2}+3^{2}+\cdots +n^{2}={1 \over 3}n^{3}+{1 \over 2}n^{2}+{1 \over 6}n\\&={1 \over 6}n(n+1)(2n+1)\end{aligned}}}"></span></dd></dl> <p>Denne formelen for summen er samtidig det <i>n</i>-te, <a href="/wiki/Pyramidetall" title="Pyramidetall">pyramidetallet</a> basert på en <a href="/wiki/Pyramide" title="Pyramide">pyramide</a> med kvadratisk grunnflate. Det kan bevises ved <a href="/wiki/Induksjon_(matematikk)" class="mw-redirect" title="Induksjon (matematikk)">induksjon</a> ved at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}(n)=S_{2}(n-1)+n^{2}={1 \over 6}n(n-1)(2n-1)+n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}(n)=S_{2}(n-1)+n^{2}={1 \over 6}n(n-1)(2n-1)+n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19cabbbb1f1cca11a63cd2549572896f9703596c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.114ex; height:5.176ex;" alt="{\displaystyle S_{2}(n)=S_{2}(n-1)+n^{2}={1 \over 6}n(n-1)(2n-1)+n^{2}}"></span></dd></dl> <p>Da formelen er riktig for <i>n</i> = 2, vil den derfor være riktig for alle større verdier av <i>n</i>. </p><p>Geometrisk tilsvarer denne rekursjonsrelasjonen at summen <i>S<sub>n</sub></i>  representerer antall kuler i en slik pyramide bestående av <i>n</i> lag med kvadrat hvor hvert kvadrat har sidelengder fra 1 til <i>n</i>. Den kan bygges opp fra en pyramide med <i>S</i><sub><i>n</i> - 1</sub>  kuler ved å tilføye en ny, kvadratisk grunnflate med <i>n</i><sup> 2</sup> kuler. </p> <div class="mw-heading mw-heading2"><h2 id="Litteratur">Litteratur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadrattall&veaction=edit&section=3" title="Rediger avsnitt: Litteratur" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadrattall&action=edit&section=3" title="Rediger kildekoden til seksjonen Litteratur"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A. Søgaard og R. Tambs Lyche, <i>Matematikk III for realgymnaset</i>, Gyldendal Norsk Forlag, Oslo (1955).</li> <li>A. Holme, <i>Matematikkens Historie 1</i>, Fagbokforlaget, Bergen (2001). <a href="/wiki/Spesial:Bokkilder/8276746780" class="internal mw-magiclink-isbn">ISBN 82-7674-678-0</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23230704">.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 .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>55 språk</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Nytt emne</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6758457cd4-5vxzs","wgBackendResponseTime":135,"wgPageParseReport":{"limitreport":{"cputime":"0.156","walltime":"0.257","ppvisitednodes":{"value":824,"limit":1000000},"postexpandincludesize":{"value":5532,"limit":2097152},"templateargumentsize":{"value":464,"limit":2097152},"expansiondepth":{"value":8,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4910,"limit":5000000},"entityaccesscount":{"value":3,"limit":400},"timingprofile":["100.00% 160.208 1 -total"," 98.87% 158.397 1 Mal:Autoritetsdata"," 52.53% 84.155 1 Mal:Navboks"," 50.40% 80.747 1 Mal:Navbox"," 0.98% 1.575 1 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