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[o]" accesskey="o" class=""><span>ಲಾಗ್ ಇನ್</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="More options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="ವೈಯಕ್ತಿಕ ಉಪಕರಣಗಳು" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">ವೈಯಕ್ತಿಕ ಉಪಕರಣಗಳು</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_kn.wikipedia.org&uselang=kn"><span>ದೇಣಿಗೆ</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%87%E0%B2%B7:CreateAccount&returnto=%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" title="ನೀವು ಹೊಸ ಖಾತೆಯನ್ನು ತೆರೆದು ಲಾಗಿನ್ ಆಗುವುದನ್ನು ಹುರಿದುಂಬಿಸುತ್ತೇವೆ; ಆದಾಗ್ಯೂ, ಇದು ಅವಶ್ಯವೇನಲ್ಲ"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>ಹೊಸ ಖಾತೆ ತೆರೆಯಿರಿ</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%87%E0%B2%B7:UserLogin&returnto=%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" title="ನೀವು ಲಾಗ್ ಇನ್ ಆಗಬೇಕೆಂದು ಕೋರುತ್ತೇವೆ, ಆದರೆ ಅದು ಖಡ್ಡಾಯ ಎನೂ ಅಲ್ಲ. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>ಲಾಗ್ ಇನ್</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"> ಲಾಗ್ ಔಟ್ ಆದ ಸಂಪಾದಕರಿಗೆ ಪುಟಗಳು <a href="/wiki/%E0%B2%B8%E0%B2%B9%E0%B2%BE%E0%B2%AF:Introduction" aria-label="Learn more about editing"><span>ಹೆಚ್ಚಿನ ಮಾಹಿತಿ</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%87%E0%B2%B7:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>ಕಾಣಿಕೆಗಳು</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%87%E0%B2%B7:MyTalk" title="ಈ ip ವಿಳಾಸದಿಂದ ಮಾಡಲಾದ ಸಂಪಾದನೆಗಳ ಬಗ್ಗೆ ಚರ್ಚೆ [n]" accesskey="n"><span>IP ಚರ್ಚಾಪುಟ</span></a></li> </ul> </div> </div> </div> </div> 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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">ಪರಿವಿಡಿ</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ಮರೆ ಮಾಡಿ</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">ಮುನ್ನುಡಿ</div> </a> </li> <li id="toc-ಬ್ರಹ್ಮಗುಪ್ತನ_ಪರಿಹಾರ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ಬ್ರಹ್ಮಗುಪ್ತನ_ಪರಿಹಾರ"> <div class="vector-toc-text"> <span class="vector-toc-numb">೧</span> <span>ಬ್ರಹ್ಮಗುಪ್ತನ ಪರಿಹಾರ</span> </div> </a> <ul id="toc-ಬ್ರಹ್ಮಗುಪ್ತನ_ಪರಿಹಾರ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ಉಲ್ಲೇಖಗಳು" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ಉಲ್ಲೇಖಗಳು"> <div class="vector-toc-text"> <span class="vector-toc-numb">೨</span> <span>ಉಲ್ಲೇಖಗಳು</span> </div> </a> <ul id="toc-ಉಲ್ಲೇಖಗಳು-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ಹೊರಗಿನ_ಕೊಂಡಿಗಳು" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ಹೊರಗಿನ_ಕೊಂಡಿಗಳು"> <div class="vector-toc-text"> <span class="vector-toc-numb">೩</span> <span>ಹೊರಗಿನ ಕೊಂಡಿಗಳು</span> </div> </a> <ul id="toc-ಹೊರಗಿನ_ಕೊಂಡಿಗಳು-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 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">ವರ್ಗಪ್ರಕೃತಿ</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="ಇನ್ನೊಂದು ಭಾಷೆಯ ಲೇಖನಕ್ಕೆ ಹೋಗಿ. ೩೦ ಭಾಷೆಗಳಲ್ಲಿ ಲಭ್ಯವಿದೆ" > <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-30" 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">೩೦ ಭಾಷೆಗಳು</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%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%A8%D9%8A%D9%84" title="معادلة بيل – ಅರೇಬಿಕ್" lang="ar" hreflang="ar" data-title="معادلة بيل" data-language-autonym="العربية" data-language-local-name="ಅರೇಬಿಕ್" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%87%E0%A6%B2_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="পেল সমীকরণ – ಬಾಂಗ್ಲಾ" lang="bn" hreflang="bn" data-title="পেল সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="ಬಾಂಗ್ಲಾ" 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/Equaci%C3%B3_de_Pell" title="Equació de Pell – ಕೆಟಲಾನ್" lang="ca" hreflang="ca" data-title="Equació de Pell" data-language-autonym="Català" data-language-local-name="ಕೆಟಲಾನ್" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pellova_rovnice" title="Pellova rovnice – ಜೆಕ್" lang="cs" hreflang="cs" data-title="Pellova rovnice" data-language-autonym="Čeština" data-language-local-name="ಜೆಕ್" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pellsche_Gleichung" title="Pellsche Gleichung – ಜರ್ಮನ್" lang="de" hreflang="de" data-title="Pellsche Gleichung" data-language-autonym="Deutsch" data-language-local-name="ಜರ್ಮನ್" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="ಉತ್ತಮ ಲೇಖನ"><a href="https://en.wikipedia.org/wiki/Pell%27s_equation" title="Pell's equation – ಇಂಗ್ಲಿಷ್" lang="en" hreflang="en" data-title="Pell's equation" data-language-autonym="English" data-language-local-name="ಇಂಗ್ಲಿಷ್" 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/Ecuaci%C3%B3n_de_Pell" title="Ecuación de Pell – ಸ್ಪ್ಯಾನಿಷ್" lang="es" hreflang="es" data-title="Ecuación de Pell" data-language-autonym="Español" data-language-local-name="ಸ್ಪ್ಯಾನಿಷ್" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pelli_v%C3%B5rrand" title="Pelli võrrand – ಎಸ್ಟೊನಿಯನ್" lang="et" hreflang="et" data-title="Pelli võrrand" data-language-autonym="Eesti" data-language-local-name="ಎಸ್ಟೊನಿಯನ್" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Pellen_ekuazio" title="Pellen ekuazio – ಬಾಸ್ಕ್" lang="eu" hreflang="eu" data-title="Pellen ekuazio" data-language-autonym="Euskara" data-language-local-name="ಬಾಸ್ಕ್" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Pelli_v%C3%B5rrand" title="Pelli võrrand – Võro" lang="vro" hreflang="vro" data-title="Pelli võrrand" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Pell-Fermat" title="Équation de Pell-Fermat – ಫ್ರೆಂಚ್" lang="fr" hreflang="fr" data-title="Équation de Pell-Fermat" data-language-autonym="Français" data-language-local-name="ಫ್ರೆಂಚ್" 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/Ecuaci%C3%B3n_de_Pell" title="Ecuación de Pell – ಗ್ಯಾಲಿಶಿಯನ್" lang="gl" hreflang="gl" data-title="Ecuación de Pell" data-language-autonym="Galego" data-language-local-name="ಗ್ಯಾಲಿಶಿಯನ್" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%A4%D7%9C" title="משוואת פל – ಹೀಬ್ರೂ" lang="he" hreflang="he" data-title="משוואת פל" data-language-autonym="עברית" data-language-local-name="ಹೀಬ್ರೂ" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%87%E0%A4%B2_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="पेल का समीकरण – ಹಿಂದಿ" lang="hi" hreflang="hi" data-title="पेल का समीकरण" data-language-autonym="हिन्दी" data-language-local-name="ಹಿಂದಿ" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pell-egyenlet" title="Pell-egyenlet – ಹಂಗೇರಿಯನ್" lang="hu" hreflang="hu" data-title="Pell-egyenlet" data-language-autonym="Magyar" data-language-local-name="ಹಂಗೇರಿಯನ್" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_Pell" title="Equazione di Pell – ಇಟಾಲಿಯನ್" lang="it" hreflang="it" data-title="Equazione di Pell" data-language-autonym="Italiano" data-language-local-name="ಇಟಾಲಿಯನ್" 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/%E3%83%9A%E3%83%AB%E6%96%B9%E7%A8%8B%E5%BC%8F" title="ペル方程式 – ಜಾಪನೀಸ್" lang="ja" hreflang="ja" data-title="ペル方程式" data-language-autonym="日本語" data-language-local-name="ಜಾಪನೀಸ್" 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/%ED%8E%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="펠 방정식 – ಕೊರಿಯನ್" lang="ko" hreflang="ko" data-title="펠 방정식" data-language-autonym="한국어" data-language-local-name="ಕೊರಿಯನ್" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vergelijking_van_Pell" title="Vergelijking van Pell – ಡಚ್" lang="nl" hreflang="nl" data-title="Vergelijking van Pell" data-language-autonym="Nederlands" data-language-local-name="ಡಚ್" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Pells_ligning" title="Pells ligning – ನಾರ್ವೆಜಿಯನ್ ಬೊಕ್ಮಲ್" lang="nb" hreflang="nb" data-title="Pells ligning" data-language-autonym="Norsk bokmål" data-language-local-name="ನಾರ್ವೆಜಿಯನ್ ಬೊಕ್ಮಲ್" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_Pella" title="Równanie Pella – ಪೊಲಿಶ್" lang="pl" hreflang="pl" data-title="Równanie Pella" data-language-autonym="Polski" data-language-local-name="ಪೊಲಿಶ್" 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/Equa%C3%A7%C3%A3o_de_Pell" title="Equação de Pell – ಪೋರ್ಚುಗೀಸ್" lang="pt" hreflang="pt" data-title="Equação de Pell" data-language-autonym="Português" data-language-local-name="ಪೋರ್ಚುಗೀಸ್" 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/Ecua%C8%9Bia_lui_Pell" title="Ecuația lui Pell – ರೊಮೇನಿಯನ್" lang="ro" hreflang="ro" data-title="Ecuația lui Pell" data-language-autonym="Română" data-language-local-name="ರೊಮೇನಿಯನ್" 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%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9F%D0%B5%D0%BB%D0%BB%D1%8F" title="Уравнение Пелля – ರಷ್ಯನ್" lang="ru" hreflang="ru" data-title="Уравнение Пелля" data-language-autonym="Русский" data-language-local-name="ರಷ್ಯನ್" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pellova_ena%C4%8Dba" title="Pellova enačba – ಸ್ಲೋವೇನಿಯನ್" lang="sl" hreflang="sl" data-title="Pellova enačba" data-language-autonym="Slovenščina" data-language-local-name="ಸ್ಲೋವೇನಿಯನ್" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pells_ekvation" title="Pells ekvation – ಸ್ವೀಡಿಷ್" lang="sv" hreflang="sv" data-title="Pells ekvation" data-language-autonym="Svenska" data-language-local-name="ಸ್ವೀಡಿಷ್" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%86%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="பெல்லின் சமன்பாடு – ತಮಿಳು" lang="ta" hreflang="ta" data-title="பெல்லின் சமன்பாடு" data-language-autonym="தமிழ்" data-language-local-name="ತಮಿಳು" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%9F%D0%B5%D0%BB%D0%BB%D1%8F" title="Рівняння Пелля – ಉಕ್ರೇನಿಯನ್" lang="uk" hreflang="uk" data-title="Рівняння Пелля" data-language-autonym="Українська" data-language-local-name="ಉಕ್ರೇನಿಯನ್" 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/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_Pell" title="Phương trình Pell – ವಿಯೆಟ್ನಾಮೀಸ್" lang="vi" hreflang="vi" data-title="Phương trình Pell" data-language-autonym="Tiếng Việt" data-language-local-name="ವಿಯೆಟ್ನಾಮೀಸ್" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BD%A9%E5%B0%94%E6%96%B9%E7%A8%8B" title="佩尔方程 – ಚೈನೀಸ್" lang="zh" hreflang="zh" data-title="佩尔方程" data-language-autonym="中文" data-language-local-name="ಚೈನೀಸ್" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q853067#sitelinks-wikipedia" title="ಇತರ ಭಾಷಾ ಕೊಂಡಿಗಳನ್ನು ಸಂಪಾದಿಸು" class="wbc-editpage">ಕೊಂಡಿಗಳನ್ನು ಸಂಪಾದಿಸಿ</a></span></div> </div> </div> </div> </header> <div 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id="siteSub" class="noprint">ವಿಕಿಪೀಡಿಯದಿಂದ, ಇದು ಮುಕ್ತ ಹಾಗೂ ಸ್ವತಂತ್ರ ವಿಶ್ವಕೋಶ</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="kn" dir="ltr"><p><b>ವರ್ಗಪ್ರಕೃತಿ</b> ಎನ್ನುವುದು <i><b>Nx<sup>2</sup>+1 = y<sup>2</sup></b></i> (<i><b>N</b></i> <a href="https://en.wikipedia.org/wiki/Integer" class="extiw" title="en:Integer">ಪೂರ್ಣಾಂಕ</a>) <a href="/wiki/%E0%B2%B8%E0%B2%AE%E0%B3%80%E0%B2%95%E0%B2%B0%E0%B2%A3" title="ಸಮೀಕರಣ">ಸಮೀಕರಣದಲ್ಲಿ</a> <i><b>x</b></i> ಮತ್ತು <i><b>y</b></i> ಗಳಿಗೆ ಹೊಂದುವ ಪೂರ್ಣಾಂಕಗಳನ್ನು ಪಡೆಯುವ ಪ್ರಕ್ರಿಯೆ. <a href="/wiki/%E0%B2%AC%E0%B3%8D%E0%B2%B0%E0%B2%B9%E0%B3%8D%E0%B2%AE%E0%B2%97%E0%B3%81%E0%B2%AA%E0%B3%8D%E0%B2%A4" title="ಬ್ರಹ್ಮಗುಪ್ತ">ಬ್ರಹ್ಮಗುಪ್ತ</a> (6-7ನೆಯ ಶತಮಾನ) ಮತ್ತು <a href="/wiki/%E0%B2%AD%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%95%E0%B2%B0%E0%B2%BE%E0%B2%9A%E0%B2%BE%E0%B2%B0%E0%B3%8D%E0%B2%AF" title="ಭಾಸ್ಕರಾಚಾರ್ಯ">ಎರಡನೆಯ ಭಾಸ್ಕರ</a> (12ನೆಯ ಶತಮಾನ) ತಮ್ಮ ತಮ್ಮ ಕಾಲಗಳಲ್ಲಿ ಈ <a href="https://en.wikipedia.org/wiki/Diophantine_equation" class="extiw" title="en:Diophantine equation">ಡಯೊಫಾಂಟೈನ್ ಸಮೀಕರಣಕ್ಕೆ</a> ಬ್ರಹ್ಮಗುಪ್ತ-ಭಾಸ್ಕರ ಸಮೀಕರಣವೆಂಬ ಹೆಸರು ಒಪ್ಪುತ್ತದೆ. <a href="https://en.wikipedia.org/wiki/John_Pell_(mathematician)" class="extiw" title="en:John Pell (mathematician)">ಜಾನ್ ಪೆಲ್</a> (1610-85) ಎಂಬಾತ ಇದೇ ರೂಪದ <i><b>x<sup>2</sup>-Dy<sup>2</sup> = 1</b></i> (ಇಲ್ಲಿ <i><b>D</b></i> ವರ್ಗಸಂಖ್ಯೆಯಲ್ಲದ ಧನ ಪೂರ್ಣಾಂಕ (positive non-square integer)) ಸಮೀಕರಣವನ್ನು ಸ್ವತಂತ್ರವಾಗಿ ಪರಿಶೀಲಿಸಿದ್ದರಿಂದ ಪಾಶ್ಚಾತ್ಯ ರಾಷ್ಟ್ರಗಳಲ್ಲಿ ವರ್ಗಪ್ರಕೃತಿ ಅಥವಾ ಬ್ರಹ್ಮಗುಪ್ತ-ಭಾಸ್ಕರ ಸಮೀಕರಣ <b>ಪೆಲ್ಲಿಯನ್ ಸಮೀಕರಣ</b>ವೆಂದೇ ಪ್ರಸಿದ್ಧವಾಗಿದೆ. </p> <div class="mw-heading mw-heading2"><h2 id="ಬ್ರಹ್ಮಗುಪ್ತನ_ಪರಿಹಾರ"><span id=".E0.B2.AC.E0.B3.8D.E0.B2.B0.E0.B2.B9.E0.B3.8D.E0.B2.AE.E0.B2.97.E0.B3.81.E0.B2.AA.E0.B3.8D.E0.B2.A4.E0.B2.A8_.E0.B2.AA.E0.B2.B0.E0.B2.BF.E0.B2.B9.E0.B2.BE.E0.B2.B0"></span>ಬ್ರಹ್ಮಗುಪ್ತನ ಪರಿಹಾರ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF&action=edit&section=1" title="ವಿಭಾಗ ಸಂಪಾದಿಸಿ: ಬ್ರಹ್ಮಗುಪ್ತನ ಪರಿಹಾರ"><span>ಬದಲಾಯಿಸಿ</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>ವರ್ಗಪ್ರಕೃತಿಗೆ ಬ್ರಹ್ಮಗುಪ್ತ ಪರಿಹಾರ ಪಡೆದ ಬಗೆ ಬಲು ಸರಳ ಮತ್ತು ಸುಂದರ. ಮೊದಲು ಅದೇ ಸಮೀಕರಣವನ್ನು </p> <pre><b>Nα<sup>2</sup> + κ = β<sup>2</sup></b>                            <b>Nα'<sup>2</sup> + κ = β'<sup>2</sup></b> </pre> <p>ಎಂದು ಬರೆಯೋಣ. ಆಗ <i><b>αβ'+α'β</b></i> ಮತ್ತು <i><b>ββ'+Nαα'</b></i> ಎಂಬವು <i><b>Nx<sup>2</sup>+κκ' = y<sup>2</sup></b></i> <a href="/wiki/%E0%B2%B8%E0%B2%AE%E0%B3%80%E0%B2%95%E0%B2%B0%E0%B2%A3%E0%B2%A6_%E0%B2%AE%E0%B3%82%E0%B2%B2" title="ಸಮೀಕರಣದ ಮೂಲ">ಸಮೀಕರಣದ ಮೂಲಗಳಾಗುತ್ತವೆ</a>. ಇದಕ್ಕೆ ಬ್ರಹ್ಮಗುಪ್ತನ ಉಪಪ್ರಮೇಯ (<a href="https://en.wikipedia.org/wiki/Lemma_(mathematics)" class="extiw" title="en:Lemma (mathematics)">ಲೆಮ್ಮ</a>) ಎಂದು ಹೆಸರು. ಇದರಲ್ಲಿ <i><b>α=α'</b></i> ಮತ್ತು <i><b>β=β'</b></i> ಎಂದು ಆದೇಶಿದಾಗ <i><b>2αβ, β<sup>2</sup>+Nα<sup>2</sup></b></i> ಎಂಬವು <i><b>Nx<sup>2</sup>+1 = y<sup>2</sup></b></i> ಸಮೀಕರಣದ ಮೂಲಗಳಾಗುತ್ತವೆ. <i><b>κ=±1</b></i> ಮತ್ತು <i><b>κ=±2</b></i> ಆದಾಗ ಇವು ಪೂರ್ಣಾಂಕಗಳು. <i><b>K=4</b></i> ಆದಾಗ ಬ್ರಹ್ಮಗುಪ್ತ <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 x={\frac {\alpha ^{2}-2}{2}},y={\frac {\alpha \beta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {\alpha ^{2}-2}{2}},y={\frac {\alpha \beta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdc08cfa577f9733d9db3502afdd05e7514298a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.753ex; height:5.676ex;" alt="{\displaystyle x={\frac {\alpha ^{2}-2}{2}},y={\frac {\alpha \beta }{2}}}"></span> ಎಂಬವನ್ನೂ, <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 x={\frac {\alpha }{2}}(\alpha ^{2}-3),y={\frac {\beta }{2}}(\alpha ^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {\alpha }{2}}(\alpha ^{2}-3),y={\frac {\beta }{2}}(\alpha ^{2}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e674f97f630b57eab1901b031af74c026ef52181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.916ex; height:5.343ex;" alt="{\displaystyle x={\frac {\alpha }{2}}(\alpha ^{2}-3),y={\frac {\beta }{2}}(\alpha ^{2}-1)}"></span> ಎಂಬವನ್ನೂ ಪಡೆಯುತ್ತಾನೆ. ಮೊದಲನೆಯ ಜೊತೆ, <i><b>α,β</b></i> ಎರಡೂ ಸರಿಸಂಖ್ಯೆಗಳಾದಾಗ, ಅಂತೆಯೇ ಎರಡನೆಯ ಜೊತೆ b ವಿಷಮಸಂಖ್ಯೆಯಾದಾಗ ಪೂರ್ಣಾಂಕಗಳನ್ನು ಕೊಡುತ್ತವೆ. </p><p>ಉದಾರಣೆಗೆ <i><b>κ=-4</b></i> ಆದಾಗ <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 x={\frac {(\alpha ^{2}+2)[(\alpha ^{2}+1)(\alpha ^{2}+3)-2)]}{2}},y={\frac {\alpha \beta (\alpha ^{2}+3)(\alpha ^{2}+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>β</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {(\alpha ^{2}+2)[(\alpha ^{2}+1)(\alpha ^{2}+3)-2)]}{2}},y={\frac {\alpha \beta (\alpha ^{2}+3)(\alpha ^{2}+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332e85af3416be5652b80721bcf3c2e4ef62eca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.179ex; height:5.843ex;" alt="{\displaystyle x={\frac {(\alpha ^{2}+2)[(\alpha ^{2}+1)(\alpha ^{2}+3)-2)]}{2}},y={\frac {\alpha \beta (\alpha ^{2}+3)(\alpha ^{2}+1)}{2}}}"></span> ಎಂಬ ಪೂರ್ಣಾಂಕಗಳು ದೊರೆಯುತ್ತವೆ.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>೧<span class="cite-bracket">]</span></a></sup> ಇಷ್ಟು ಸಂಕೀರ್ಣ ಪ್ರಶ್ನೆಯನ್ನು 628ರಷ್ಟು ಹಿಂದೆಯೇ ಭಾರತೀಯ ಗಣಿತವಿದ ಬ್ರಹ್ಮಗುಪ್ತ ಸಾಧಿಸಿದುದು ಅಂದಿನ <a href="/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ">ಗಣಿತ</a> ಚಿಂತನೆಯ ಹಿರಿಮಟ್ಟಕ್ಕೆ ಒಂದು ಸೂಚ್ಯಂಕ. ಮುಂದೆ ಭಾಸ್ಕರ 1150 ರಲ್ಲಿ ಈ ಸಾಧನೆಯನ್ನು ಉತ್ತಮೀಕರಿಸಿದ. ವರ್ಗಪ್ರಕೃತಿಯ ಪೂರ್ಣಸಿದ್ಧಾಂತವನ್ನು <a href="/wiki/%E0%B2%9C%E0%B3%8B%E0%B2%B8%E0%B3%86%E0%B2%AB%E0%B3%8D_%E0%B2%B2%E0%B3%82%E0%B2%AF%E0%B2%BF_%E0%B2%B2%E0%B2%97%E0%B3%8D%E0%B2%B0%E0%B2%BE%E0%B2%82%E0%B2%9C%E0%B3%8D" title="ಜೋಸೆಫ್ ಲೂಯಿ ಲಗ್ರಾಂಜ್">ಲಗ್ರಾಂಜ್</a> (1776-1813) 1766 ರಲ್ಲಿ ಸ್ಥಾಪಿಸಿದ.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>೨<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="ಉಲ್ಲೇಖಗಳು"><span id=".E0.B2.89.E0.B2.B2.E0.B3.8D.E0.B2.B2.E0.B3.87.E0.B2.96.E0.B2.97.E0.B2.B3.E0.B3.81"></span>ಉಲ್ಲೇಖಗಳು</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF&action=edit&section=2" title="ವಿಭಾಗ ಸಂಪಾದಿಸಿ: ಉಲ್ಲೇಖಗಳು"><span>ಬದಲಾಯಿಸಿ</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1246530">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1247490">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFDatta_and_Singh1962" class="citation book cs1">Datta and Singh (1962). <i>History of Hindu Mathematics : A Source Book Parts I and II</i>. Asia Publishing House. pp. 157–160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%87%E0%B2%B7:BookSources/8180903907" title="ವಿಶೇಷ:BookSources/8180903907"><bdi>8180903907</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Hindu+Mathematics+%3A+A+Source+Book+Parts+I+and+II&rft.pages=157-160&rft.pub=Asia+Publishing+House&rft.date=1962&rft.isbn=8180903907&rft.au=Datta+and+Singh&rfr_id=info%3Asid%2Fkn.wikipedia.org%3A%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1247490"><cite class="citation cs2">O'Connor, John J.; <a href="/w/index.php?title=Edmund_F._Robertson&action=edit&redlink=1" class="new" title="Edmund F. Robertson (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell.html">"Pell's equation"</a>, <i><a href="/w/index.php?title=MacTutor_History_of_Mathematics_archive&action=edit&redlink=1" class="new" title="MacTutor History of Mathematics archive (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">MacTutor History of Mathematics archive</a></i>, <a href="/w/index.php?title=University_of_St_Andrews&action=edit&redlink=1" class="new" title="University of St Andrews (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pell%27s+equation&rft.btitle=MacTutor+History+of+Mathematics+archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FPell.html&rfr_id=info%3Asid%2Fkn.wikipedia.org%3A%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="ಹೊರಗಿನ_ಕೊಂಡಿಗಳು"><span id=".E0.B2.B9.E0.B3.8A.E0.B2.B0.E0.B2.97.E0.B2.BF.E0.B2.A8_.E0.B2.95.E0.B3.8A.E0.B2.82.E0.B2.A1.E0.B2.BF.E0.B2.97.E0.B2.B3.E0.B3.81"></span>ಹೊರಗಿನ ಕೊಂಡಿಗಳು</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF&action=edit&section=3" title="ವಿಭಾಗ ಸಂಪಾದಿಸಿ: ಹೊರಗಿನ ಕೊಂಡಿಗಳು"><span>ಬದಲಾಯಿಸಿ</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Pell's_equation"><a href="/w/index.php?title=Eric_W._Weisstein&action=edit&redlink=1" class="new" title="Eric W. Weisstein (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">Weisstein, Eric W.</a>, "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PellEquation.html">Pell's equation</a>", <i><a href="/w/index.php?title=MathWorld&action=edit&redlink=1" class="new" title="MathWorld (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">MathWorld</a></i>.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1247490"><cite class="citation cs2">O'Connor, John J.; <a href="/w/index.php?title=Edmund_F._Robertson&action=edit&redlink=1" class="new" title="Edmund F. Robertson (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell.html">"Pell's equation"</a>, <i><a href="/w/index.php?title=MacTutor_History_of_Mathematics_archive&action=edit&redlink=1" class="new" title="MacTutor History of Mathematics archive (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">MacTutor History of Mathematics archive</a></i>, <a href="/w/index.php?title=University_of_St_Andrews&action=edit&redlink=1" class="new" title="University of St Andrews (ಪುಟವು ಇನ್ನೂ ಸೃಷ್ಟಿತವಾಗಿಲ್ಲ)">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pell%27s+equation&rft.btitle=MacTutor+History+of+Mathematics+archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FPell.html&rfr_id=info%3Asid%2Fkn.wikipedia.org%3A%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.jakebakermaths.org.uk/maths/jshtmlpellsolverbigintegerv10.html">Pell equation solver</a> (<i>n</i> has no upper limit)</li> <li><a rel="nofollow" class="external text" href="http://www.numbertheory.org/php/pell.html">Pell equation solver</a> (<i>n</i> < 10<sup>10</sup>, can also return the solution to <i>x</i><sup>2</sup> − <i>ny</i><sup>2</sup> = ±1, ±2, ±3, and ±4)</li></ul>    </div><noscript><img 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