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conjectures subsection</span> </button> <ul id="toc-Resolution_of_conjectures-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Disproof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Disproof"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Disproof</span> </div> </a> <ul id="toc-Disproof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independent_conjectures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independent_conjectures"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Independent conjectures</span> </div> </a> <ul id="toc-Independent_conjectures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conditional_proofs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conditional_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Conditional proofs</span> </div> </a> <ul id="toc-Conditional_proofs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Important_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Important_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Important examples</span> </div> </a> <button aria-controls="toc-Important_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Important examples subsection</span> </button> <ul id="toc-Important_examples-sublist" class="vector-toc-list"> <li id="toc-Fermat's_Last_Theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fermat's_Last_Theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fermat's Last Theorem</span> </div> </a> <ul id="toc-Fermat's_Last_Theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Four_color_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Four_color_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Four color theorem</span> </div> </a> <ul id="toc-Four_color_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hauptvermutung" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hauptvermutung"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Hauptvermutung</span> </div> </a> <ul id="toc-Hauptvermutung-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weil_conjectures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weil_conjectures"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Weil conjectures</span> </div> </a> <ul id="toc-Weil_conjectures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poincaré_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poincaré_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Poincaré conjecture</span> </div> </a> <ul id="toc-Poincaré_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemann_hypothesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemann_hypothesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Riemann hypothesis</span> </div> </a> <ul id="toc-Riemann_hypothesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-P_versus_NP_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#P_versus_NP_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>P versus NP problem</span> </div> </a> <ul id="toc-P_versus_NP_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_conjectures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_conjectures"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Other conjectures</span> </div> </a> <ul id="toc-Other_conjectures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_other_sciences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_other_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In other sciences</span> </div> </a> <ul id="toc-In_other_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Conjecture</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="Go to an article in another language. Available in 45 languages" > <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-45" 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">45 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AF%D8%B3%D9%8A%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="حدسية (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="حدسية (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Conxetura" title="Conxetura – Asturian" lang="ast" hreflang="ast" data-title="Conxetura" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипотеза (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Гипотеза (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjectura" title="Conjectura – Catalan" lang="ca" hreflang="ca" data-title="Conjectura" data-language-autonym="Català" data-language-local-name="Catalan" 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%93%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипотеза (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Гипотеза (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" 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/Domn%C4%9Bnka" title="Domněnka – Czech" lang="cs" hreflang="cs" data-title="Domněnka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Formodning_(matematik)" title="Formodning (matematik) – Danish" lang="da" hreflang="da" data-title="Formodning (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Vermutung_(Mathematik)" title="Vermutung (Mathematik) – German" lang="de" hreflang="de" data-title="Vermutung (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" 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%95%CE%B9%CE%BA%CE%B1%CF%83%CE%AF%CE%B1" title="Εικασία – Greek" lang="el" hreflang="el" data-title="Εικασία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjetura" title="Conjetura – Spanish" lang="es" hreflang="es" data-title="Conjetura" data-language-autonym="Español" data-language-local-name="Spanish" 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/Konjekto_(matematiko)" title="Konjekto (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Konjekto (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aieru_(zientzia)" title="Aieru (zientzia) – Basque" lang="eu" hreflang="eu" data-title="Aieru (zientzia)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%AF%D8%B3" title="حدس – Persian" lang="fa" hreflang="fa" data-title="حدس" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Conjecture" title="Conjecture – French" lang="fr" hreflang="fr" data-title="Conjecture" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tuairim%C3%ADocht" title="Tuairimíocht – Irish" lang="ga" hreflang="ga" data-title="Tuairimíocht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Baralachas" title="Baralachas – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Baralachas" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Conxectura" title="Conxectura – Galician" lang="gl" hreflang="gl" data-title="Conxectura" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B6%94%EC%B8%A1" title="추측 – Korean" lang="ko" hreflang="ko" data-title="추측" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%BA%D5%B8%D5%A9%D5%A5%D5%A6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Հիպոթեզ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Հիպոթեզ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%9F%E0%A4%95%E0%A4%B2" title="अटकल – Hindi" lang="hi" hreflang="hi" data-title="अटकल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Konjektur" title="Konjektur – Indonesian" lang="id" hreflang="id" data-title="Konjektur" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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/Congettura" title="Congettura – Italian" lang="it" hreflang="it" data-title="Congettura" data-language-autonym="Italiano" data-language-local-name="Italian" 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%94%D7%A9%D7%A2%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="השערה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="השערה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" 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/Coniectura" title="Coniectura – Latin" lang="la" hreflang="la" data-title="Coniectura" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Prielaida" title="Prielaida – Lithuanian" lang="lt" hreflang="lt" data-title="Prielaida" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sejt%C3%A9s" title="Sejtés – Hungarian" lang="hu" hreflang="hu" data-title="Sejtés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Konjektur" title="Konjektur – Malay" lang="ms" hreflang="ms" data-title="Konjektur" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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/Vermoeden" title="Vermoeden – Dutch" lang="nl" hreflang="nl" data-title="Vermoeden" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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/%E4%BA%88%E6%83%B3_(%E6%95%B0%E5%AD%A6)" title="予想 (数学) – Japanese" lang="ja" hreflang="ja" data-title="予想 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przypuszczenie" title="Przypuszczenie – Polish" lang="pl" hreflang="pl" data-title="Przypuszczenie" data-language-autonym="Polski" data-language-local-name="Polish" 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/Conjectura" title="Conjectura – Portuguese" lang="pt" hreflang="pt" data-title="Conjectura" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипотеза (математика) – Russian" lang="ru" hreflang="ru" data-title="Гипотеза (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Cungittura" title="Cungittura – Sicilian" lang="scn" hreflang="scn" data-title="Cungittura" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Conjecture" title="Conjecture – Simple English" lang="en-simple" hreflang="en-simple" data-title="Conjecture" 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/Domnienka" title="Domnienka – Slovak" lang="sk" hreflang="sk" data-title="Domnienka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%95%D8%B2%D9%86%D8%AF%DB%95" title="مەزندە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="مەزندە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Hipoteza_(matematika)" title="Hipoteza (matematika) – Serbian" lang="sr" hreflang="sr" data-title="Hipoteza (matematika)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Konjektuuri" title="Konjektuuri – Finnish" lang="fi" hreflang="fi" data-title="Konjektuuri" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/F%C3%B6rmodan" title="Förmodan – Swedish" lang="sv" hreflang="sv" data-title="Förmodan" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link 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class="noprint">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Proposition in mathematics that is unproven</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For text reconstruction, see <a href="/wiki/Conjecture_(textual_criticism)" title="Conjecture (textual criticism)">Conjecture (textual criticism)</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RiemannCriticalLine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/350px-RiemannCriticalLine.svg.png" decoding="async" width="350" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/525px-RiemannCriticalLine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/700px-RiemannCriticalLine.svg.png 2x" data-file-width="933" data-file-height="434" /></a><figcaption>The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(<i>s</i>) = 1/2. The first non-trivial zeros can be seen at Im(<i>s</i>) = ±14.135, ±21.022 and ±25.011. The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, a famous conjecture, says that all non-trivial zeros of the zeta function lie along the critical line.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>conjecture</b> is a <a href="/wiki/Consequent" title="Consequent">conclusion</a> or a <a href="/wiki/Proposition" title="Proposition">proposition</a> that is proffered on a tentative basis without <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a>.<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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Some conjectures, such as the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> or <a href="/wiki/Fermat%27s_conjecture" class="mw-redirect" title="Fermat's conjecture">Fermat's conjecture</a> (now a <a href="/wiki/Theorem" title="Theorem">theorem</a>, proven in 1995 by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Resolution_of_conjectures">Resolution of conjectures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=1" title="Edit section: Resolution of conjectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Proof">Proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=2" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formal mathematics is based on <i>provable</i> truth. In mathematics, any number of cases supporting a <a href="/wiki/Universally_quantified" class="mw-redirect" title="Universally quantified">universally quantified</a> conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single <a href="/wiki/Counterexample" title="Counterexample">counterexample</a> could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a>, which concerns whether or not certain <a href="/wiki/Sequence" title="Sequence">sequences</a> of <a href="/wiki/Integer" title="Integer">integers</a> terminate, has been tested for all integers up to 1.2 × 10<sup>12</sup> (1.2 trillion). However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample. </p><p>Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved. That evidence may be of various kinds, such as verification of consequences of it or strong interconnections with known results.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>A conjecture is considered proven only when it has been shown that it is logically impossible for it to be false. There are various methods of doing so; see <a href="/wiki/Mathematical_proof#Methods_of_proof" title="Mathematical proof">methods of mathematical proof</a> for more details. </p><p>One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as "<a href="/wiki/Proof_by_exhaustion" title="Proof by exhaustion">brute force</a>": in this approach, all possible cases are considered and shown not to give counterexamples. In some occasions, the number of cases is quite large, in which case a brute-force proof may require as a practical matter the use of a computer algorithm to check all the cases. For example, the validity of the 1976 and 1997 brute-force proofs of the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> by computer was initially doubted, but was eventually confirmed in 2005 by <a href="/wiki/Theorem-proving" class="mw-redirect" title="Theorem-proving">theorem-proving</a> software. </p><p>When a conjecture has been <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a>, it is no longer a conjecture but a <a href="/wiki/Theorem" title="Theorem">theorem</a>. Many important theorems were once conjectures, such as the <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">Geometrization theorem</a> (which resolved the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>), <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>, and others. </p> <div class="mw-heading mw-heading3"><h3 id="Disproof">Disproof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=3" title="Edit section: Disproof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conjectures disproven through counterexample are sometimes referred to as <i>false conjectures</i> (cf. the <a href="/wiki/P%C3%B3lya_conjecture" title="Pólya conjecture">Pólya conjecture</a> and <a href="/wiki/Euler%27s_sum_of_powers_conjecture" title="Euler's sum of powers conjecture">Euler's sum of powers conjecture</a>). In the case of the latter, the first counterexample found for the n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is actually smaller. </p> <div class="mw-heading mw-heading3"><h3 id="Independent_conjectures">Independent conjectures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=4" title="Edit section: Independent conjectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Not every conjecture ends up being proven true or false. The <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>, which tries to ascertain the relative <a href="/wiki/Cardinal_number" title="Cardinal number">cardinality</a> of certain <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>, was eventually shown to be <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> from the generally accepted set of <a href="/wiki/Zermelo%E2%80%93Fraenkel_axioms" class="mw-redirect" title="Zermelo–Fraenkel axioms">Zermelo–Fraenkel axioms</a> of set theory. It is therefore possible to adopt this statement, or its negation, as a new <a href="/wiki/Axiom" title="Axiom">axiom</a> in a consistent manner (much as <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> can be taken either as true or false in an axiomatic system for geometry). </p><p>In this case, if a proof uses this statement, researchers will often look for a new proof that <i>does not</i> require the hypothesis (in the same way that it is desirable that statements in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> be proved using only the axioms of neutral geometry, i.e. without the parallel postulate). The one major exception to this in practice is the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, as the majority of researchers usually do not worry whether a result requires it—unless they are studying this axiom in particular. </p> <div class="mw-heading mw-heading2"><h2 id="Conditional_proofs">Conditional proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=5" title="Edit section: Conditional proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sometimes, a conjecture is called a <i>hypothesis</i> when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is a conjecture from <a href="/wiki/Number_theory" title="Number theory">number theory</a> that — amongst other things — makes predictions about the distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. Few number theorists doubt that the Riemann hypothesis is true. In fact, in anticipation of its eventual proof, some have even proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called <i><a href="/wiki/Conditional_proof" title="Conditional proof">conditional proofs</a></i>: the conjectures assumed appear in the hypotheses of the theorem, for the time being. </p><p>These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. </p> <div class="mw-heading mw-heading2"><h2 id="Important_examples">Important examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=6" title="Edit section: Important examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fermat's_Last_Theorem"><span id="Fermat.27s_Last_Theorem"></span>Fermat's Last Theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=7" title="Edit section: Fermat's Last Theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a></div> <p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> (sometimes called <b>Fermat's conjecture</b>, especially in older texts) states that no three <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> <a href="/wiki/Integer" title="Integer">integers</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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, <i><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></i>, and <i><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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span></i> can satisfy the equation <i><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 a^{n}+b^{n}=c^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}+b^{n}=c^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2e31ced64b8cef38ab186ec86755ecc47c861f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.828ex; height:2.509ex;" alt="{\displaystyle a^{n}+b^{n}=c^{n}}"></span></i> for any integer value of <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i> greater than two. </p><p>This theorem was first conjectured by <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> in 1637 in the margin of a copy of <i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i>, where he claimed that he had a proof that was too large to fit in the margin.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem" class="mw-redirect" title="Wiles' proof of Fermat's Last Theorem">The first successful proof</a> was released in 1994 by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> in the 19th century, and the proof of the <a href="/wiki/Modularity_theorem" title="Modularity theorem">modularity theorem</a> in the 20th century. It is among the most notable theorems in the <a href="/wiki/History_of_mathematics" title="History of mathematics">history of mathematics</a>, and prior to its proof it was in the <i><a href="/wiki/Guinness_Book_of_World_Records" class="mw-redirect" title="Guinness Book of World Records">Guinness Book of World Records</a></i> for "most difficult mathematical problems".<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Four_color_theorem">Four color theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=8" title="Edit section: Four color theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Four_color_theorem" title="Four color theorem">Four color theorem</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Map_of_United_States_vivid_colors_shown.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Map_of_United_States_vivid_colors_shown.png/220px-Map_of_United_States_vivid_colors_shown.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Map_of_United_States_vivid_colors_shown.png/330px-Map_of_United_States_vivid_colors_shown.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Map_of_United_States_vivid_colors_shown.png/440px-Map_of_United_States_vivid_colors_shown.png 2x" data-file-width="800" data-file-height="495" /></a><figcaption>A four-coloring of a map of the states of the United States (ignoring lakes).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a>, or the four color map theorem, states that given any separation of a plane into <a href="https://en.wiktionary.org/wiki/contiguity" class="extiw" title="wikt:contiguity">contiguous</a> regions, producing a figure called a <i>map</i>, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two regions are called <i>adjacent</i> if they share a common boundary that is not a corner, where corners are the points shared by three or more regions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a <a href="/wiki/Four_Corners_Monument" title="Four Corners Monument">point</a> that also belongs to Arizona and Colorado, are not. </p><p><a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a> mentioned the problem in his lectures as early as 1840.<sup id="cite_ref-rouse_ball_1960_9-0" class="reference"><a href="#cite_note-rouse_ball_1960-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The conjecture was first proposed on October 23, 1852<sup id="cite_ref-MacKenzie_10-0" class="reference"><a href="#cite_note-MacKenzie-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> when <a href="/wiki/Francis_Guthrie" title="Francis Guthrie">Francis Guthrie</a>, while trying to color the map of counties of England, noticed that only four different colors were needed. The <a href="/wiki/Five_color_theorem" title="Five color theorem">five color theorem</a>, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century;<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false <a href="/wiki/Counterexample" title="Counterexample">counterexamples</a> have appeared since the first statement of the four color theorem in 1852. </p><p>The four color theorem was ultimately proven in 1976 by <a href="/wiki/Kenneth_Appel" title="Kenneth Appel">Kenneth Appel</a> and <a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Wolfgang Haken</a>. It was the first major <a href="/wiki/Theorem" title="Theorem">theorem</a> to be <a href="/wiki/Computer-assisted_proof#Theorems_proved_with_the_help_of_computer_programs" title="Computer-assisted proof">proved using a computer</a>. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example). Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by mathematicians at all because the <a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">computer-assisted proof</a> was infeasible for a human to check by hand.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> However, the proof has since then gained wider acceptance, although doubts still remain.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hauptvermutung">Hauptvermutung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=9" title="Edit section: Hauptvermutung"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hauptvermutung" title="Hauptvermutung">Hauptvermutung</a></div> <p>The <a href="/wiki/Hauptvermutung" title="Hauptvermutung">Hauptvermutung</a> (German for main conjecture) of <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a> is the conjecture that any two <a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">triangulations</a> of a <a href="/wiki/Triangulable_space" class="mw-redirect" title="Triangulable space">triangulable space</a> have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by <a href="/wiki/Ernst_Steinitz" title="Ernst Steinitz">Steinitz</a> and <a href="/wiki/Heinrich_Tietze" title="Heinrich Tietze">Tietze</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>This conjecture is now known to be false. The non-manifold version was disproved by <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> in 1961 using <a href="/wiki/Analytic_torsion" title="Analytic torsion">Reidemeister torsion</a>. </p><p>The <a href="/wiki/Manifold" title="Manifold">manifold</a> version is true in <a href="/wiki/Dimension" title="Dimension">dimensions</a> <span class="nowrap"><i>m</i> ≤ 3</span>. The cases <span class="nowrap"><i>m</i> = 2 and 3</span> were proved by <a href="/wiki/Tibor_Rad%C3%B3" title="Tibor Radó">Tibor Radó</a> and <a href="/wiki/Edwin_E._Moise" title="Edwin E. Moise">Edwin E. Moise</a><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> in the 1920s and 1950s, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Weil_conjectures">Weil conjectures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=10" title="Edit section: Weil conjectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a> were some highly influential proposals by <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a> (<a href="#CITEREFWeil1949">1949</a>) on the <a href="/wiki/Generating_function" title="Generating function">generating functions</a> (known as <a href="/wiki/Local_zeta-function" class="mw-redirect" title="Local zeta-function">local zeta-functions</a>) derived from counting the number of points on <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. </p><p>A variety <i>V</i> over a finite field with <i>q</i> elements has a finite number of <a href="/wiki/Rational_point" title="Rational point">rational points</a>, as well as points over every finite field with <i>q</i><sup><i>k</i></sup> elements containing that field. The generating function has coefficients derived from the numbers <i>N</i><sub><i>k</i></sub> of points over the (essentially unique) field with <i>q</i><sup><i>k</i></sup> elements. </p><p>Weil conjectured that such <i>zeta-functions</i> should be <a href="/wiki/Rational_function" title="Rational function">rational functions</a>, should satisfy a form of <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a>, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>. The rationality was proved by <a href="#CITEREFDwork1960">Dwork (1960)</a>, the functional equation by <a href="#CITEREFGrothendieck1965">Grothendieck (1965)</a>, and the analogue of the Riemann hypothesis was proved by <a href="#CITEREFDeligne1974">Deligne (1974)</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Poincaré_conjecture"><span id="Poincar.C3.A9_conjecture"></span>Poincaré conjecture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=11" title="Edit section: Poincaré conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></div><p> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> is a <a href="/wiki/Theorem" title="Theorem">theorem</a> about the <a href="/wiki/Characterization_(mathematics)" title="Characterization (mathematics)">characterization</a> of the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>, which is the hypersphere that bounds the <a href="/wiki/Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a> in four-dimensional space. The conjecture states that: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style></p><blockquote class="templatequote"><p>Every <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>, <a href="/wiki/Closed_manifold" title="Closed manifold">closed</a> 3-<a href="/wiki/Manifold" title="Manifold">manifold</a> is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the 3-sphere.</p></blockquote><p> An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called <a href="/wiki/Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a>: if a 3-manifold is <i>homotopy equivalent</i> to the 3-sphere, then it is necessarily <i>homeomorphic</i> to it. </p><p>Originally conjectured by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> in 1904, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a <a href="/wiki/Closed_manifold" title="Closed manifold">closed</a> <a href="/wiki/3-manifold" title="3-manifold">3-manifold</a>). The Poincaré conjecture claims that if such a space has the additional property that each <a href="/wiki/Path_(topology)" title="Path (topology)">loop</a> in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An <a href="/wiki/Generalized_Poincar%C3%A9_conjecture" title="Generalized Poincaré conjecture">analogous result</a> has been known in higher dimensions for some time. </p><p>After nearly a century of effort by mathematicians, <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> presented a proof of the conjecture in three papers made available in 2002 and 2003 on <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>. The proof followed on from the program of <a href="/wiki/Richard_S._Hamilton" title="Richard S. Hamilton">Richard S. Hamilton</a> to use the <a href="/wiki/Ricci_flow" title="Ricci flow">Ricci flow</a> to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called <i>Ricci flow with surgery</i> to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Perelman completed this portion of the proof. Several teams of mathematicians have verified that Perelman's proof is correct. </p><p>The Poincaré conjecture, before being proven, was one of the most important open questions in <a href="/wiki/Topology" title="Topology">topology</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Riemann_hypothesis">Riemann hypothesis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=12" title="Edit section: Riemann hypothesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></div> <p>In mathematics, the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, proposed by <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (<a href="#CITEREFRiemann1859">1859</a>), is a conjecture that the non-trivial <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">zeros</a> of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> all have <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> 1/2. The name is also used for some closely related analogues, such as the <a href="/wiki/Riemann_hypothesis_for_curves_over_finite_fields" class="mw-redirect" title="Riemann hypothesis for curves over finite fields">Riemann hypothesis for curves over finite fields</a>. </p><p>The Riemann hypothesis implies results about the distribution of <a href="/wiki/Prime_numbers" class="mw-redirect" title="Prime numbers">prime numbers</a>. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The Riemann hypothesis, along with the <a href="/wiki/Goldbach_conjecture" class="mw-redirect" title="Goldbach conjecture">Goldbach conjecture</a>, is part of <a href="/wiki/Hilbert%27s_eighth_problem" title="Hilbert's eighth problem">Hilbert's eighth problem</a> in <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s list of <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">23 unsolved problems</a>; it is also one of the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>. </p> <div class="mw-heading mw-heading3"><h3 id="P_versus_NP_problem">P versus NP problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=13" title="Edit section: P versus NP problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></div> <p>The <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a> is a major <a href="/wiki/List_of_unsolved_problems_in_computer_science" title="List of unsolved problems in computer science">unsolved problem in computer science</a>. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjectured that the answer is no. It was essentially first mentioned in a 1956 letter written by <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> to <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>. Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The precise statement of the P=NP problem was introduced in 1971 by <a href="/wiki/Stephen_Cook" title="Stephen Cook">Stephen Cook</a> in his seminal paper "The complexity of theorem proving procedures"<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and is considered by many to be the most important open problem in the field.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It is one of the seven <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a> selected by the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> to carry a US$1,000,000 prize for the first correct solution. </p> <div class="mw-heading mw-heading3"><h3 id="Other_conjectures">Other conjectures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=14" title="Edit section: Other conjectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a></li> <li>The <a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">twin prime conjecture</a></li> <li>The <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a></li> <li>The <a href="/wiki/Manin_conjecture" title="Manin conjecture">Manin conjecture</a></li> <li>The <a href="/wiki/Maldacena_conjecture" class="mw-redirect" title="Maldacena conjecture">Maldacena conjecture</a></li> <li>The <a href="/wiki/Euler%27s_sum_of_powers_conjecture" title="Euler's sum of powers conjecture">Euler conjecture</a>, proposed by Euler in the 18th century but for which counterexamples for a number of exponents (starting with n=4) were found beginning in the mid 20th century</li> <li>The <a href="/wiki/Second_Hardy%E2%80%93Littlewood_conjecture" title="Second Hardy–Littlewood conjecture">Hardy-Littlewood conjectures</a> are a pair of conjectures concerning the distribution of prime numbers, the first of which expands upon the aforementioned twin prime conjecture. Neither one has either been proven or disproven, but it <i>has</i> been proven that both cannot simultaneously be true (i.e., at least one must be false). It has not been proven which one is false, but it is widely believed that the first conjecture is true and the second one is false.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Langlands_program" title="Langlands program">Langlands program</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> is a far-reaching web of these ideas of '<a href="/wiki/Unifying_conjecture" class="mw-redirect" title="Unifying conjecture">unifying conjectures</a>' that link different subfields of mathematics (e.g. between <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>). Some of these conjectures have since been proved.</li></ul> <div class="mw-heading mw-heading2"><h2 id="In_other_sciences">In other sciences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=15" title="Edit section: In other sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Karl_Popper" title="Karl Popper">Karl Popper</a> pioneered the use of the term "conjecture" in <a href="/wiki/Philosophy_of_science" title="Philosophy of science">scientific philosophy</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Conjecture is related to <a href="/wiki/Hypothesis" title="Hypothesis">hypothesis</a>, which in <a href="/wiki/Science" title="Science">science</a> refers to a testable conjecture. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bold_hypothesis" title="Bold hypothesis">Bold hypothesis</a></li> <li><a href="/wiki/Futures_studies" title="Futures studies">Futures studies</a></li> <li><a href="/wiki/Hypotheticals" class="mw-redirect" title="Hypotheticals">Hypotheticals</a></li> <li><a href="/wiki/List_of_conjectures" title="List of conjectures">List of conjectures</a></li> <li><a href="/wiki/Ramanujan_machine" title="Ramanujan machine">Ramanujan machine</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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 mw-references-columns"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.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.id-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.id-lock-limited a,.mw-parser-output .id-lock-registration.id-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.id-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}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.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:#085;margin-left:0.3em}.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}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/conjecture">"Definition of CONJECTURE"</a>. <i>www.merriam-webster.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.merriam-webster.com&rft.atitle=Definition+of+CONJECTURE&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fconjecture&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Oxford Dictionary of English</i> (2010 ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+Dictionary+of+English&rft.edition=2010&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz1995" class="citation book cs1">Schwartz, JL (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JyKelnvECc4C&q=%22although+counterpoint+between+the+particular+and+the+general%22&pg=PA93"><i>Shuttling between the particular and the general: reflections on the role of conjecture and hypothesis in the generation of knowledge in science and mathematics</i></a>. Oxford University Press. p. 93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195115772" title="Special:BookSources/9780195115772"><bdi>9780195115772</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Shuttling+between+the+particular+and+the+general%3A+reflections+on+the+role+of+conjecture+and+hypothesis+in+the+generation+of+knowledge+in+science+and+mathematics.&rft.pages=93&rft.pub=Oxford+University+Press&rft.date=1995&rft.isbn=9780195115772&rft.aulast=Schwartz&rft.aufirst=JL&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJyKelnvECc4C%26q%3D%2522although%2Bcounterpoint%2Bbetween%2Bthe%2Bparticular%2Band%2Bthe%2Bgeneral%2522%26pg%3DPA93&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FermatsLastTheorem.html">"Fermat's Last Theorem"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Fermat%27s+Last+Theorem&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FFermatsLastTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklin2016" class="citation journal cs1">Franklin, James (2016). <a rel="nofollow" class="external text" href="https://web.maths.unsw.edu.au/~jim/logicalprobabilitymathintelldraft.pdf">"Logical probability and the strength of mathematical conjectures"</a> <span class="cs1-format">(PDF)</span>. <i>Mathematical Intelligencer</i>. <b>38</b> (3): 14–19. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-015-9612-3">10.1007/s00283-015-9612-3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:30291085">30291085</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170309031840/http://web.maths.unsw.edu.au/~jim/logicalprobabilitymathintelldraft.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2017-03-09<span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Intelligencer&rft.atitle=Logical+probability+and+the+strength+of+mathematical+conjectures&rft.volume=38&rft.issue=3&rft.pages=14-19&rft.date=2016&rft_id=info%3Adoi%2F10.1007%2Fs00283-015-9612-3&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A30291085%23id-name%3DS2CID&rft.aulast=Franklin&rft.aufirst=James&rft_id=https%3A%2F%2Fweb.maths.unsw.edu.au%2F~jim%2Flogicalprobabilitymathintelldraft.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1988" class="citation cs2">Ore, Oystein (1988) [1948], <a rel="nofollow" class="external text" href="https://archive.org/details/numbertheoryitsh0000orey/page/203"><i>Number Theory and Its History</i></a>, Dover, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/numbertheoryitsh0000orey/page/203">203–204</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65620-5" title="Special:BookSources/978-0-486-65620-5"><bdi>978-0-486-65620-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+and+Its+History&rft.pages=203-204&rft.pub=Dover&rft.date=1988&rft.isbn=978-0-486-65620-5&rft.aulast=Ore&rft.aufirst=Oystein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumbertheoryitsh0000orey%2Fpage%2F203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"Science and Technology". <i>The Guinness Book of World Records</i>. Guinness Publishing Ltd. 1995.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Science+and+Technology&rft.btitle=The+Guinness+Book+of+World+Records&rft.pub=Guinness+Publishing+Ltd.&rft.date=1995&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorges_Gonthier2008" class="citation journal cs1"><a href="/wiki/Georges_Gonthier" title="Georges Gonthier">Georges Gonthier</a> (December 2008). "Formal Proof—The Four-Color Theorem". <i>Notices of the AMS</i>. <b>55</b> (11): 1382–1393. <q>From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=Formal+Proof%E2%80%94The+Four-Color+Theorem&rft.volume=55&rft.issue=11&rft.pages=1382-1393&rft.date=2008-12&rft.au=Georges+Gonthier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-rouse_ball_1960-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-rouse_ball_1960_9-0">^</a></b></span> <span class="reference-text"><a href="/wiki/W._W._Rouse_Ball" title="W. W. Rouse Ball">W. W. Rouse Ball</a> (1960) <i>The Four Color Theorem</i>, in Mathematical Recreations and Essays, Macmillan, New York, pp 222-232.</span> </li> <li id="cite_note-MacKenzie-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-MacKenzie_10-0">^</a></b></span> <span class="reference-text">Donald MacKenzie, <i>Mechanizing Proof: Computing, Risk, and Trust</i> (MIT Press, 2004) p103</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeawood1890" class="citation journal cs1">Heawood, P. J. (1890). "Map-Colour Theorems". <i>Quarterly Journal of Mathematics</i>. <b>24</b>. Oxford: 332–338.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quarterly+Journal+of+Mathematics&rft.atitle=Map-Colour+Theorems&rft.volume=24&rft.pages=332-338&rft.date=1890&rft.aulast=Heawood&rft.aufirst=P.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwart1980" class="citation journal cs1">Swart, E. R. (1980). "The Philosophical Implications of the Four-Color Problem". <i>The American Mathematical Monthly</i>. <b>87</b> (9): 697–702. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2321855">10.2307/2321855</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2321855">2321855</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Philosophical+Implications+of+the+Four-Color+Problem&rft.volume=87&rft.issue=9&rft.pages=697-702&rft.date=1980&rft.issn=0002-9890&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2321855%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2321855&rft.aulast=Swart&rft.aufirst=E.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2014" class="citation book cs1">Wilson, Robin (2014). <i>Four colors suffice : how the map problem was solved</i> (Revised color ed.). Princeton, New Jersey: Princeton University Press. pp. 216–222. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691158228" title="Special:BookSources/9780691158228"><bdi>9780691158228</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/847985591">847985591</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Four+colors+suffice+%3A+how+the+map+problem+was+solved&rft.place=Princeton%2C+New+Jersey&rft.pages=216-222&rft.edition=Revised+color&rft.pub=Princeton+University+Press&rft.date=2014&rft_id=info%3Aoclcnum%2F847985591&rft.isbn=9780691158228&rft.aulast=Wilson&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/haupt/">"Triangulation and the Hauptvermutung"</a>. <i>www.maths.ed.ac.uk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.maths.ed.ac.uk&rft.atitle=Triangulation+and+the+Hauptvermutung&rft_id=https%3A%2F%2Fwww.maths.ed.ac.uk%2F~v1ranick%2Fhaupt%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1961" class="citation journal cs1">Milnor, John W. (1961). "Two complexes which are homeomorphic but combinatorially distinct". <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. <b>74</b> (2): 575–590. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970299">10.2307/1970299</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970299">1970299</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0133127">0133127</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Two+complexes+which+are+homeomorphic+but+combinatorially+distinct&rft.volume=74&rft.issue=2&rft.pages=575-590&rft.date=1961&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D133127%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970299%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1970299&rft.aulast=Milnor&rft.aufirst=John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoise1977" class="citation book cs1">Moise, Edwin E. (1977). <i>Geometric Topology in Dimensions 2 and 3</i>. New York: New York : Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90220-3" title="Special:BookSources/978-0-387-90220-3"><bdi>978-0-387-90220-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Topology+in+Dimensions+2+and+3&rft.place=New+York&rft.pub=New+York+%3A+Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-90220-3&rft.aulast=Moise&rft.aufirst=Edwin+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1997" class="citation journal cs1"><a href="/wiki/Richard_S._Hamilton" title="Richard S. Hamilton">Hamilton, Richard S.</a> (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FCAG.1997.v5.n1.a1">"Four-manifolds with positive isotropic curvature"</a>. <i>Communications in Analysis and Geometry</i>. <b>5</b> (1): 1–92. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FCAG.1997.v5.n1.a1">10.4310/CAG.1997.v5.n1.a1</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1456308">1456308</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0892.53018">0892.53018</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Analysis+and+Geometry&rft.atitle=Four-manifolds+with+positive+isotropic+curvature&rft.volume=5&rft.issue=1&rft.pages=1-92&rft.date=1997&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0892.53018%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1456308%23id-name%3DMR&rft_id=info%3Adoi%2F10.4310%2FCAG.1997.v5.n1.a1&rft.aulast=Hamilton&rft.aufirst=Richard+S.&rft_id=https%3A%2F%2Fdoi.org%2F10.4310%252FCAG.1997.v5.n1.a1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBombieri2000" class="citation web cs1">Bombieri, Enrico (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151222090027/http://www.claymath.org/sites/default/files/official_problem_description.pdf">"The Riemann Hypothesis – official problem description"</a> <span class="cs1-format">(PDF)</span>. <i>Clay Mathematics Institute</i>. 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Retrieved <span class="nowrap">2019-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Clay+Mathematics+Institute&rft.atitle=The+Riemann+Hypothesis+%E2%80%93+official+problem+description&rft.date=2000&rft.aulast=Bombieri&rft.aufirst=Enrico&rft_id=http%3A%2F%2Fwww.claymath.org%2Fsites%2Fdefault%2Ffiles%2Fofficial_problem_description.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Juris Hartmanis 1989, <a rel="nofollow" class="external text" href="http://ecommons.library.cornell.edu/bitstream/1813/6910/1/89-994.pdf">Gödel, von Neumann, and the P = NP problem</a>, Bulletin of the European Association for Theoretical Computer Science, vol. 38, pp. 101–107</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCook1971" class="citation book cs1"><a href="/wiki/Stephen_Cook" title="Stephen Cook">Cook, Stephen</a> (1971). <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=805047">"The complexity of theorem proving procedures"</a>. <i>Proceedings of the Third Annual ACM Symposium on Theory of Computing</i>. pp. 151–158. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F800157.805047">10.1145/800157.805047</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781450374644" title="Special:BookSources/9781450374644"><bdi>9781450374644</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7573663">7573663</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+complexity+of+theorem+proving+procedures&rft.btitle=Proceedings+of+the+Third+Annual+ACM+Symposium+on+Theory+of+Computing&rft.pages=151-158&rft.date=1971&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7573663%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F800157.805047&rft.isbn=9781450374644&rft.aulast=Cook&rft.aufirst=Stephen&rft_id=http%3A%2F%2Fportal.acm.org%2Fcitation.cfm%3Fcoll%3DGUIDE%26dl%3DGUIDE%26id%3D805047&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="/wiki/Lance_Fortnow" title="Lance Fortnow">Lance Fortnow</a>, <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20110224135332/http://www.cs.uchicago.edu/~fortnow/papers/pnp-cacm.pdf"><i>The status of the <b>P</b> versus <b>NP</b> problem</i></a>, Communications of the ACM 52 (2009), no. 9, pp. 78–86. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1562164.1562186">10.1145/1562164.1562186</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichards1974" class="citation journal cs1">Richards, Ian (1974). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1974-13434-8">"On the Incompatibility of Two Conjectures Concerning Primes"</a>. <i>Bull. Amer. Math. Soc</i>. <b>80</b>: 419–438. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1974-13434-8">10.1090/S0002-9904-1974-13434-8</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=On+the+Incompatibility+of+Two+Conjectures+Concerning+Primes&rft.volume=80&rft.pages=419-438&rft.date=1974&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1974-13434-8&rft.aulast=Richards&rft.aufirst=Ian&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1974-13434-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanglands1967" class="citation cs2">Langlands, Robert (1967), <a rel="nofollow" class="external text" href="http://publications.ias.edu/rpl/section/21"><i>Letter to Prof. Weil</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Letter+to+Prof.+Weil&rft.date=1967&rft.aulast=Langlands&rft.aufirst=Robert&rft_id=http%3A%2F%2Fpublications.ias.edu%2Frpl%2Fsection%2F21&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopper2004" class="citation book cs1">Popper, Karl (2004). <i>Conjectures and refutations : the growth of scientific knowledge</i>. London: Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-28594-1" title="Special:BookSources/0-415-28594-1"><bdi>0-415-28594-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conjectures+and+refutations+%3A+the+growth+of+scientific+knowledge&rft.place=London&rft.pub=Routledge&rft.date=2004&rft.isbn=0-415-28594-1&rft.aulast=Popper&rft.aufirst=Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=18" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeligne1974" class="citation cs2"><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Deligne, Pierre</a> (1974), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=PMIHES_1974__43__273_0">"La conjecture de Weil. I"</a>, <i><a href="/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a></i>, <b>43</b> (43): 273–307, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02684373">10.1007/BF02684373</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1618-1913">1618-1913</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0340258">0340258</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123139343">123139343</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+Math%C3%A9matiques+de+l%27IH%C3%89S&rft.atitle=La+conjecture+de+Weil.+I&rft.volume=43&rft.issue=43&rft.pages=273-307&rft.date=1974&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123139343%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0340258%23id-name%3DMR&rft.issn=1618-1913&rft_id=info%3Adoi%2F10.1007%2FBF02684373&rft.aulast=Deligne&rft.aufirst=Pierre&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DPMIHES_1974__43__273_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDwork1960" class="citation cs2"><a href="/wiki/Bernard_Dwork" title="Bernard Dwork">Dwork, Bernard</a> (1960), "On the rationality of the zeta function of an algebraic variety", <i><a href="/wiki/American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a></i>, <b>82</b> (3), American Journal of Mathematics, Vol. 82, No. 3: 631–648, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372974">10.2307/2372974</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372974">2372974</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0140494">0140494</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=On+the+rationality+of+the+zeta+function+of+an+algebraic+variety&rft.volume=82&rft.issue=3&rft.pages=631-648&rft.date=1960&rft.issn=0002-9327&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0140494%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2372974%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2372974&rft.aulast=Dwork&rft.aufirst=Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieck1965" class="citation cs2"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a> (1995) [1965], "Formule de Lefschetz et rationalité des fonctions L", <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=SB_1964-1966__9__41_0"><i>Séminaire Bourbaki</i></a>, vol. 9, Paris: <a href="/wiki/Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France" class="mw-redirect" title="Société Mathématique de France">Société Mathématique de France</a>, pp. 41–55, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1608788">1608788</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Formule+de+Lefschetz+et+rationalit%C3%A9+des+fonctions+L&rft.btitle=S%C3%A9minaire+Bourbaki&rft.place=Paris&rft.pages=41-55&rft.pub=Soci%C3%A9t%C3%A9+Math%C3%A9matique+de+France&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1608788%23id-name%3DMR&rft.aulast=Grothendieck&rft.aufirst=Alexander&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DSB_1964-1966__9__41_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConjecture" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjecture&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output 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