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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Origins"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Origins</span> </div> </a> <ul id="toc-Origins-sublist" class="vector-toc-list"> <li id="toc-Dawn_of_arithmetic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dawn_of_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Dawn of arithmetic</span> </div> </a> <ul id="toc-Dawn_of_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_Greece_and_the_early_Hellenistic_period" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Classical_Greece_and_the_early_Hellenistic_period"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Classical Greece and the early Hellenistic period</span> </div> </a> <ul id="toc-Classical_Greece_and_the_early_Hellenistic_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diophantus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Diophantus"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Diophantus</span> </div> </a> <ul id="toc-Diophantus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Āryabhaṭa,_Brahmagupta,_Bhāskara" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Āryabhaṭa,_Brahmagupta,_Bhāskara"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.4</span> <span>Āryabhaṭa, Brahmagupta, Bhāskara</span> </div> </a> <ul id="toc-Āryabhaṭa,_Brahmagupta,_Bhāskara-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_in_the_Islamic_golden_age" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Arithmetic_in_the_Islamic_golden_age"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.5</span> <span>Arithmetic in the Islamic golden age</span> </div> </a> <ul id="toc-Arithmetic_in_the_Islamic_golden_age-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Western_Europe_in_the_Middle_Ages" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Western_Europe_in_the_Middle_Ages"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.6</span> <span>Western Europe in the Middle Ages</span> </div> </a> <ul id="toc-Western_Europe_in_the_Middle_Ages-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Early_modern_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_modern_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Early modern number theory</span> </div> </a> <ul id="toc-Early_modern_number_theory-sublist" class="vector-toc-list"> <li id="toc-Fermat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fermat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Fermat</span> </div> </a> <ul id="toc-Fermat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Euler</span> </div> </a> <ul id="toc-Euler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrange,_Legendre,_and_Gauss" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lagrange,_Legendre,_and_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.3</span> <span>Lagrange, Legendre, and Gauss</span> </div> </a> <ul id="toc-Lagrange,_Legendre,_and_Gauss-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Maturity_and_division_into_subfields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maturity_and_division_into_subfields"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Maturity and division into subfields</span> </div> </a> <ul id="toc-Maturity_and_division_into_subfields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_subdivisions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Main_subdivisions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Main subdivisions</span> </div> </a> <button aria-controls="toc-Main_subdivisions-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 Main subdivisions subsection</span> </button> <ul id="toc-Main_subdivisions-sublist" class="vector-toc-list"> <li id="toc-Elementary_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementary_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Elementary number theory</span> </div> </a> <ul id="toc-Elementary_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytic_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Analytic number theory</span> </div> </a> <ul id="toc-Analytic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Algebraic number theory</span> </div> </a> <ul id="toc-Algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diophantine_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diophantine_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Diophantine geometry</span> </div> </a> <ul id="toc-Diophantine_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_subfields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_subfields"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Other subfields</span> </div> </a> <button aria-controls="toc-Other_subfields-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 Other subfields subsection</span> </button> <ul id="toc-Other_subfields-sublist" class="vector-toc-list"> <li id="toc-Probabilistic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probabilistic_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Probabilistic number theory</span> </div> </a> <ul id="toc-Probabilistic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_combinatorics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_combinatorics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Arithmetic combinatorics</span> </div> </a> <ul id="toc-Arithmetic_combinatorics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Computational number theory</span> </div> </a> <ul id="toc-Computational_number_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prizes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Prizes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Prizes</span> </div> </a> <ul id="toc-Prizes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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">Number theory</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 114 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-114" 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">114 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Getalteorie" title="Getalteorie – Afrikaans" lang="af" hreflang="af" data-title="Getalteorie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Zahlentheorie" title="Zahlentheorie – Alemannic" lang="gsw" hreflang="gsw" data-title="Zahlentheorie" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF" title="نظرية الأعداد – Arabic" lang="ar" hreflang="ar" data-title="نظرية الأعداد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Teor%C3%ADa_de_numeros" title="Teoría de numeros – Aragonese" lang="an" hreflang="an" data-title="Teoría de numeros" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="সংখ্যাতত্ত্ব – Assamese" lang="as" hreflang="as" data-title="সংখ্যাতত্ত্ব" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmberos" title="Teoría de númberos – Asturian" lang="ast" hreflang="ast" data-title="Teoría de númberos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99dl%C9%99r_n%C9%99z%C9%99riyy%C9%99si" title="Ədədlər nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Ədədlər nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="সংখ্যাতত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="সংখ্যাতত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-l%C5%ABn" title="Sò͘-lūn – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-lūn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D0%B0%D0%BD%D0%B4%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Һандар теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Һандар теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D0%B0%D1%9E" title="Тэорыя лікаў – Belarusian" lang="be" hreflang="be" data-title="Тэорыя лікаў" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D0%B0%D1%9E" title="Тэорыя лікаў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэорыя лікаў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Teorya_nin_bilang" title="Teorya nin bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Teorya nin bilang" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B0%D1%82%D0%B0" title="Теория на числата – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на числата" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorija_brojeva" title="Teorija brojeva – Bosnian" lang="bs" hreflang="bs" data-title="Teorija brojeva" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Damkaniezh_an_nivero%C3%B9" title="Damkaniezh an niveroù – Breton" lang="br" hreflang="br" data-title="Damkaniezh an niveroù" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_nombres" title="Teoria de nombres – Catalan" lang="ca" hreflang="ca" data-title="Teoria de nombres" 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%A5%D0%B8%D1%81%D0%B5%D0%BF%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Хисепсен теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Хисепсен теорийĕ" 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/Teorie_%C4%8D%C3%ADsel" title="Teorie čísel – Czech" lang="cs" hreflang="cs" data-title="Teorie čísel" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_rhifau" title="Damcaniaeth rhifau – Welsh" lang="cy" hreflang="cy" data-title="Damcaniaeth rhifau" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Talteori" title="Talteori – Danish" lang="da" hreflang="da" data-title="Talteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%86%D8%B6%D8%B1%D9%8A%D8%A9_%D8%AF_%D9%84%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF" title="نضرية د لأعداد – Moroccan Arabic" lang="ary" hreflang="ary" data-title="نضرية د لأعداد" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zahlentheorie" title="Zahlentheorie – German" lang="de" hreflang="de" data-title="Zahlentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arvuteooria" title="Arvuteooria – Estonian" lang="et" hreflang="et" data-title="Arvuteooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD" title="Θεωρία αριθμών – Greek" lang="el" hreflang="el" data-title="Θεωρία αριθμών" 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/Teor%C3%ADa_de_n%C3%BAmeros" title="Teoría de números – Spanish" lang="es" hreflang="es" data-title="Teoría de números" 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/Nombroteorio" title="Nombroteorio – Esperanto" lang="eo" hreflang="eo" data-title="Nombroteorio" 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/Zenbakien_teoria" title="Zenbakien teoria – Basque" lang="eu" hreflang="eu" data-title="Zenbakien teoria" 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/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="نظریه اعداد – Persian" lang="fa" hreflang="fa" data-title="نظریه اعداد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Number_theory" title="Number theory – Fiji Hindi" lang="hif" hreflang="hif" data-title="Number theory" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_nombres" title="Théorie des nombres – French" lang="fr" hreflang="fr" data-title="Théorie des nombres" 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/Uimhirtheoiric" title="Uimhirtheoiric – Irish" lang="ga" hreflang="ga" data-title="Uimhirtheoiric" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmeros" title="Teoría de números – Galician" lang="gl" hreflang="gl" data-title="Teoría de números" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E8%AB%96" title="數論 – Gan" lang="gan" hreflang="gan" data-title="數論" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE_%E0%AA%B8%E0%AA%BF%E0%AA%A6%E0%AB%8D%E0%AA%A7%E0%AA%BE%E0%AA%82%E0%AA%A4" title="સંખ્યા સિદ્ધાંત – Gujarati" lang="gu" hreflang="gu" data-title="સંખ્યા સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EB%A1%A0" title="수론 – Korean" lang="ko" hreflang="ko" data-title="수론" 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/%D4%B9%D5%BE%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Թվերի տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Թվերի տեսություն" 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%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="संख्या सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="संख्या सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_brojeva" title="Teorija brojeva – Croatian" lang="hr" hreflang="hr" data-title="Teorija brojeva" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Teorio_di_nombri" title="Teorio di nombri – Ido" lang="io" hreflang="io" data-title="Teorio di nombri" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_bilangan" title="Teori bilangan – Indonesian" lang="id" hreflang="id" data-title="Teori bilangan" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theoria_de_numeros" title="Theoria de numeros – Interlingua" lang="ia" hreflang="ia" data-title="Theoria de numeros" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Talnafr%C3%A6%C3%B0i" title="Talnafræði – Icelandic" lang="is" hreflang="is" data-title="Talnafræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dei_numeri" title="Teoria dei numeri – Italian" lang="it" hreflang="it" data-title="Teoria dei numeri" 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%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D" title="תורת המספרים – Hebrew" lang="he" hreflang="he" data-title="תורת המספרים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/T%C3%A9ori_wilangan" title="Téori wilangan – Javanese" lang="jv" hreflang="jv" data-title="Téori wilangan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%B8%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%A7%E0%B2%BE%E0%B2%82%E0%B2%A4" title="ಸಂಖ್ಯಾಸಿದ್ಧಾಂತ – Kannada" lang="kn" hreflang="kn" data-title="ಸಂಖ್ಯಾಸಿದ್ಧಾಂತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="რიცხვთა თეორია – Georgian" lang="ka" hreflang="ka" data-title="რიცხვთა თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BD%D0%B4%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Сандар теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Сандар теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Nadharia_ya_namba" title="Nadharia ya namba – Swahili" lang="sw" hreflang="sw" data-title="Nadharia ya namba" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/T%C3%A9ori_di_s%C3%A9_nonm" title="Téori di sé nonm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Téori di sé nonm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/B%C3%AErdoza_jimare" title="Bîrdoza jimare – Kurdish" lang="ku" hreflang="ku" data-title="Bîrdoza jimare" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Theoria_numerorum" title="Theoria numerorum – Latin" lang="la" hreflang="la" data-title="Theoria numerorum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skait%C4%BCu_teorija" title="Skaitļu teorija – Latvian" lang="lv" hreflang="lv" data-title="Skaitļu teorija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Zuelentheorie" title="Zuelentheorie – Luxembourgish" lang="lb" hreflang="lb" data-title="Zuelentheorie" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skai%C4%8Di%C5%B3_teorija" title="Skaičių teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Skaičių teorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/nacycmaci" title="nacycmaci – Lojban" lang="jbo" hreflang="jbo" data-title="nacycmaci" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teoria_di_numer" title="Teoria di numer – Lombard" lang="lmo" hreflang="lmo" data-title="Teoria di numer" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Számelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%B1%D1%80%D0%BE%D0%B5%D0%B2%D0%B8%D1%82%D0%B5" title="Теорија на броевите – Macedonian" lang="mk" hreflang="mk" data-title="Теорија на броевите" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="സംഖ്യാസിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="സംഖ്യാസിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Teorija_tan-numri" title="Teorija tan-numri – Maltese" lang="mt" hreflang="mt" data-title="Teorija tan-numri" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="अंकशास्त्र – Marathi" lang="mr" hreflang="mr" data-title="अंकशास्त्र" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%A3%E1%83%94%E1%83%A4%E1%83%98%E1%83%A8_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="რიცხუეფიშ თეორია – Mingrelian" lang="xmf" hreflang="xmf" data-title="რიცხუეფიშ თეორია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D9%87_%D8%A7%D9%84%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="نظريه الاعداد – Egyptian Arabic" lang="arz" hreflang="arz" data-title="نظريه الاعداد" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_nombor" title="Teori nombor – Malay" lang="ms" hreflang="ms" data-title="Teori nombor" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D0%BD%D1%8B_%D0%BE%D0%BD%D0%BE%D0%BB" title="Тооны онол – Mongolian" lang="mn" hreflang="mn" data-title="Тооны онол" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="ကိန်းသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="ကိန်းသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Naba_icavacava" title="Naba icavacava – Fijian" lang="fj" hreflang="fj" data-title="Naba icavacava" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Getaltheorie" title="Getaltheorie – Dutch" lang="nl" hreflang="nl" data-title="Getaltheorie" 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/%E6%95%B0%E8%AB%96" title="数論 – Japanese" lang="ja" hreflang="ja" data-title="数論" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Taalenteorii" title="Taalenteorii – Northern Frisian" lang="frr" hreflang="frr" data-title="Taalenteorii" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tallteori" title="Tallteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tallteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Talteori" title="Talteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Talteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teoria_dels_nombres" title="Teoria dels nombres – Occitan" lang="oc" hreflang="oc" data-title="Teoria dels nombres" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sonlar_nazariyasi" title="Sonlar nazariyasi – Uzbek" lang="uz" hreflang="uz" data-title="Sonlar nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A9%B0%E0%A8%95_%E0%A8%B8%E0%A8%BF%E0%A8%A7%E0%A8%BE%E0%A8%82%E0%A8%A4" title="ਅੰਕ ਸਿਧਾਂਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਅੰਕ ਸਿਧਾਂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%D9%85%D8%A8%D8%B1_%D8%AA%DA%BE%DB%8C%D9%88%D8%B1%DB%8C" title="نمبر تھیوری – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نمبر تھیوری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Nomba_tiori" title="Nomba tiori – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nomba tiori" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teor%C3%ACa_dij_n%C3%B9mer" title="Teorìa dij nùmer – Piedmontese" lang="pms" hreflang="pms" data-title="Teorìa dij nùmer" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Tallentheorie" title="Tallentheorie – Low German" lang="nds" hreflang="nds" data-title="Tallentheorie" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_liczb" title="Teoria liczb – Polish" lang="pl" hreflang="pl" data-title="Teoria liczb" 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/Teoria_dos_n%C3%BAmeros" title="Teoria dos números – Portuguese" lang="pt" hreflang="pt" data-title="Teoria dos números" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_numerelor" title="Teoria numerelor – Romanian" lang="ro" hreflang="ro" data-title="Teoria numerelor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Теория чисел – Russian" lang="ru" hreflang="ru" data-title="Теория чисел" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D0%B0%D0%B0%D0%BC%D0%B0%D0%B9_%D1%83%D0%BB%D0%B0%D1%85%D0%B0%D0%BD_%D1%83%D0%BE%D0%BF%D1%81%D0%B0%D0%B9_%D1%82%D2%AF%D2%A5%D1%8D%D1%82%D1%8D%D1%8D%D1%87%D1%87%D0%B8" title="Саамай улахан уопсай түҥэтээччи – Yakut" lang="sah" hreflang="sah" data-title="Саамай улахан уопсай түҥэтээччи" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%8D" title="संख्याशास्त्रम् – Sanskrit" lang="sa" hreflang="sa" data-title="संख्याशास्त्रम्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Nummer_theory" title="Nummer theory – Scots" lang="sco" hreflang="sco" data-title="Nummer theory" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_numrave" title="Teoria e numrave – Albanian" lang="sq" hreflang="sq" data-title="Teoria e numrave" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiur%C3%ACa_d%C3%AE_n%C3%B9mmura" title="Tiurìa dî nùmmura – Sicilian" lang="scn" hreflang="scn" data-title="Tiurìa dî nùmmura" 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/Number_theory" title="Number theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Number theory" 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/Te%C3%B3ria_%C4%8D%C3%ADsel" title="Teória čísel – Slovak" lang="sk" hreflang="sk" data-title="Teória čísel" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_%C5%A1tevil" title="Teorija števil – Slovenian" lang="sl" hreflang="sl" data-title="Teorija števil" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%DA%98%D9%85%D8%A7%D8%B1%DB%95" title="تیۆریی ژمارە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆریی ژمارە" 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/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B0" title="Теорија бројева – Serbian" lang="sr" hreflang="sr" data-title="Теорија бројева" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_brojeva" title="Teorija brojeva – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Teorija brojeva" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lukuteoria" title="Lukuteoria – Finnish" lang="fi" hreflang="fi" data-title="Lukuteoria" 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/Talteori" title="Talteori – Swedish" lang="sv" hreflang="sv" data-title="Talteori" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_bilang" title="Teorya ng bilang – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="எண் கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="எண் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99" title="ทฤษฎีจำนวน – Thai" lang="th" hreflang="th" data-title="ทฤษฎีจำนวน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Say%C4%B1lar_teorisi" title="Sayılar teorisi – Turkish" lang="tr" hreflang="tr" data-title="Sayılar teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Sanlar_teori%C3%BDasy" title="Sanlar teoriýasy – Turkmen" lang="tk" hreflang="tk" data-title="Sanlar teoriýasy" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Теорія чисел – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія чисел" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%B9%D8%AF%D8%AF" title="نظریۂ عدد – Urdu" lang="ur" hreflang="ur" data-title="نظریۂ عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91" title="Lý thuyết số – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Numateor" title="Numateor – Volapük" lang="vo" hreflang="vo" data-title="Numateor" data-language-autonym="Volapük" data-language-local-name="Volapük" class="interlanguage-link-target"><span>Volapük</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Arvoteooria" title="Arvoteooria – Võro" lang="vro" hreflang="vro" data-title="Arvoteooria" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Teyorya_hin_ihap" title="Teyorya hin ihap – Waray" lang="war" hreflang="war" data-title="Teyorya hin ihap" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E8%AE%BA" title="数论 – Wu" lang="wuu" hreflang="wuu" data-title="数论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%95%D7%9E%D7%A2%D7%A8%D7%9F_%D7%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2" title="נומערן טעאריע – Yiddish" lang="yi" hreflang="yi" data-title="נומערן טעאריע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E8%AB%96" title="數論 – Cantonese" lang="yue" hreflang="yue" data-title="數論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Skaitliu_teuor%C4%97j%C4%97" title="Skaitliu teuorėjė – Samogitian" lang="sgs" 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematics of integer properties</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 the book by André Weil, see <a href="/wiki/Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre" title="Number Theory: An Approach Through History from Hammurapi to Legendre"><i>Number Theory: An Approach Through History from Hammurapi to Legendre</i></a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Numerology" title="Numerology">Numerology</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Spirale_Ulam_150.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/249px-Spirale_Ulam_150.jpg" decoding="async" width="249" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/374px-Spirale_Ulam_150.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/499px-Spirale_Ulam_150.jpg 2x" data-file-width="750" data-file-height="752" /></a><figcaption>The distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> is a central point of study in number theory. 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abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Number theory</b> (or <b><a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a></b> or <b>higher arithmetic</b> in older usage) is a branch of <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a> devoted primarily to the study of the <a href="/wiki/Integer" title="Integer">integers</a> and <a href="/wiki/Arithmetic_function" title="Arithmetic function">arithmetic functions</a>. German mathematician <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."<sup id="cite_ref-FOOTNOTELong19721_1-0" class="reference"><a href="#cite_note-FOOTNOTELong19721-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Number theorists study <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> as well as the properties of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> constructed from integers (for example, <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>), or defined as generalizations of the integers (for example, <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>). </p><p>Integers can be considered either in themselves or as solutions to equations (<a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a>). Questions in number theory are often best understood through the study of <a href="/wiki/Complex_analysis" title="Complex analysis">analytical</a> objects (for example, the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>) that encode properties of the integers, primes or other number-theoretic objects in some fashion (<a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>). One may also study <a href="/wiki/Real_number" title="Real number">real numbers</a> in relation to rational numbers; for example, as approximated by the latter (<a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a>). </p><p>The older term for number theory is <i>arithmetic</i>. By the early twentieth century, it had been superseded by <i>number theory</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> (The word <i><a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a></i> is used by the general public to mean "<a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary calculations</a>"; it has also acquired other meanings in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, as in <i><a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a></i>, and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, as in <i><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a></i>.) The use of the term <i>arithmetic</i> for <i>number theory</i> regained some ground in the second half of the 20th century, arguably in part due to French influence.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> In particular, <i>arithmetical</i> is commonly preferred as an adjective to <i>number-theoretic</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Origins">Origins</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=2" title="Edit section: Origins"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Dawn_of_arithmetic">Dawn of arithmetic</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=3" title="Edit section: Dawn of arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Plimpton_322.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/220px-Plimpton_322.jpg" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/330px-Plimpton_322.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/440px-Plimpton_322.jpg 2x" data-file-width="1246" data-file-height="863" /></a><figcaption>The Plimpton 322 tablet</figcaption></figure> <p>The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> (<a href="/wiki/Larsa" title="Larsa">Larsa, Mesopotamia</a>, ca. 1800 BC) contains a list of "<a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a>", that is, integers <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,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}"></span> such that <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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span>. The triples are too many and too large to have been obtained by <a href="/wiki/Brute_force_method" class="mw-redirect" title="Brute force method">brute force</a>. The heading over the first column reads: "The <i>takiltum</i> of the diagonal which has been subtracted such that the width..."<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The table's layout suggests<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> that it was constructed by means of what amounts, in modern language, to the <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9be9111eb813cf2c8b31c740ddff47308c1e1a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.984ex; height:6.509ex;" alt="{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}"></span></dd></dl> <p>which is implicit in routine <a href="/wiki/Old_Babylonian_language" class="mw-redirect" title="Old Babylonian language">Old Babylonian</a> exercises.<sup id="cite_ref-FOOTNOTEvan_der_Waerden1961184_6-0" class="reference"><a href="#cite_note-FOOTNOTEvan_der_Waerden1961184-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> If some other method was used,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the triples were first constructed and then reordered by <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/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c/a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e52b3c6d3960cfc6efa96a811ec2feb5efa2b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.399ex; height:2.843ex;" alt="{\displaystyle c/a}"></span>, presumably for actual use as a "table", for example, with a view to applications. </p><p>It is not known what these applications may have been, or whether there could have been any; <a href="/wiki/Babylonian_astronomy" title="Babylonian astronomy">Babylonian astronomy</a>, for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems.<sup id="cite_ref-FOOTNOTEFriberg1981302_8-0" class="reference"><a href="#cite_note-FOOTNOTEFriberg1981302-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p><p>While evidence of Babylonian number theory is only survived by the Plimpton 322 tablet, some authors assert that <a href="/wiki/Babylonian_mathematics#Algebra" title="Babylonian mathematics">Babylonian algebra</a> was exceptionally well developed and included the foundations of modern <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>.<sup id="cite_ref-FOOTNOTEvan_der_Waerden196143_10-0" class="reference"><a href="#cite_note-FOOTNOTEvan_der_Waerden196143-10"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Late Neoplatonic sources<sup id="cite_ref-vanderW2_11-0" class="reference"><a href="#cite_note-vanderW2-11"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> state that <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> learned mathematics from the Babylonians. Much earlier sources<sup id="cite_ref-stanencyc_12-0" class="reference"><a href="#cite_note-stanencyc-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> state that <a href="/wiki/Thales" class="mw-redirect" title="Thales">Thales</a> and Pythagoras traveled and studied in <a href="/wiki/Egypt" title="Egypt">Egypt</a>. </p><p>In book nine of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, propositions 21–34 are very probably influenced by <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagorean teachings</a>;<sup id="cite_ref-Becker_13-0" class="reference"><a href="#cite_note-Becker-13"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> it is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that <a href="/wiki/Square_root_of_2" title="Square root of 2"><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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span></a> is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>.<sup id="cite_ref-FOOTNOTEBecker1936_14-0" class="reference"><a href="#cite_note-FOOTNOTEBecker1936-14"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagorean mystics</a> gave great importance to the odd and the even.<sup id="cite_ref-FOOTNOTEvan_der_Waerden1961109_15-0" class="reference"><a href="#cite_note-FOOTNOTEvan_der_Waerden1961109-15"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The discovery that <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> is irrational is credited to the early Pythagoreans (pre-<a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a>).<sup id="cite_ref-Thea_16-0" class="reference"><a href="#cite_note-Thea-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to <a href="/wiki/Hippasus_of_Metapontum" class="mw-redirect" title="Hippasus of Metapontum">Hippasus</a>, who was expelled or split from the Pythagorean sect.<sup id="cite_ref-FOOTNOTEvon_Fritz2004_17-0" class="reference"><a href="#cite_note-FOOTNOTEvon_Fritz2004-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> This forced a distinction between <i><a href="/wiki/Number" title="Number">numbers</a></i> (integers and the rationals—the subjects of arithmetic), on the one hand, and <i>lengths</i> and <i>proportions</i> (which may be identified with real numbers, whether rational or not), on the other hand. </p><p>The Pythagorean tradition spoke also of so-called <a href="/wiki/Polygonal_number" title="Polygonal number">polygonal</a> or <a href="/wiki/Figurate_numbers" class="mw-redirect" title="Figurate numbers">figurate numbers</a>.<sup id="cite_ref-FOOTNOTEHeath192176_18-0" class="reference"><a href="#cite_note-FOOTNOTEHeath192176-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> While <a href="/wiki/Square_number" title="Square number">square numbers</a>, <a href="/wiki/Cubic_number" class="mw-redirect" title="Cubic number">cubic numbers</a>, etc., are seen now as more natural than <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a>, <a href="/wiki/Pentagonal_number" title="Pentagonal number">pentagonal numbers</a>, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the <a href="/wiki/Early_modern_period" title="Early modern period">early modern period</a> (17th to early 19th centuries). </p><p>The <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a> appears as an exercise<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> in <i><a href="/wiki/Sunzi_Suanjing" title="Sunzi Suanjing">Sunzi Suanjing</a></i> (3rd, 4th or 5th century CE).<sup id="cite_ref-YongSe_20-0" class="reference"><a href="#cite_note-YongSe-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> (There is one important step glossed over in Sunzi's solution:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> it is the problem that was later solved by <a href="/wiki/%C4%80ryabha%E1%B9%ADa" class="mw-redirect" title="Āryabhaṭa">Āryabhaṭa</a>'s <a href="/wiki/Ku%E1%B9%AD%E1%B9%ADaka" title="Kuṭṭaka">Kuṭṭaka</a> – see <a href="#Āryabhaṭa,_Brahmagupta,_Bhāskara">below</a>.) The result was later generalized with a complete solution called <i>Da-yan-shu</i> (<span title="Chinese-language text"><span lang="zh">大衍術</span></span>) in <a href="/wiki/Qin_Jiushao" title="Qin Jiushao">Qin Jiushao</a>'s 1247 <i><a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a></i> <sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> which was translated into English in early 19th century by British missionary <a href="/wiki/Alexander_Wylie_(missionary)" title="Alexander Wylie (missionary)">Alexander Wylie</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>There is also some numerical mysticism in Chinese mathematics,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>note 5<span class="cite-bracket">]</span></a></sup> but, unlike that of the Pythagoreans, it seems to have led nowhere. </p> <div class="mw-heading mw-heading4"><h4 id="Classical_Greece_and_the_early_Hellenistic_period">Classical Greece and the early Hellenistic period</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=4" title="Edit section: Classical Greece and the early Hellenistic period"><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">Further information: <a href="/wiki/Ancient_Greek_mathematics" class="mw-redirect" title="Ancient Greek mathematics">Ancient Greek mathematics</a></div> <p>Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early <a href="/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a>.<sup id="cite_ref-FOOTNOTEBoyerMerzbach199182_25-0" class="reference"><a href="#cite_note-FOOTNOTEBoyerMerzbach199182-25"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In the case of number theory, this means, by and large, <i><a href="/wiki/Plato" title="Plato">Plato</a></i> and <i>Euclid</i>, respectively. </p><p>While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. </p><p><a href="/wiki/Eusebius" title="Eusebius">Eusebius</a>, PE X, chapter 4 mentions of <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>: </p> <blockquote><p>"In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad."<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and Cicero repeats this claim: <i>Platonem ferunt didicisse Pythagorea omnia</i> ("They say Plato learned all things Pythagorean").<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By <i>arithmetic</i> he meant, in part, theorising on number, rather than what <i>arithmetic</i> or <i>number theory</i> have come to mean.) It is through one of Plato's dialogues—namely, <a href="/wiki/Theaetetus_(dialogue)" title="Theaetetus (dialogue)"><i>Theaetetus</i></a>—that it is known that <a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a> had proven that <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 {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c30069c6e8ec13aa767b2a0fe071bba15bedcc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.67ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}}"></span> are irrational. <a href="/wiki/Theaetetus_of_Athens" class="mw-redirect" title="Theaetetus of Athens">Theaetetus</a> was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurables</a>, and was thus arguably a pioneer in the study of <a href="/wiki/Number_systems" class="mw-redirect" title="Number systems">number systems</a>. (Book X of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a> is described by <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a> as being largely based on Theaetetus's work.) </p><p>Euclid devoted part of his <i>Elements</i> to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's <i>Elements</i>). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>; <i>Elements</i>, Prop. VII.2) and the first known proof of the <a href="/wiki/Infinitude_of_primes" class="mw-redirect" title="Infinitude of primes">infinitude of primes</a> (<i>Elements</i>, Prop. IX.20). </p><p>In 1773, <a href="/wiki/Gotthold_Ephraim_Lessing" title="Gotthold Ephraim Lessing">Lessing</a> published an <a href="/wiki/Epigram" title="Epigram">epigram</a> he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> to <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a>.<sup id="cite_ref-FOOTNOTEVardi1998305–319_29-0" class="reference"><a href="#cite_note-FOOTNOTEVardi1998305–319-29"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWeil198417–24_30-0" class="reference"><a href="#cite_note-FOOTNOTEWeil198417–24-30"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The epigram proposed what has become known as <a href="/wiki/Archimedes%27s_cattle_problem" title="Archimedes's cattle problem">Archimedes's cattle problem</a>; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a>). As far as it is known, such equations were first successfully treated by the <a href="#Āryabhaṭa,_Brahmagupta,_Bhāskara">Indian school</a>. It is not known whether Archimedes himself had a method of solution. </p> <div class="mw-heading mw-heading4"><h4 id="Diophantus">Diophantus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=5" title="Edit section: Diophantus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diophantus-cover.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/170px-Diophantus-cover.png" decoding="async" width="170" height="266" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/255px-Diophantus-cover.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/340px-Diophantus-cover.png 2x" data-file-width="828" data-file-height="1295" /></a><figcaption>Title page of the 1621 edition of <a href="/wiki/Diophantus_of_Alexandria" class="mw-redirect" title="Diophantus of Alexandria">Diophantus of Alexandria</a>'s <i>Arithmetica</i>, translated into <a href="/wiki/Latin" title="Latin">Latin</a> by <a href="/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac" class="mw-redirect" title="Claude Gaspard Bachet de Méziriac">Claude Gaspard Bachet de Méziriac</a></figcaption></figure> <p>Very little is known about <a href="/wiki/Diophantus_of_Alexandria" class="mw-redirect" title="Diophantus of Alexandria">Diophantus of Alexandria</a>; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's <i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i> survive in the original Greek and four more survive in an Arabic translation. The <i>Arithmetica</i> is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form <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 f(x,y)=z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3d49febe8c5fb88fb25d4376433469c13bdbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.85ex; height:3.176ex;" alt="{\displaystyle f(x,y)=z^{2}}"></span> or <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 f(x,y,z)=w^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=w^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1004a1f588de9d793e07ee911e4378555096b606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.546ex; height:3.176ex;" alt="{\displaystyle f(x,y,z)=w^{2}}"></span>. Thus, nowadays, a <i><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a></i> a <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a> to which rational or integer solutions are sought. </p> <div class="mw-heading mw-heading4"><h4 id="Āryabhaṭa,_Brahmagupta,_Bhāskara"><span id=".C4.80ryabha.E1.B9.ADa.2C_Brahmagupta.2C_Bh.C4.81skara"></span>Āryabhaṭa, Brahmagupta, Bhāskara</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=6" title="Edit section: Āryabhaṭa, Brahmagupta, Bhāskara"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While Greek astronomy probably influenced Indian learning, to the point of introducing <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>,<sup id="cite_ref-FOOTNOTEPlofker2008119_31-0" class="reference"><a href="#cite_note-FOOTNOTEPlofker2008119-31"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> it seems to be the case that Indian mathematics is otherwise an indigenous tradition;<sup id="cite_ref-Plofbab_32-0" class="reference"><a href="#cite_note-Plofbab-32"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> in particular, there is no evidence that Euclid's Elements reached India before the 18th century.<sup id="cite_ref-FOOTNOTEMumford2010387_33-0" class="reference"><a href="#cite_note-FOOTNOTEMumford2010387-33"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences <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\equiv a_{1}{\bmod {m}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\equiv a_{1}{\bmod {m}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c663611bcfb301bc7826f1b93ccd106dec042744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.553ex; height:2.509ex;" alt="{\displaystyle n\equiv a_{1}{\bmod {m}}_{1}}"></span>, <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\equiv a_{2}{\bmod {m}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\equiv a_{2}{\bmod {m}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c47219ea418a20f473f14a09b48959cbe71125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.553ex; height:2.509ex;" alt="{\displaystyle n\equiv a_{2}{\bmod {m}}_{2}}"></span> could be solved by a method he called <i>kuṭṭaka</i>, or <i>pulveriser</i>;<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India.<sup id="cite_ref-FOOTNOTEMumford2010388_35-0" class="reference"><a href="#cite_note-FOOTNOTEMumford2010388-35"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Āryabhaṭa seems to have had in mind applications to astronomical calculations.<sup id="cite_ref-FOOTNOTEPlofker2008119_31-1" class="reference"><a href="#cite_note-FOOTNOTEPlofker2008119-31"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell equation</a>, in which <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the <a href="/wiki/Chakravala_method" title="Chakravala method">chakravala</a>, or "cyclic method") for solving Pell's equation was finally found by <a href="/wiki/Jayadeva_(mathematician)" title="Jayadeva (mathematician)">Jayadeva</a> (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a>'s Bīja-gaṇita (twelfth century).<sup id="cite_ref-FOOTNOTEPlofker2008194_36-0" class="reference"><a href="#cite_note-FOOTNOTEPlofker2008194-36"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Indian mathematics remained largely unknown in Europe until the late eighteenth century;<sup id="cite_ref-FOOTNOTEPlofker2008283_37-0" class="reference"><a href="#cite_note-FOOTNOTEPlofker2008283-37"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Brahmagupta and Bhāskara's work was translated into English in 1817 by <a href="/wiki/Henry_Thomas_Colebrooke" title="Henry Thomas Colebrooke">Henry Colebrooke</a>.<sup id="cite_ref-FOOTNOTEColebrooke1817_38-0" class="reference"><a href="#cite_note-FOOTNOTEColebrooke1817-38"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Arithmetic_in_the_Islamic_golden_age">Arithmetic in the Islamic golden age</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=7" title="Edit section: Arithmetic in the Islamic golden age"><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">Further information: <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Mathematics in medieval Islam</a> and <a href="/wiki/Islamic_Golden_Age" title="Islamic Golden Age">Islamic Golden Age</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hevelius_Selenographia_frontispiece.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Hevelius_Selenographia_frontispiece.png/170px-Hevelius_Selenographia_frontispiece.png" decoding="async" width="170" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Hevelius_Selenographia_frontispiece.png/255px-Hevelius_Selenographia_frontispiece.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Hevelius_Selenographia_frontispiece.png/340px-Hevelius_Selenographia_frontispiece.png 2x" data-file-width="864" data-file-height="1456" /></a><figcaption><a href="/wiki/Al-Haytham" class="mw-redirect" title="Al-Haytham">Al-Haytham</a> as seen by the West: on the frontispiece of <i><a href="/wiki/Selenographia" class="mw-redirect" title="Selenographia">Selenographia</a></i> Alhasen [<i><a href="/wiki/Sic" title="Sic">sic</a></i>] represents knowledge through reason and Galileo knowledge through the senses.</figcaption></figure> <p>In the early ninth century, the caliph <a href="/wiki/Al-Ma%27mun" title="Al-Ma'mun">Al-Ma'mun</a> ordered translations of many Greek mathematical works and at least one Sanskrit work (the <i>Sindhind</i>, which may<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> or may not<sup id="cite_ref-Plofnot_40-0" class="reference"><a href="#cite_note-Plofnot-40"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> be Brahmagupta's <a href="/wiki/Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta">Brāhmasphuṭasiddhānta</a>). Diophantus's main work, the <i>Arithmetica</i>, was translated into Arabic by <a href="/wiki/Qusta_ibn_Luqa" title="Qusta ibn Luqa">Qusta ibn Luqa</a> (820–912). Part of the treatise <i>al-Fakhri</i> (by <a href="/wiki/Al-Karaji" title="Al-Karaji">al-Karajī</a>, 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> knew<sup id="cite_ref-FOOTNOTERashed1980305–321_41-0" class="reference"><a href="#cite_note-FOOTNOTERashed1980305–321-41"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> what would later be called <a href="/wiki/Wilson%27s_theorem" title="Wilson's theorem">Wilson's theorem</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Western_Europe_in_the_Middle_Ages">Western Europe in the Middle Ages</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=8" title="Edit section: Western Europe in the Middle Ages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other than a treatise on squares in arithmetic progression by <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' <i>Arithmetica</i>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Early_modern_number_theory">Early modern number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=9" title="Edit section: Early modern number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Fermat">Fermat</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=10" title="Edit section: Fermat"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pierre_de_Fermat.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pierre_de_Fermat.png/170px-Pierre_de_Fermat.png" decoding="async" width="170" height="301" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pierre_de_Fermat.png/255px-Pierre_de_Fermat.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pierre_de_Fermat.png/340px-Pierre_de_Fermat.png 2x" data-file-width="566" data-file-height="1003" /></a><figcaption><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></figcaption></figure> <p><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> (1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.<sup id="cite_ref-FOOTNOTEWeil198445–46_43-0" class="reference"><a href="#cite_note-FOOTNOTEWeil198445–46-43"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> In his notes and letters, he scarcely wrote any proofs—he had no models in the area.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>Over his lifetime, Fermat made the following contributions to the field: </p> <ul><li>One of Fermat's first interests was <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a> (which appear in Euclid, <i>Elements</i> IX) and <a href="/wiki/Amicable_numbers" title="Amicable numbers">amicable numbers</a>;<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>note 6<span class="cite-bracket">]</span></a></sup> these topics led him to work on integer <a href="/wiki/Divisor" title="Divisor">divisors</a>, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></li> <li>In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> (1640):<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> if <i>a</i> is not divisible by a prime <i>p</i>, then <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^{p-1}\equiv 1{\bmod {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{p-1}\equiv 1{\bmod {p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ec1de006b31585777b0bd54505db183dc0281e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.148ex; height:3.009ex;" alt="{\displaystyle a^{p-1}\equiv 1{\bmod {p}}.}"></span><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>note 7<span class="cite-bracket">]</span></a></sup></li> <li>If <i>a</i> and <i>b</i> are <a href="/wiki/Coprime_integers" title="Coprime integers">coprime</a>, then <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^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fc28103c5d2aa9276728469f82c9f415f4b257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.176ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}}"></span> is not divisible by any prime congruent to −1 modulo 4;<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> and every prime congruent to 1 modulo 4 can be written in the form <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^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fc28103c5d2aa9276728469f82c9f415f4b257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.176ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}}"></span>.<sup id="cite_ref-FOOTNOTETanneryHenry1891Vol._II,_p._213_51-0" class="reference"><a href="#cite_note-FOOTNOTETanneryHenry1891Vol._II,_p._213-51"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the <a href="/wiki/Proof_by_infinite_descent" title="Proof by infinite descent">method of infinite descent</a>.<sup id="cite_ref-FOOTNOTETanneryHenry1891Vol._II,_p._423_52-0" class="reference"><a href="#cite_note-FOOTNOTETanneryHenry1891Vol._II,_p._423-52"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li> <li>In 1657, Fermat posed the problem of solving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-Ny^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-Ny^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa2a18557e1553bfe01d2ed1dfeafed52b42c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.764ex; height:3.009ex;" alt="{\displaystyle x^{2}-Ny^{2}=1}"></span> as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker.<sup id="cite_ref-FOOTNOTEWeil198492_53-0" class="reference"><a href="#cite_note-FOOTNOTEWeil198492-53"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.</li> <li>Fermat stated and proved (by infinite descent) in the appendix to <i>Observations on Diophantus</i> (Obs. XLV)<sup id="cite_ref-FOOTNOTETanneryHenry1891Vol._I,_pp._340–341_54-0" class="reference"><a href="#cite_note-FOOTNOTETanneryHenry1891Vol._I,_pp._340–341-54"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+y^{4}=z^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+y^{4}=z^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc85e410302961db48dfb90dfb4f27372961922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:3.009ex;" alt="{\displaystyle x^{4}+y^{4}=z^{4}}"></span> has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}+y^{3}=z^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}+y^{3}=z^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff0aa9dde87ce8348763c196bd299ad434b822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:3.009ex;" alt="{\displaystyle x^{3}+y^{3}=z^{3}}"></span> has no non-trivial solutions, and that this could also be proven by infinite descent.<sup id="cite_ref-FOOTNOTEWeil1984115_55-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984115-55"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> The first known proof is due to Euler (1753; indeed by infinite descent).<sup id="cite_ref-FOOTNOTEWeil1984115–116_56-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984115–116-56"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></li> <li>Fermat claimed (<a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>) to have shown there are no solutions to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}+y^{n}=z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}+y^{n}=z^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34dcc46ebfb58e1b91a6c0caa1470e76139543a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.175ex; height:2.676ex;" alt="{\displaystyle x^{n}+y^{n}=z^{n}}"></span> for all <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\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}"></span>; this claim appears in his annotations in the margins of his copy of Diophantus.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Euler">Euler</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=11" title="Edit section: Euler"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Leonhard_Euler.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/170px-Leonhard_Euler.jpg" decoding="async" width="170" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/255px-Leonhard_Euler.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/340px-Leonhard_Euler.jpg 2x" data-file-width="4672" data-file-height="6040" /></a><figcaption>Leonhard Euler</figcaption></figure> <p>The interest of <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>note 8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Goldbach</a>, pointed him towards some of Fermat's work on the subject.<sup id="cite_ref-FOOTNOTEWeil19842,_172_58-0" class="reference"><a href="#cite_note-FOOTNOTEWeil19842,_172-58"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEVaradarajan20069_59-0" class="reference"><a href="#cite_note-FOOTNOTEVaradarajan20069-59"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> This has been called the "rebirth" of modern number theory,<sup id="cite_ref-FOOTNOTEWeil19841–2_60-0" class="reference"><a href="#cite_note-FOOTNOTEWeil19841–2-60"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> after Fermat's relative lack of success in getting his contemporaries' attention for the subject.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Euler's work on number theory includes the following:<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i>Proofs for Fermat's statements.</i> This includes <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> (generalised by Euler to non-prime moduli); the fact that <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 p=x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=x^{2}+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe433e9ba9c42efe1bc9fd51bb439c698b8f6843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.796ex; height:3.009ex;" alt="{\displaystyle p=x^{2}+y^{2}}"></span> if and only if <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 p\equiv 1{\bmod {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\equiv 1{\bmod {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d504d3c492a1b5a4fcf14acb15f857fb0ea705ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.363ex; height:2.509ex;" alt="{\displaystyle p\equiv 1{\bmod {4}}}"></span>; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> (1770), soon improved by Euler himself<sup id="cite_ref-FOOTNOTEWeil1984178–179_63-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984178–179-63"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup>); the lack of non-zero integer solutions to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+y^{4}=z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+y^{4}=z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35aa613c26db550f2733d251ae8cbcb906831ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:3.009ex;" alt="{\displaystyle x^{4}+y^{4}=z^{2}}"></span> (implying the case <i>n=4</i> of Fermat's last theorem, the case <i>n=3</i> of which Euler also proved by a related method).</li> <li><i><a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a></i>, first misnamed by Euler.<sup id="cite_ref-Eulpell_64-0" class="reference"><a href="#cite_note-Eulpell-64"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> He wrote on the link between <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">continued fractions</a> and Pell's equation.<sup id="cite_ref-FOOTNOTEWeil1984183_65-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984183-65"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></li> <li><i>First steps towards analytic number theory.</i> In his work of sums of four squares, <a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)">partitions</a>, <a href="/wiki/Pentagonal_numbers" class="mw-redirect" title="Pentagonal numbers">pentagonal numbers</a>, and the <a href="/wiki/Distribution_(number_theory)" title="Distribution (number theory)">distribution</a> of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, most of his work is restricted to the formal manipulation of <a href="/wiki/Power_series" title="Power series">power series</a>. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup></li> <li><i>Quadratic forms</i>. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+Ny^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+Ny^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0d4c34b673da9926f76d0642c4ba92e09e882d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.503ex; height:3.009ex;" alt="{\displaystyle x^{2}+Ny^{2}}"></span>, some of it prefiguring <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>.<sup id="cite_ref-FOOTNOTEVaradarajan200644–47_67-0" class="reference"><a href="#cite_note-FOOTNOTEVaradarajan200644–47-67"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWeil1984177–179_68-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984177–179-68"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEEdwards1983285–291_69-0" class="reference"><a href="#cite_note-FOOTNOTEEdwards1983285–291-69"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></li> <li><i>Diophantine equations</i>. Euler worked on some Diophantine equations of genus 0 and 1.<sup id="cite_ref-FOOTNOTEVaradarajan200655–56_70-0" class="reference"><a href="#cite_note-FOOTNOTEVaradarajan200655–56-70"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWeil1984179–181_71-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984179–181-71"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy.<sup id="cite_ref-FOOTNOTEWeil1984181_72-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984181-72"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> He did notice there was a connection between Diophantine problems and <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integrals</a>,<sup id="cite_ref-FOOTNOTEWeil1984181_72-1" class="reference"><a href="#cite_note-FOOTNOTEWeil1984181-72"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> whose study he had himself initiated.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Andrew_wiles1-3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Andrew_wiles1-3.jpg/170px-Andrew_wiles1-3.jpg" decoding="async" width="170" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Andrew_wiles1-3.jpg/255px-Andrew_wiles1-3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Andrew_wiles1-3.jpg/340px-Andrew_wiles1-3.jpg 2x" data-file-width="1986" data-file-height="2502" /></a><figcaption>"Here was a problem, that I, a ten-year-old, could understand, and I knew from that moment that I would never let it go. I had to solve it."<sup id="cite_ref-pbs_73-0" class="reference"><a href="#cite_note-pbs-73"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> —Sir <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a> about <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">his proof</a> of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Lagrange,_Legendre,_and_Gauss"><span id="Lagrange.2C_Legendre.2C_and_Gauss"></span>Lagrange, Legendre, and Gauss</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=12" title="Edit section: Lagrange, Legendre, and Gauss"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Disqvisitiones-800.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/150px-Disqvisitiones-800.jpg" decoding="async" width="150" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/225px-Disqvisitiones-800.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/300px-Disqvisitiones-800.jpg 2x" data-file-width="478" data-file-height="800" /></a><figcaption>Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition</figcaption></figure> <p><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the <a href="/wiki/Lagrange%27s_four-square_theorem" title="Lagrange's four-square theorem">four-square theorem</a> and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> in full generality (as opposed to <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 mX^{2}+nY^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mX^{2}+nY^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af5c33baa55ad20007ec7c02f6351cdb98980ec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.281ex; height:2.843ex;" alt="{\displaystyle mX^{2}+nY^{2}}"></span>)—defining their equivalence relation, showing how to put them in reduced form, etc. </p><p><a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> and <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a>. He gave a full treatment of the equation <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 ax^{2}+by^{2}+cz^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+by^{2}+cz^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7c75ed5d8d24057cd1160d2dfc659972e903af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.919ex; height:3.009ex;" alt="{\displaystyle ax^{2}+by^{2}+cz^{2}=0}"></span><sup id="cite_ref-FOOTNOTEWeil1984327–328_74-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984327–328-74"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> and worked on quadratic forms along the lines later developed fully by Gauss.<sup id="cite_ref-FOOTNOTEWeil1984332–334_75-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984332–334-75"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> In his old age, he was the first to prove Fermat's Last Theorem for <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=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb41e9a10a8fd7179b9170149a8d70949ba5d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}"></span> (completing work by <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>, and crediting both him and <a href="/wiki/Sophie_Germain" title="Sophie Germain">Sophie Germain</a>).<sup id="cite_ref-FOOTNOTEWeil1984337–338_76-0" class="reference"><a href="#cite_note-FOOTNOTEWeil1984337–338-76"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_Friedrich_Gauss.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/220px-Carl_Friedrich_Gauss.jpg" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/330px-Carl_Friedrich_Gauss.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/440px-Carl_Friedrich_Gauss.jpg 2x" data-file-width="917" data-file-height="1180" /></a><figcaption>Carl Friedrich Gauss</figcaption></figure> <p>In his <i>Disquisitiones Arithmeticae</i> (1798), Carl Friedrich Gauss (1777–1855) proved the law of <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a> and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (<a href="/wiki/Congruences" class="mw-redirect" title="Congruences">congruences</a>) and devoted a section to computational matters, including primality tests.<sup id="cite_ref-FOOTNOTEGoldsteinSchappacher200714_77-0" class="reference"><a href="#cite_note-FOOTNOTEGoldsteinSchappacher200714-77"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> The last section of the <i>Disquisitiones</i> established a link between <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> and number theory: </p> <blockquote><p>The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>In this way, Gauss arguably made a first foray towards both <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a>'s work and <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Maturity_and_division_into_subfields">Maturity and division into subfields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=13" title="Edit section: Maturity and division into subfields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:ErnstKummer.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/ErnstKummer.jpg/170px-ErnstKummer.jpg" decoding="async" width="170" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2f/ErnstKummer.jpg 1.5x" data-file-width="202" data-file-height="319" /></a><figcaption><a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a></figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Peter_Gustav_Lejeune_Dirichlet.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Peter_Gustav_Lejeune_Dirichlet.jpg/170px-Peter_Gustav_Lejeune_Dirichlet.jpg" decoding="async" width="170" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Peter_Gustav_Lejeune_Dirichlet.jpg/255px-Peter_Gustav_Lejeune_Dirichlet.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Peter_Gustav_Lejeune_Dirichlet.jpg/340px-Peter_Gustav_Lejeune_Dirichlet.jpg 2x" data-file-width="493" data-file-height="545" /></a><figcaption><a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a></figcaption></figure> <p>Starting early in the nineteenth century, the following developments gradually took place: </p> <ul><li>The rise to self-consciousness of number theory (or <i>higher arithmetic</i>) as a field of study.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup></li> <li>The development of much of modern mathematics necessary for basic modern number theory: <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>—accompanied by greater rigor in analysis and abstraction in algebra.</li> <li>The rough subdivision of number theory into its modern subfields—in particular, <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic</a> and algebraic number theory.</li></ul> <p>Algebraic number theory may be said to start with the study of reciprocity and <a href="/wiki/Root_of_unity" title="Root of unity">cyclotomy</a>, but truly came into its own with the development of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> and early ideal theory and <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuation</a> theory; see below. A conventional starting point for analytic number theory is <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a> (1837),<sup id="cite_ref-FOOTNOTEApostol19767_80-0" class="reference"><a href="#cite_note-FOOTNOTEApostol19767-80"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEDavenportMontgomery20001_81-0" class="reference"><a href="#cite_note-FOOTNOTEDavenportMontgomery20001-81"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> whose proof introduced <a href="/wiki/L-functions" class="mw-redirect" title="L-functions">L-functions</a> and involved some asymptotic analysis and a limiting process on a real variable.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> The first use of analytic ideas in number theory actually goes back to Euler (1730s),<sup id="cite_ref-FOOTNOTEIwaniecKowalski20041_83-0" class="reference"><a href="#cite_note-FOOTNOTEIwaniecKowalski20041-83"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEVaradarajan2006sections_2.5,_3.1_and_6.1_84-0" class="reference"><a href="#cite_note-FOOTNOTEVaradarajan2006sections_2.5,_3.1_and_6.1-84"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> who used formal power series and non-rigorous (or implicit) limiting arguments. The use of <i>complex</i> analysis in number theory comes later: the work of <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (1859) on the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">zeta function</a> is the canonical starting point;<sup id="cite_ref-FOOTNOTEGranville2008322–348_85-0" class="reference"><a href="#cite_note-FOOTNOTEGranville2008322–348-85"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Jacobi%27s_four-square_theorem" title="Jacobi's four-square theorem">Jacobi's four-square theorem</a> (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (<a href="/wiki/Modular_form" title="Modular form">modular forms</a>).<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p><p>The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on. </p> <div class="mw-heading mw-heading2"><h2 id="Main_subdivisions">Main subdivisions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=14" title="Edit section: Main subdivisions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Elementary_number_theory">Elementary number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=15" title="Edit section: Elementary number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i><a href="/wiki/Elementary_proof" title="Elementary proof">elementary</a></i> generally denotes a method that does not use <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>. For example, the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős</a> and <a href="/wiki/Atle_Selberg" title="Atle Selberg">Selberg</a>.<sup id="cite_ref-FOOTNOTEGoldfeld2003_87-0" class="reference"><a href="#cite_note-FOOTNOTEGoldfeld2003-87"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> The term is somewhat ambiguous: for example, proofs based on complex <a href="/wiki/Tauberian_theorem" class="mw-redirect" title="Tauberian theorem">Tauberian theorems</a> (for example, <a href="/wiki/Wiener%E2%80%93Ikehara_theorem" title="Wiener–Ikehara theorem">Wiener–Ikehara</a>) are often seen as quite enlightening but not elementary, in spite of using <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, rather than complex analysis as such. Here as elsewhere, an <i>elementary</i> proof may be longer and more difficult for most readers than a non-elementary one. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Paul_Erdos_with_Terence_Tao.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Paul_Erdos_with_Terence_Tao.jpg/270px-Paul_Erdos_with_Terence_Tao.jpg" decoding="async" width="270" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Paul_Erdos_with_Terence_Tao.jpg/405px-Paul_Erdos_with_Terence_Tao.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c4/Paul_Erdos_with_Terence_Tao.jpg 2x" data-file-width="515" data-file-height="348" /></a><figcaption>Number theorists <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> in 1985, when Erdős was 72 and Tao was 10</figcaption></figure> <p>Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Analytic_number_theory">Analytic number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=16" title="Edit section: Analytic number theory"><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/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_zeta.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Complex_zeta.jpg/220px-Complex_zeta.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Complex_zeta.jpg/330px-Complex_zeta.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Complex_zeta.jpg/440px-Complex_zeta.jpg 2x" data-file-width="821" data-file-height="820" /></a><figcaption><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> ζ(<i>s</i>) in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. The color of a point <i>s</i> gives the value of ζ(<i>s</i>): dark colors denote values close to zero and hue gives the value's <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:ModularGroup-FundamentalDomain.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/220px-ModularGroup-FundamentalDomain.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/330px-ModularGroup-FundamentalDomain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/440px-ModularGroup-FundamentalDomain.svg.png 2x" data-file-width="401" data-file-height="181" /></a><figcaption>The action of the <a href="/wiki/Modular_group" title="Modular group">modular group</a> on the <a href="/wiki/Upper_half_plane" class="mw-redirect" title="Upper half plane">upper half plane</a>. The region in grey is the standard <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a>.</figcaption></figure> <p><i>Analytic number theory</i> may be defined </p> <ul><li>in terms of its tools, as the study of the integers by means of tools from <a href="/wiki/Real_analysis" title="Real analysis">real</a> and <a href="/wiki/Complex_analysis" title="Complex analysis">complex</a> analysis;<sup id="cite_ref-FOOTNOTEApostol19767_80-1" class="reference"><a href="#cite_note-FOOTNOTEApostol19767-80"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> or</li> <li>in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup></li></ul> <p>Some subjects generally considered to be part of analytic number theory, for example, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a>,<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>note 9<span class="cite-bracket">]</span></a></sup> are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis,<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>note 10<span class="cite-bracket">]</span></a></sup> yet it does belong to analytic number theory. </p><p>The following are examples of problems in analytic number theory: the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>, the <a href="/wiki/Goldbach_conjecture" class="mw-redirect" title="Goldbach conjecture">Goldbach conjecture</a> (or the <a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">twin prime conjecture</a>, or the <a href="/wiki/Hardy%E2%80%93Littlewood_conjecture" class="mw-redirect" title="Hardy–Littlewood conjecture">Hardy–Littlewood conjectures</a>), the <a href="/wiki/Waring_problem" class="mw-redirect" title="Waring problem">Waring problem</a> and the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>. Some of the most important tools of analytic number theory are the <a href="/wiki/Circle_method" class="mw-redirect" title="Circle method">circle method</a>, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve methods</a> and <a href="/wiki/L-functions" class="mw-redirect" title="L-functions">L-functions</a> (or, rather, the study of their properties). The theory of <a href="/wiki/Modular_form" title="Modular form">modular forms</a> (and, more generally, <a href="/wiki/Automorphic_forms" class="mw-redirect" title="Automorphic forms">automorphic forms</a>) also occupies an increasingly central place in the toolbox of analytic number theory.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> </p><p>One may ask analytic questions about <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> (generalizations of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question <a href="/wiki/Landau_prime_ideal_theorem" title="Landau prime ideal theorem">can be answered</a> by means of an examination of <a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta functions</a>, which are generalizations of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, a key analytic object at the roots of the subject.<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> This is an example of a general procedure in analytic number theory: deriving information about the distribution of a <a href="/wiki/Sequence" title="Sequence">sequence</a> (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.<sup id="cite_ref-:0_94-0" class="reference"><a href="#cite_note-:0-94"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_number_theory">Algebraic number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=17" title="Edit section: Algebraic number theory"><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/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></div> <p>An <i>algebraic number</i> is any <a href="/wiki/Complex_number" title="Complex number">complex number</a> that is a solution to some polynomial equation <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 f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}"></span> with rational coefficients; for example, every solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9503fed2affe4705838d28dd802e295c5df1d2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.718ex; height:3.176ex;" alt="{\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0}"></span> (say) is an algebraic number. Fields of algebraic numbers are also called <i><a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a></i>, or shortly <i><a href="/wiki/Number_field" class="mw-redirect" title="Number field">number fields</a></i>. Algebraic number theory studies algebraic number fields.<sup id="cite_ref-FOOTNOTEMilne20172_95-0" class="reference"><a href="#cite_note-FOOTNOTEMilne20172-95"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. </p><p>It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in <i>Disquisitiones arithmeticae</i> can be restated in terms of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> and <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norms</a> in quadratic fields. (A <i>quadratic field</i> consists of all numbers of the form <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+b{\sqrt {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6c46254eee2978b88135fbd1d75e89d836dbcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.22ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {d}}}"></span>, where <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> and <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> are rational numbers and <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is a fixed rational number whose square root is not rational.) For that matter, the 11th-century <a href="/wiki/Chakravala_method" title="Chakravala method">chakravala method</a> amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such. </p><p>The grounds of the subject were set in the late nineteenth century, when <i>ideal numbers</i>, the <i>theory of ideals</i> and <i>valuation theory</i> were introduced; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and <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 {\sqrt {-5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6cc38ada9b0a58b1d3248d62f616e40341e8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-5}}}"></span>, the number <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 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"></span> can be factorised both as <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 6=2\cdot 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=2\cdot 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bd35c05d4411ba15b1890e917fc1370381a8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 6=2\cdot 3}"></span> and <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 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec35824d17526a7edf178c52be24823bb3091f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.698ex; height:3.009ex;" alt="{\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})}"></span>; all of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>, <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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\sqrt {-5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\sqrt {-5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4625b87f83984dd8727b066cc0f19f801185b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.909ex; height:3.009ex;" alt="{\displaystyle 1+{\sqrt {-5}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\sqrt {-5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\sqrt {-5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1831393106d87542d5282bf65d7ec460390ee09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.909ex; height:3.009ex;" alt="{\displaystyle 1-{\sqrt {-5}}}"></span> are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer</a>) seems to have come from the study of higher reciprocity laws,<sup id="cite_ref-FOOTNOTEEdwards200079_96-0" class="reference"><a href="#cite_note-FOOTNOTEEdwards200079-96"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> that is, generalisations of <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>. </p><p>Number fields are often studied as extensions of smaller number fields: a field <i>L</i> is said to be an <i>extension</i> of a field <i>K</i> if <i>L</i> contains <i>K</i>. (For example, the complex numbers <i>C</i> are an extension of the reals <i>R</i>, and the reals <i>R</i> are an extension of the rationals <i>Q</i>.) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions <i>L</i> of <i>K</i> such that the <a href="/wiki/Galois_group" title="Galois group">Galois group</a><sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>note 11<span class="cite-bracket">]</span></a></sup> Gal(<i>L</i>/<i>K</i>) of <i>L</i> over <i>K</i> is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>—are relatively well understood. Their classification was the object of the programme of <a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>, which was initiated in the late 19th century (partly by <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a> and <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a>) and carried out largely in 1900–1950. </p><p>An example of an active area of research in algebraic number theory is <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>. The <a href="/wiki/Langlands_program" title="Langlands program">Langlands program</a>, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. </p> <div class="mw-heading mw-heading3"><h3 id="Diophantine_geometry">Diophantine geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=18" title="Edit section: Diophantine geometry"><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/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></div> <p>The central problem of <i>Diophantine geometry</i> is to determine when a <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a> has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object. </p><p>For example, an equation in two variables defines a curve in the plane. More generally, an equation or system of equations in two or more variables defines a <a href="/wiki/Algebraic_curve" title="Algebraic curve">curve</a>, a <a href="/wiki/Algebraic_surface" title="Algebraic surface">surface</a>, or some other such object in <span class="texhtml"><i>n</i></span>-dimensional space. In Diophantine geometry, one asks whether there are any <i>rational points</i> (points all of whose coordinates are rationals) or <i>integral points</i> (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is whether there are finitely or infinitely many rational points on a given curve or surface. </p><p>An example here may be helpful. Consider the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean equation</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 x^{2}+y^{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf613d4a7485ac7b6ae29f066158d58b6bc2e719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.347ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1,}"></span> one would like to know its rational solutions; that is, its solutions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> such that <i>x</i> and <i>y</i> are both rational. This is the same as asking for all integer solutions to <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^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></span>; any solution to the latter equation gives us a solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d355166b22e17a160f930cee7316594fcb8f80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.827ex; height:2.843ex;" alt="{\displaystyle x=a/c}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1ee1bcb03c82c95a3e8587b13db23cb274c434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle y=b/c}"></span> to the former. It is also the same as asking for all points with rational coordinates on the curve described by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"></span> (a circle of radius 1 centered on the origin). </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ECClines-3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/220px-ECClines-3.svg.png" decoding="async" width="220" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/330px-ECClines-3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/ECClines-3.svg/440px-ECClines-3.svg.png 2x" data-file-width="335" data-file-height="190" /></a><figcaption>Two examples of <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a>, that is, curves of genus 1 having at least one rational point. Such curves have always infinitely many rational points.</figcaption></figure> <p>The rephrasing of questions on equations in terms of points on curves is felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve (that is, rational or integer solutions to an equation <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 f(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle f(x,y)=0}"></span>, where <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is a polynomial in two variables) depends crucially on the <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> of the curve.<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>note 12<span class="cite-bracket">]</span></a></sup> A major achievement of this approach is <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a>, for which other geometrical notions are just as crucial. </p><p>There is also the closely linked area of <a href="/wiki/Diophantine_approximations" class="mw-redirect" title="Diophantine approximations">Diophantine approximations</a>: given a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call <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/q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8286e068898b57215dbd608e8c97520a4ce3db0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.462ex; height:2.843ex;" alt="{\displaystyle a/q}"></span> (with <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 \gcd(a,q)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(a,q)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf09a62819517d7404f4839223e5944d455f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.891ex; height:2.843ex;" alt="{\displaystyle \gcd(a,q)=1}"></span>) a good approximation to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-a/q|<{\frac {1}{q^{c}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-a/q|<{\frac {1}{q^{c}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57fea4b868d34d02adcc805fd4669fec9caa3c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.884ex; height:5.676ex;" alt="{\displaystyle |x-a/q|<{\frac {1}{q^{c}}}}"></span>, where <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> is large. This question is of special interest if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is an algebraic number. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of <a href="/wiki/Glossary_of_arithmetic_and_diophantine_geometry#H" title="Glossary of arithmetic and diophantine geometry">height</a>) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in <a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">transcendental number theory</a>: if a number can be approximated better than any algebraic number, then it is a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>. It is by this argument that <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> have been shown to be transcendental. </p><p>Diophantine geometry should not be confused with the <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a>, which is a collection of graphical methods for answering certain questions in algebraic number theory. <i>Arithmetic geometry</i>, however, is a contemporary term for much the same domain as that covered by the term <i>Diophantine geometry</i>. The term <i>arithmetic geometry</i> is arguably used most often when one wishes to emphasize the connections to modern algebraic geometry (for example, in <a href="/wiki/Faltings%27s_theorem" title="Faltings's theorem">Faltings's theorem</a>) rather than to techniques in Diophantine approximations. </p> <div class="mw-heading mw-heading2"><h2 id="Other_subfields">Other subfields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=19" title="Edit section: Other subfields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as explained below, algorithms in number theory have a long history, arguably predating the formal concept of proof. However, the modern study of <a href="/wiki/Computability" title="Computability">computability</a> began only in the 1930s and 1940s, while <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> emerged in the 1970s. </p> <div class="mw-heading mw-heading3"><h3 id="Probabilistic_number_theory">Probabilistic number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=20" title="Edit section: Probabilistic number theory"><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/Probabilistic_number_theory" title="Probabilistic number theory">Probabilistic number theory</a></div> <p>Probabilistic number theory starts with questions such as the following ones: Take an integer <span class="texhtml mvar" style="font-style:italic;">n</span> at random between one and a million. How likely is it to be prime? (this is just another way of asking how many primes there are between one and a million). How many prime divisors will <span class="texhtml mvar" style="font-style:italic;">n</span> have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average? </p><p>Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite. </p><p>It is sometimes said that <a href="/wiki/Probabilistic_combinatorics" class="mw-redirect" title="Probabilistic combinatorics">probabilistic combinatorics</a> uses the fact that whatever happens with probability greater than <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one. </p><p>At times, a non-rigorous, probabilistic approach leads to a number of <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> algorithms and open problems, notably <a href="/wiki/Cram%C3%A9r%27s_conjecture" title="Cramér's conjecture">Cramér's conjecture</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_combinatorics">Arithmetic combinatorics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=21" title="Edit section: Arithmetic combinatorics"><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 articles: <a href="/wiki/Arithmetic_combinatorics" title="Arithmetic combinatorics">Arithmetic combinatorics</a> and <a href="/wiki/Additive_number_theory" title="Additive number theory">Additive number theory</a></div> <p>Arithmetic combinatorics starts with questions like the following ones: Does a fairly "thick" <a href="/wiki/Infinite_set" title="Infinite set">infinite set</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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> contain many elements in arithmetic progression: <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>, <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+b,a+2b,a+3b,\ldots ,a+10b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>10</mn> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b,a+2b,a+3b,\ldots ,a+10b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fae7f0c412e074931fb8f7c39e3ff32642bd577" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.167ex; height:2.509ex;" alt="{\displaystyle a+b,a+2b,a+3b,\ldots ,a+10b}"></span>, say? Should it be possible to write large integers as sums of elements of <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>? </p><p>These questions are characteristic of <i>arithmetic combinatorics</i>. This is a presently coalescing field; it subsumes <i><a href="/wiki/Additive_number_theory" title="Additive number theory">additive number theory</a></i> (which concerns itself with certain very specific sets <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of arithmetic significance, such as the primes or the squares) and, arguably, some of the <i>geometry of numbers</i>, together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>, <a href="/wiki/Finite_group_theory" class="mw-redirect" title="Finite group theory">finite group theory</a>, <a href="/wiki/Model_theory" title="Model theory">model theory</a>, and other fields. The term <i>additive combinatorics</i> is also used; however, the sets <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> being studied need not be sets of integers, but rather subsets of non-commutative <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, in which case the growth of <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+A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c6ebc8e6d1bbf9f61da8a7e870949abb404c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.327ex; height:2.343ex;" alt="{\displaystyle A+A}"></span> and <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>·<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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> may be compared. </p> <div class="mw-heading mw-heading3"><h3 id="Computational_number_theory">Computational number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=22" title="Edit section: Computational number theory"><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/Computational_number_theory" title="Computational number theory">Computational number theory</a></div><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Computer_History_Museum_(4145886786).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Computer_History_Museum_%284145886786%29.jpg/220px-Computer_History_Museum_%284145886786%29.jpg" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Computer_History_Museum_%284145886786%29.jpg/330px-Computer_History_Museum_%284145886786%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Computer_History_Museum_%284145886786%29.jpg/440px-Computer_History_Museum_%284145886786%29.jpg 2x" data-file-width="3164" data-file-height="4752" /></a><figcaption>A <a href="/wiki/Lehmer_sieve" title="Lehmer sieve">Lehmer sieve</a>, a primitive <a href="/wiki/Digital_computer" class="mw-redirect" title="Digital computer">digital computer</a> used to find <a href="/wiki/Prime_number" title="Prime number">primes</a> and solve simple <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a></figcaption></figure><p>While the word <i>algorithm</i> goes back only to certain readers of <a href="/wiki/Al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Al-Khwārizmī">al-Khwārizmī</a>, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period. </p><p>An early case is that of what is now called the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a>) it appears as Proposition 2 of Book VII in <i>Elements</i>, together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation <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 ax+by=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a39c048e6c827a005e417be0e1bd5ac314e6fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle ax+by=c}"></span>, or, what is the same, for finding the quantities whose existence is assured by the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>) it first appears in the works of <a href="/wiki/Aryabhata#Indeterminate_equations" title="Aryabhata">Āryabhaṭa</a> (5th–6th century CE) as an algorithm called <i>kuṭṭaka</i> ("pulveriser"), without a proof of correctness. </p><p>There are two main questions: "Can this be computed?" and "Can it be computed rapidly?" Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. Fast algorithms for <a href="/wiki/Primality_test" title="Primality test">testing primality</a> are now known, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring. </p><p>The difficulty of a computation can be useful: modern protocols for <a href="/wiki/Cryptography" title="Cryptography">encrypting messages</a> (for example, <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a>) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems. </p><p>Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>, that there is no <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> which can solve all Diophantine equations.<sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> In particular, this means that, given a <a href="/wiki/Computably_enumerable" class="mw-redirect" title="Computably enumerable">computably enumerable</a> set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (i.e., Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. It cannot be proven that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.) </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=23" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number-theorist <a href="/wiki/Leonard_Dickson" class="mw-redirect" title="Leonard Dickson">Leonard Dickson</a> (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> In 1974, <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".<sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> Elementary number theory is taught in <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a> courses for <a href="/wiki/Computer_scientist" title="Computer scientist">computer scientists</a>. It also has applications to the continuous in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>.<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> </p><p>Number theory has now several modern applications spanning diverse areas such as: </p> <ul><li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a>: Public-key encryption schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.<sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Computer_science" title="Computer science">Computer science</a>: The <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.<sup id="cite_ref-104" class="reference"><a href="#cite_note-104"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Physics" title="Physics">Physics</a>: The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> has connections to the distribution of prime numbers and has been studied for its potential implications in physics.<sup id="cite_ref-105" class="reference"><a href="#cite_note-105"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Error_correction_code" title="Error correction code">Error correction codes</a>: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.<sup id="cite_ref-106" class="reference"><a href="#cite_note-106"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup></li> <li>Communications: The design of cellular telephone networks requires knowledge of the theory of <a href="/wiki/Modular_form" title="Modular form">modular forms</a>, which is a part of analytic number theory.<sup id="cite_ref-107" class="reference"><a href="#cite_note-107"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup></li> <li>Study of musical scales: the concept of "<a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>", which is the basis for most modern Western music, involves dividing the <a href="/wiki/Octave" title="Octave">octave</a> into 12 equal parts.<sup id="cite_ref-108" class="reference"><a href="#cite_note-108"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> This has been studied using number theory and in particular the properties of the 12th root of 2.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Prizes">Prizes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=24" title="Edit section: Prizes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> awards the <i><a href="/wiki/Cole_Prize" title="Cole Prize">Cole Prize in Number Theory</a></i>. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the <i><a href="/wiki/Fermat_Prize" title="Fermat Prize">Fermat Prize</a></i>. </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=Number_theory&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Algebraic_function_field" title="Algebraic function field">Algebraic function field</a></li> <li><a href="/wiki/Finite_field" title="Finite field">Finite field</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number">p-adic number</a></li> <li><a href="/wiki/List_of_number_theoretic_algorithms" class="mw-redirect" title="List of number theoretic algorithms">List of number theoretic algorithms</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=26" title="Edit section: Notes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Already in 1921, <a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">T. L. Heath</a> had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers." (<a href="#CITEREFHeath1921">Heath 1921</a>, p. 13)</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">Take, for example, <a href="#CITEREFSerre1996">Serre 1996</a>. In 1952, <a href="/wiki/Harold_Davenport" title="Harold Davenport">Davenport</a> still had to specify that he meant <i>The Higher Arithmetic</i>. <a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy</a> and Wright wrote in the introduction to <i><a href="/wiki/An_Introduction_to_the_Theory_of_Numbers" title="An Introduction to the Theory of Numbers">An Introduction to the Theory of Numbers</a></i> (1938): "We proposed at one time to change [the title] to <i>An introduction to arithmetic</i>, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." (<a href="#CITEREFHardyWright2008">Hardy & Wright 2008</a>)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFRobson2001">Robson 2001</a>, p. 201. This is controversial. See <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a>. Robson's article is written polemically (<a href="#CITEREFRobson2001">Robson 2001</a>, p. 202) with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" (<a href="#CITEREFRobson2001">Robson 2001</a>, p. 167); at the same time, it settles to the conclusion that <blockquote><p>[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (<a href="#CITEREFRobson2001">Robson 2001</a>, p. 202). </p></blockquote> <p>Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".(<a href="#CITEREFRobson2001">Robson 2001</a>, pp. 199–200) </p> </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"><i>Sunzi Suanjing</i>, Ch. 3, Problem 26, in <a href="#CITEREFLamAng2004">Lam & Ang 2004</a>, pp. 219–220:<blockquote> <p>[26] Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. <i>Answer</i>: 23.<br /> </p> <p><i>Method</i>: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When [a number] exceeds 106, the result is obtained by subtracting 105.</p></blockquote></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">See, for example, <i>Sunzi Suanjing</i>, Ch. 3, Problem 36, in <a href="#CITEREFLamAng2004">Lam & Ang 2004</a>, pp. 223–224:<blockquote> <p>[36] Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. <i>Answer</i>: Male.<br /> </p> <p><i>Method</i>: Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female.</p></blockquote> <p>This is the last problem in Sunzi's otherwise matter-of-fact treatise. </p> </span></li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean (and hence mystical) <a href="/wiki/Nicomachus_of_Gerasa" class="mw-redirect" title="Nicomachus of Gerasa">Nicomachus</a> (ca. 100 CE), who wrote a primitive but influential "<a href="/wiki/Introduction_to_Arithmetic" class="mw-redirect" title="Introduction to Arithmetic">Introduction to Arithmetic</a>". See <a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, Ch. IV.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Here, as usual, given two integers <i>a</i> and <i>b</i> and a non-zero integer <i>m</i>, we write <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\equiv b{\bmod {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b{\bmod {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e71bfeb1edd0b7451eff65e6c7f3c8c3eaa944b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.047ex; height:2.176ex;" alt="{\displaystyle a\equiv b{\bmod {m}}}"></span> (read "<i>a</i> is congruent to <i>b</i> modulo <i>m</i>") to mean that <i>m</i> divides <i>a</i> − <i>b</i>, or, what is the same, <i>a</i> and <i>b</i> leave the same residue when divided by <i>m</i>. This notation is actually much later than Fermat's; it first appears in section 1 of <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>'s <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a>. Fermat's little theorem is a consequence of the <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">fact</a> that the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of an element of a group divides the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo <i>p</i>, that is, given <i>a</i> not divisible by a prime <i>p</i>, there is an integer <i>x</i> such that <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 xa\equiv 1{\bmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>a</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xa\equiv 1{\bmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b88601064cc54c2ab369f4a06bd6a7a9f0fa74d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.671ex; height:2.509ex;" alt="{\displaystyle xa\equiv 1{\bmod {p}}}"></span>); this fact (which, in modern language, makes the residues mod <i>p</i> into a group, and which was already known to Āryabhaṭa; see <a href="#Indian_school:_Āryabhaṭa,_Brahmagupta,_Bhāskara">above</a>) was familiar to Fermat thanks to its rediscovery by <a href="/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac" class="mw-redirect" title="Claude Gaspard Bachet de Méziriac">Bachet</a> (<a href="#CITEREFWeil1984">Weil 1984</a>, p. 7). Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text">Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way (<a href="#CITEREFWeil1984">Weil 1984</a>, pp. 159, 161). (There were already some recognisable features of professional <i>practice</i>, viz., seeking correspondents, visiting foreign colleagues, building private libraries (<a href="#CITEREFWeil1984">Weil 1984</a>, pp. 160–161). Matters started to shift in the late 17th century (<a href="#CITEREFWeil1984">Weil 1984</a>, p. 161); scientific academies were founded in England (the <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>, 1662) and France (the <a href="/wiki/French_Academy_of_Sciences" title="French Academy of Sciences">Académie des sciences</a>, 1666) and <a href="/wiki/Russian_Academy_of_Sciences" title="Russian Academy of Sciences">Russia</a> (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 (<a href="#CITEREFWeil1984">Weil 1984</a>, p. 163 and <a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 7). In this context, the term <i>amateur</i> usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy (<a href="#CITEREFTruesdell1984">Truesdell 1984</a>, p. xv); cited in <a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 9). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.</span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, <a href="#CITEREFIwaniecKowalski2004">Iwaniec & Kowalski 2004</a> or <a href="#CITEREFMontgomeryVaughan2007">Montgomery & Vaughan 2007</a></span> </li> <li id="cite_note-91"><span class="mw-cite-backlink"><b><a href="#cite_ref-91">^</a></b></span> <span class="reference-text">This is the case for small sieves (in particular, some combinatorial sieves such as the <a href="/wiki/Brun_sieve" title="Brun sieve">Brun sieve</a>) rather than for <a href="/wiki/Large_sieve" title="Large sieve">large sieves</a>; the study of the latter now includes ideas from <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.</span> </li> <li id="cite_note-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-97">^</a></b></span> <span class="reference-text">The Galois group of an extension <i>L/K</i> consists of the operations (<a href="/wiki/Isomorphisms" class="mw-redirect" title="Isomorphisms">isomorphisms</a>) that send elements of L to other elements of L while leaving all elements of K fixed. Thus, for instance, <i>Gal(C/R)</i> consists of two elements: the identity element (taking every element <i>x</i> + <i>iy</i> of <i>C</i> to itself) and complex conjugation (the map taking each element <i>x</i> + <i>iy</i> to <i>x</i> − <i>iy</i>). The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a>; in modern language, the main outcome of his work is that an equation <i>f</i>(<i>x</i>) = 0 can be solved by radicals (that is, <i>x</i> can be expressed in terms of the four basic operations together with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation <i>f</i>(<i>x</i>) = 0 has a Galois group that is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a> in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)</span> </li> <li id="cite_note-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-98">^</a></b></span> <span class="reference-text">The <i>genus</i> can be defined as follows: allow the variables in <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 f(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle f(x,y)=0}"></span> to be complex numbers; then <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 f(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle f(x,y)=0}"></span> defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of doughnut-like holes in the surface is called the <i>genus</i> of the curve of equation <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 f(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle f(x,y)=0}"></span>.</span> </li> </ol></div> <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=Number_theory&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTELong19721-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELong19721_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLong1972">Long 1972</a>, p. 1.</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"><a href="#CITEREFNeugebauer_&_Sachs1945">Neugebauer & Sachs 1945</a>, p. 40. The term <i>takiltum</i> is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".<a href="#CITEREFRobson2001">Robson 2001</a>, p. 192</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"><a href="#CITEREFRobson2001">Robson 2001</a>, p. 189. Other sources give the modern formula <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 (p^{2}-q^{2},2pq,p^{2}+q^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>2</mn> <mi>p</mi> <mi>q</mi> <mo>,</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p^{2}-q^{2},2pq,p^{2}+q^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e679a80d8e3df3cfd591b045627eecb10262f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.674ex; height:3.176ex;" alt="{\displaystyle (p^{2}-q^{2},2pq,p^{2}+q^{2})}"></span>. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.(<a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 79)</span> </li> <li id="cite_note-FOOTNOTEvan_der_Waerden1961184-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_der_Waerden1961184_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 184.</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">Neugebauer (<a href="#CITEREFNeugebauer1969">Neugebauer 1969</a>, pp. 36–40) discusses the table in detail and mentions in passing Euclid's method in modern notation (<a href="#CITEREFNeugebauer1969">Neugebauer 1969</a>, p. 39).</span> </li> <li id="cite_note-FOOTNOTEFriberg1981302-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFriberg1981302_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFriberg1981">Friberg 1981</a>, p. 302.</span> </li> <li id="cite_note-FOOTNOTEvan_der_Waerden196143-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_der_Waerden196143_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 43.</span> </li> <li id="cite_note-vanderW2-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-vanderW2_11-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Iamblichus" title="Iamblichus">Iamblichus</a>, <i>Life of Pythagoras</i>,(trans., for example, <a href="#CITEREFGuthrie1987">Guthrie 1987</a>) cited in <a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 108. See also <a href="/wiki/Porphyry_(philosopher)" title="Porphyry (philosopher)">Porphyry</a>, <i>Life of Pythagoras</i>, paragraph 6, in <a href="#CITEREFGuthrie1987">Guthrie 1987</a> Van der Waerden (<a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, pp. 87–90) sustains the view that Thales knew Babylonian mathematics.</span> </li> <li id="cite_note-stanencyc-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-stanencyc_12-0">^</a></b></span> <span class="reference-text">Herodotus (II. 81) and Isocrates (<i>Busiris</i> 28), cited in: <a href="#CITEREFHuffman2011">Huffman 2011</a>. On Thales, see Eudemus ap. Proclus, 65.7, (for example, <a href="#CITEREFMorrow1992">Morrow 1992</a>, p. 52) cited in: <a href="#CITEREFO'Grady2004">O'Grady 2004</a>, p. 1. Proclus was using a work by <a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus of Rhodes</a> (now lost), the <i>Catalogue of Geometers</i>. See also introduction, <a href="#CITEREFMorrow1992">Morrow 1992</a>, p. xxx on Proclus's reliability.</span> </li> <li id="cite_note-Becker-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Becker_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBecker1936">Becker 1936</a>, p. 533, cited in: <a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 108.</span> </li> <li id="cite_note-FOOTNOTEBecker1936-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBecker1936_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBecker1936">Becker 1936</a>.</span> </li> <li id="cite_note-FOOTNOTEvan_der_Waerden1961109-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_der_Waerden1961109_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_der_Waerden1961">van der Waerden 1961</a>, p. 109.</span> </li> <li id="cite_note-Thea-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Thea_16-0">^</a></b></span> <span class="reference-text">Plato, <i>Theaetetus</i>, p. 147 B, (for example, <a href="#CITEREFJowett1871">Jowett 1871</a>), cited in <a href="#CITEREFvon_Fritz2004">von Fritz 2004</a>, p. 212: "Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit;..." <i>See also</i> <a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a>.</span> </li> <li id="cite_note-FOOTNOTEvon_Fritz2004-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvon_Fritz2004_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvon_Fritz2004">von Fritz 2004</a>.</span> </li> <li id="cite_note-FOOTNOTEHeath192176-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHeath192176_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeath1921">Heath 1921</a>, p. 76.</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"><i>Sunzi Suanjing</i>, Chapter 3, Problem 26. This can be found in <a href="#CITEREFLamAng2004">Lam & Ang 2004</a>, pp. 219–220, which contains a full translation of the <i>Suan Ching</i> (based on <a href="#CITEREFQian1963">Qian 1963</a>). See also the discussion in <a href="#CITEREFLamAng2004">Lam & Ang 2004</a>, pp. 138–140.</span> </li> <li id="cite_note-YongSe-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-YongSe_20-0">^</a></b></span> <span class="reference-text">The date of the text has been narrowed down to 220–420 CE (Yan Dunjie) or 280–473 CE (Wang Ling) through internal evidence (= taxation systems assumed in the text). See <a href="#CITEREFLamAng2004">Lam & Ang 2004</a>, pp. 27–28.</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"><a href="#CITEREFDauben2007">Dauben 2007</a>, p. 310</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"><a href="#CITEREFLibbrecht1973">Libbrecht 1973</a></span> </li> <li id="cite_note-FOOTNOTEBoyerMerzbach199182-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoyerMerzbach199182_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyerMerzbach1991">Boyer & Merzbach 1991</a>, p. 82.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</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="http://www.tertullian.org/fathers/eusebius_pe_10_book10.htm">"Eusebius of Caesarea: Praeparatio Evangelica (Preparation for the Gospel). Tr. E.H. Gifford (1903) – Book 10"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161211194042/http://www.tertullian.org/fathers/eusebius_pe_10_book10.htm">Archived</a> from the original on 2016-12-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-02-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Eusebius+of+Caesarea%3A+Praeparatio+Evangelica+%28Preparation+for+the+Gospel%29.+Tr.+E.H.+Gifford+%281903%29+%E2%80%93+Book+10&rft_id=http%3A%2F%2Fwww.tertullian.org%2Ffathers%2Feusebius_pe_10_book10.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Metaphysics, 1.6.1 (987a)</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Tusc. Disput. 1.17.39.</span> </li> <li id="cite_note-FOOTNOTEVardi1998305–319-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVardi1998305–319_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVardi1998">Vardi 1998</a>, pp. 305–319.</span> </li> <li id="cite_note-FOOTNOTEWeil198417–24-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil198417–24_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 17–24.</span> </li> <li id="cite_note-FOOTNOTEPlofker2008119-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEPlofker2008119_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEPlofker2008119_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFPlofker2008">Plofker 2008</a>, p. 119.</span> </li> <li id="cite_note-Plofbab-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Plofbab_32-0">^</a></b></span> <span class="reference-text">Any early contact between Babylonian and Indian mathematics remains conjectural (<a href="#CITEREFPlofker2008">Plofker 2008</a>, p. 42).</span> </li> <li id="cite_note-FOOTNOTEMumford2010387-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMumford2010387_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMumford2010">Mumford 2010</a>, p. 387.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: <a href="#CITEREFPlofker2008">Plofker 2008</a>, pp. 134–140. See also <a href="#CITEREFClark1930">Clark 1930</a>, pp. 42–50. A slightly more explicit description of the kuṭṭaka was later given in <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>, <i>Brāhmasphuṭasiddhānta</i>, XVIII, 3–5 (in <a href="#CITEREFColebrooke1817">Colebrooke 1817</a>, p. 325, cited in <a href="#CITEREFClark1930">Clark 1930</a>, p. 42).</span> </li> <li id="cite_note-FOOTNOTEMumford2010388-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMumford2010388_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMumford2010">Mumford 2010</a>, p. 388.</span> </li> <li id="cite_note-FOOTNOTEPlofker2008194-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPlofker2008194_36-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPlofker2008">Plofker 2008</a>, p. 194.</span> </li> <li id="cite_note-FOOTNOTEPlofker2008283-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPlofker2008283_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPlofker2008">Plofker 2008</a>, p. 283.</span> </li> <li id="cite_note-FOOTNOTEColebrooke1817-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEColebrooke1817_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFColebrooke1817">Colebrooke 1817</a>.</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="#CITEREFColebrooke1817">Colebrooke 1817</a>, p. lxv, cited in <a href="#CITEREFHopkins1990">Hopkins 1990</a>, p. 302. See also the preface in <a href="#CITEREFSachauBīrūni1888">Sachau & Bīrūni 1888</a> cited in <a href="#CITEREFSmith1958">Smith 1958</a>, pp. 168</span> </li> <li id="cite_note-Plofnot-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-Plofnot_40-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPingree1968">Pingree 1968</a>, pp. 97–125, and <a href="#CITEREFPingree1970">Pingree 1970</a>, pp. 103–123, cited in <a href="#CITEREFPlofker2008">Plofker 2008</a>, p. 256.</span> </li> <li id="cite_note-FOOTNOTERashed1980305–321-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERashed1980305–321_41-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRashed1980">Rashed 1980</a>, pp. 305–321.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><a href="/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac" class="mw-redirect" title="Claude Gaspard Bachet de Méziriac">Bachet</a>, 1621, following a first attempt by <a href="/wiki/Guilielmus_Xylander" class="mw-redirect" title="Guilielmus Xylander">Xylander</a>, 1575</span> </li> <li id="cite_note-FOOTNOTEWeil198445–46-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil198445–46_43-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 45–46.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 118. This was more so in number theory than in other areas (remark in <a href="#CITEREFMahoney1994">Mahoney 1994</a>, p. 284). Bachet's own proofs were "ludicrously clumsy" (<a href="#CITEREFWeil1984">Weil 1984</a>, p. 33).</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><a href="#CITEREFMahoney1994">Mahoney 1994</a>, pp. 48, 53–54. The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, <a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. II, pp. 72, 74, cited in <a href="#CITEREFMahoney1994">Mahoney 1994</a>, p. 54.</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFaulknerHosch2017" class="citation book cs1">Faulkner, Nicholas; Hosch, William L. (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5tFFDwAAQBAJ"><i>Numbers and Measurements</i></a>. Encyclopaedia Britannica. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1538300428" title="Special:BookSources/978-1538300428"><bdi>978-1538300428</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301144254/https://books.google.com/books?id=5tFFDwAAQBAJ">Archived</a> from the original on 2023-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers+and+Measurements&rft.pub=Encyclopaedia+Britannica&rft.date=2017&rft.isbn=978-1538300428&rft.aulast=Faulkner&rft.aufirst=Nicholas&rft.au=Hosch%2C+William+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5tFFDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. II, p. 209, Letter XLVI from Fermat to Frenicle, 1640, cited in <a href="#CITEREFWeil1984">Weil 1984</a>, p. 56</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. II, p. 204, cited in <a href="#CITEREFWeil1984">Weil 1984</a>, p. 63. All of the following citations from Fermat's <i>Varia Opera</i> are taken from <a href="#CITEREFWeil1984">Weil 1984</a>, Chap. II. The standard Tannery & Henry work includes a revision of Fermat's posthumous <i>Varia Opera Mathematica</i> originally prepared by his son (<a href="#CITEREFFermat1679">Fermat 1679</a>).</span> </li> <li id="cite_note-FOOTNOTETanneryHenry1891Vol._II,_p._213-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETanneryHenry1891Vol._II,_p._213_51-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. II, p. 213.</span> </li> <li id="cite_note-FOOTNOTETanneryHenry1891Vol._II,_p._423-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETanneryHenry1891Vol._II,_p._423_52-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. II, p. 423.</span> </li> <li id="cite_note-FOOTNOTEWeil198492-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil198492_53-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 92.</span> </li> <li id="cite_note-FOOTNOTETanneryHenry1891Vol._I,_pp._340–341-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETanneryHenry1891Vol._I,_pp._340–341_54-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTanneryHenry1891">Tannery & Henry 1891</a>, Vol. I, pp. 340–341.</span> </li> <li id="cite_note-FOOTNOTEWeil1984115-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984115_55-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 115.</span> </li> <li id="cite_note-FOOTNOTEWeil1984115–116-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984115–116_56-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 115–116.</span> </li> <li id="cite_note-FOOTNOTEWeil19842,_172-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil19842,_172_58-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 2, 172.</span> </li> <li id="cite_note-FOOTNOTEVaradarajan20069-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVaradarajan20069_59-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 9.</span> </li> <li id="cite_note-FOOTNOTEWeil19841–2-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil19841–2_60-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 1–2.</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 2 and <a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 37</span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 39 and <a href="#CITEREFWeil1984">Weil 1984</a>, pp. 176–189</span> </li> <li id="cite_note-FOOTNOTEWeil1984178–179-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984178–179_63-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 178–179.</span> </li> <li id="cite_note-Eulpell-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-Eulpell_64-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 174. Euler was generous in giving credit to others (<a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, p. 14), not always correctly.</span> </li> <li id="cite_note-FOOTNOTEWeil1984183-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984183_65-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 183.</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, pp. 45–55; see also chapter III.</span> </li> <li id="cite_note-FOOTNOTEVaradarajan200644–47-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVaradarajan200644–47_67-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, pp. 44–47.</span> </li> <li id="cite_note-FOOTNOTEWeil1984177–179-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984177–179_68-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 177–179.</span> </li> <li id="cite_note-FOOTNOTEEdwards1983285–291-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEdwards1983285–291_69-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdwards1983">Edwards 1983</a>, pp. 285–291.</span> </li> <li id="cite_note-FOOTNOTEVaradarajan200655–56-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVaradarajan200655–56_70-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, pp. 55–56.</span> </li> <li id="cite_note-FOOTNOTEWeil1984179–181-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984179–181_71-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 179–181.</span> </li> <li id="cite_note-FOOTNOTEWeil1984181-72"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWeil1984181_72-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWeil1984181_72-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, p. 181.</span> </li> <li id="cite_note-pbs-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-pbs_73-0">^</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.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html">"Andrew Wiles on Solving Fermat"</a>. <a href="/wiki/WGBH-TV" title="WGBH-TV">WGBH</a>. November 2000. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160317012127/http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html">Archived</a> from the original on 17 March 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">16 March</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Andrew+Wiles+on+Solving+Fermat&rft.pub=WGBH&rft.date=2000-11&rft_id=https%3A%2F%2Fwww.pbs.org%2Fwgbh%2Fnova%2Fphysics%2Fandrew-wiles-fermat.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEWeil1984327–328-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984327–328_74-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 327–328.</span> </li> <li id="cite_note-FOOTNOTEWeil1984332–334-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984332–334_75-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 332–334.</span> </li> <li id="cite_note-FOOTNOTEWeil1984337–338-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeil1984337–338_76-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeil1984">Weil 1984</a>, pp. 337–338.</span> </li> <li id="cite_note-FOOTNOTEGoldsteinSchappacher200714-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGoldsteinSchappacher200714_77-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldsteinSchappacher2007">Goldstein & Schappacher 2007</a>, p. 14.</span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text">From the preface of <i>Disquisitiones Arithmeticae</i>; the translation is taken from <a href="#CITEREFGoldsteinSchappacher2007">Goldstein & Schappacher 2007</a>, p. 16</span> </li> <li id="cite_note-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-79">^</a></b></span> <span class="reference-text">See the discussion in section 5 of <a href="#CITEREFGoldsteinSchappacher2007">Goldstein & Schappacher 2007</a>. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in <a href="#CITEREFWeil1984">Weil 1984</a>, p. 25).</span> </li> <li id="cite_note-FOOTNOTEApostol19767-80"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEApostol19767_80-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEApostol19767_80-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFApostol1976">Apostol 1976</a>, p. 7.</span> </li> <li id="cite_note-FOOTNOTEDavenportMontgomery20001-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDavenportMontgomery20001_81-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDavenportMontgomery2000">Davenport & Montgomery 2000</a>, p. 1.</span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text">See the proof in <a href="#CITEREFDavenportMontgomery2000">Davenport & Montgomery 2000</a>, section 1</span> </li> <li id="cite_note-FOOTNOTEIwaniecKowalski20041-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEIwaniecKowalski20041_83-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFIwaniecKowalski2004">Iwaniec & Kowalski 2004</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEVaradarajan2006sections_2.5,_3.1_and_6.1-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVaradarajan2006sections_2.5,_3.1_and_6.1_84-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2006">Varadarajan 2006</a>, sections 2.5, 3.1 and 6.1.</span> </li> <li id="cite_note-FOOTNOTEGranville2008322–348-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGranville2008322–348_85-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGranville2008">Granville 2008</a>, pp. 322–348.</span> </li> <li id="cite_note-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-86">^</a></b></span> <span class="reference-text">See the comment on the importance of modularity in <a href="#CITEREFIwaniecKowalski2004">Iwaniec & Kowalski 2004</a>, p. 1</span> </li> <li id="cite_note-FOOTNOTEGoldfeld2003-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGoldfeld2003_87-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldfeld2003">Goldfeld 2003</a>.</span> </li> <li id="cite_note-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-88">^</a></b></span> <span class="reference-text">See, for example, the initial comment in <a href="#CITEREFIwaniecKowalski2004">Iwaniec & Kowalski 2004</a>, p. 1.</span> </li> <li id="cite_note-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-89">^</a></b></span> <span class="reference-text"><a href="#CITEREFGranville2008">Granville 2008</a>, section 1: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for <i>good approximations</i>."</span> </li> <li id="cite_note-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-92">^</a></b></span> <span class="reference-text">See the remarks in the introduction to <a href="#CITEREFIwaniecKowalski2004">Iwaniec & Kowalski 2004</a>, p. 1: "However much stronger...".</span> </li> <li id="cite_note-93"><span class="mw-cite-backlink"><b><a href="#cite_ref-93">^</a></b></span> <span class="reference-text"><a href="#CITEREFGranville2008">Granville 2008</a>, section 3: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]"</span> </li> <li id="cite_note-:0-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_94-0">^</a></b></span> <span class="reference-text">See, for example, <a href="#CITEREFMontgomeryVaughan2007">Montgomery & Vaughan 2007</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEMilne20172-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMilne20172_95-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilne2017">Milne 2017</a>, p. 2.</span> </li> <li id="cite_note-FOOTNOTEEdwards200079-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEdwards200079_96-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdwards2000">Edwards 2000</a>, p. 79.</span> </li> <li id="cite_note-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-99">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisMatiyasevichRobinson1976" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a>; <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Matiyasevich, Yuri</a>; <a href="/wiki/Julia_Robinson" title="Julia Robinson">Robinson, Julia</a> (1976). "Hilbert's Tenth Problem: Diophantine Equations: Positive Aspects of a Negative Solution". In <a href="/wiki/Felix_Browder" title="Felix Browder">Felix E. Browder</a> (ed.). <i>Mathematical Developments Arising from Hilbert Problems</i>. <a href="/wiki/Proceedings_of_Symposia_in_Pure_Mathematics" class="mw-redirect" title="Proceedings of Symposia in Pure Mathematics">Proceedings of Symposia in Pure Mathematics</a>. Vol. XXVIII.2. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. pp. 323–378. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1428-4" title="Special:BookSources/978-0-8218-1428-4"><bdi>978-0-8218-1428-4</bdi></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:0346.02026">0346.02026</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hilbert%27s+Tenth+Problem%3A+Diophantine+Equations%3A+Positive+Aspects+of+a+Negative+Solution&rft.btitle=Mathematical+Developments+Arising+from+Hilbert+Problems&rft.series=Proceedings+of+Symposia+in+Pure+Mathematics&rft.pages=323-378&rft.pub=American+Mathematical+Society&rft.date=1976&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0346.02026%23id-name%3DZbl&rft.isbn=978-0-8218-1428-4&rft.aulast=Davis&rft.aufirst=Martin&rft.au=Matiyasevich%2C+Yuri&rft.au=Robinson%2C+Julia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span> Reprinted in <i>The Collected Works of Julia Robinson</i>, <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a>, editor, pp. 269–378, American Mathematical Society 1996.</span> </li> <li id="cite_note-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-100">^</a></b></span> <span class="reference-text"><i>The Unreasonable Effectiveness of Number Theory</i>, Stefan Andrus Burr, George E. Andrews, American Mathematical Soc., 1992, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5501-0" title="Special:BookSources/978-0-8218-5501-0">978-0-8218-5501-0</a></span> </li> <li id="cite_note-101"><span class="mw-cite-backlink"><b><a href="#cite_ref-101">^</a></b></span> <span class="reference-text">Computer science and its relation to mathematics" DE Knuth – The American Mathematical Monthly, 1974</span> </li> <li id="cite_note-102"><span class="mw-cite-backlink"><b><a href="#cite_ref-102">^</a></b></span> <span class="reference-text">"Applications of number theory to numerical analysis", Lo-keng Hua, Luogeng Hua, Yuan Wang, Springer-Verlag, 1981, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-10382-0" title="Special:BookSources/978-3-540-10382-0">978-3-540-10382-0</a></span> </li> <li id="cite_note-103"><span class="mw-cite-backlink"><b><a href="#cite_ref-103">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/9781351664110"><i>An Introduction to Number Theory with Cryptography</i></a> (2nd ed.). Chapman and Hall/CRC. 2018. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781351664110">10.1201/9781351664110</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-66411-0" title="Special:BookSources/978-1-351-66411-0"><bdi>978-1-351-66411-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301144259/https://www.taylorfrancis.com/books/mono/10.1201/9781351664110/introduction-number-theory-cryptography-james-kraft-lawrence-washington">Archived</a> from the original on 2023-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Number+Theory+with+Cryptography&rft.edition=2nd&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2018&rft_id=info%3Adoi%2F10.1201%2F9781351664110&rft.isbn=978-1-351-66411-0&rft_id=https%3A%2F%2Fwww.taylorfrancis.com%2Fbooks%2F9781351664110&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-104"><span class="mw-cite-backlink"><b><a href="#cite_ref-104">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrishna2017" class="citation book cs1">Krishna, Hari (2017). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1004350753"><i>Digital Signal Processing Algorithms : Number Theory, Convolution, Fast Fourier Transforms, and Applications</i></a>. London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-45497-1" title="Special:BookSources/978-1-351-45497-1"><bdi>978-1-351-45497-1</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/1004350753">1004350753</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301144250/https://www.worldcat.org/title/1004350753">Archived</a> from the original on 2023-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing+Algorithms+%3A+Number+Theory%2C+Convolution%2C+Fast+Fourier+Transforms%2C+and+Applications&rft.place=London&rft.date=2017&rft_id=info%3Aoclcnum%2F1004350753&rft.isbn=978-1-351-45497-1&rft.aulast=Krishna&rft.aufirst=Hari&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1004350753&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-105"><span class="mw-cite-backlink"><b><a href="#cite_ref-105">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchumayerHutchinson2011" class="citation journal cs1">Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis". <i>Reviews of Modern Physics</i>. <b>83</b> (2): 307–330. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.3116">1101.3116</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011RvMP...83..307S">2011RvMP...83..307S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.83.307">10.1103/RevModPhys.83.307</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:119290777">119290777</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Physics+of+the+Riemann+Hypothesis&rft.volume=83&rft.issue=2&rft.pages=307-330&rft.date=2011&rft_id=info%3Aarxiv%2F1101.3116&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119290777%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.83.307&rft_id=info%3Abibcode%2F2011RvMP...83..307S&rft.aulast=Schumayer&rft.aufirst=Daniel&rft.au=Hutchinson%2C+David+A.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-106"><span class="mw-cite-backlink"><b><a href="#cite_ref-106">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaylis2018" class="citation book cs1">Baylis, John (2018). <a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/9781351449847"><i>Error-Correcting Codes: A Mathematical Introduction</i></a>. Routledge. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9780203756676">10.1201/9780203756676</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-203-75667-6" title="Special:BookSources/978-0-203-75667-6"><bdi>978-0-203-75667-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301144305/https://www.taylorfrancis.com/books/mono/10.1201/9780203756676/error-correcting-codes-baylis">Archived</a> from the original on 2023-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Error-Correcting+Codes%3A+A+Mathematical+Introduction&rft.pub=Routledge&rft.date=2018&rft_id=info%3Adoi%2F10.1201%2F9780203756676&rft.isbn=978-0-203-75667-6&rft.aulast=Baylis&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.taylorfrancis.com%2Fbooks%2F9781351449847&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-107"><span class="mw-cite-backlink"><b><a href="#cite_ref-107">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLivné2001" class="citation cs2">Livné, R. (2001), Ciliberto, Ciro; Hirzebruch, Friedrich; Miranda, Rick; Teicher, Mina (eds.), <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-94-010-1011-5_13">"Communication Networks and Hilbert Modular Forms"</a>, <i>Applications of Algebraic Geometry to Coding Theory, Physics and Computation</i>, Dordrecht: Springer Netherlands, pp. 255–270, <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%2F978-94-010-1011-5_13">10.1007/978-94-010-1011-5_13</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-0005-8" title="Special:BookSources/978-1-4020-0005-8"><bdi>978-1-4020-0005-8</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230301144239/https://link.springer.com/chapter/10.1007/978-94-010-1011-5_13">archived</a> from the original on 2023-03-01<span class="reference-accessdate">, retrieved <span class="nowrap">2023-02-22</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applications+of+Algebraic+Geometry+to+Coding+Theory%2C+Physics+and+Computation&rft.atitle=Communication+Networks+and+Hilbert+Modular+Forms&rft.pages=255-270&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F978-94-010-1011-5_13&rft.isbn=978-1-4020-0005-8&rft.aulast=Livn%C3%A9&rft.aufirst=R.&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2F978-94-010-1011-5_13&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> <li id="cite_note-108"><span class="mw-cite-backlink"><b><a href="#cite_ref-108">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartwrightGonzalezPiroStanzial2002" class="citation journal cs1">Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1076/jnmr.31.1.51.8099">"Aesthetics, Dynamics, and Musical Scales: A Golden Connection"</a>. <i>Journal of New Music Research</i>. <b>31</b> (1): 51–58. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1076%2Fjnmr.31.1.51.8099">10.1076/jnmr.31.1.51.8099</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10261%2F18003">10261/18003</a></span>. <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/0929-8215">0929-8215</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:12232457">12232457</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+New+Music+Research&rft.atitle=Aesthetics%2C+Dynamics%2C+and+Musical+Scales%3A+A+Golden+Connection&rft.volume=31&rft.issue=1&rft.pages=51-58&rft.date=2002-03-01&rft_id=info%3Ahdl%2F10261%2F18003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12232457%23id-name%3DS2CID&rft.issn=0929-8215&rft_id=info%3Adoi%2F10.1076%2Fjnmr.31.1.51.8099&rft.aulast=Cartwright&rft.aufirst=Julyan+H.+E.&rft.au=Gonzalez%2C+Diego+L.&rft.au=Piro%2C+Oreste&rft.au=Stanzial%2C+Domenico&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1076%2Fjnmr.31.1.51.8099&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=28" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDauben2007" class="citation cs2">Dauben, Joseph W. (2007), "Chapter 3: Chinese Mathematics", in Katz, Victor J. (ed.), <i>The Mathematics of Egypt, Mesopotamia, China, India and Islam : A Sourcebook</i>, Princeton University Press, pp. 187–384, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+3%3A+Chinese+Mathematics&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India+and+Islam+%3A+A+Sourcebook&rft.pages=187-384&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rft.aulast=Dauben&rft.aufirst=Joseph+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1976" class="citation book cs1"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Il64dZELHEIC"><i>Introduction to analytic number theory</i></a>. <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90163-3" title="Special:BookSources/978-0-387-90163-3"><bdi>978-0-387-90163-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+analytic+number+theory&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1976&rft.isbn=978-0-387-90163-3&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIl64dZELHEIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLibbrecht1973" class="citation cs2">Libbrecht, Ulrich (1973), <i>Chinese Mathematics in the Thirteenth Century: the "Shu-shu Chiu-chang" of Ch'in Chiu-shao</i>, Dover Publications Inc, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44619-6" title="Special:BookSources/978-0-486-44619-6"><bdi>978-0-486-44619-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chinese+Mathematics+in+the+Thirteenth+Century%3A+the+%22Shu-shu+Chiu-chang%22+of+Ch%27in+Chiu-shao&rft.pub=Dover+Publications+Inc&rft.date=1973&rft.isbn=978-0-486-44619-6&rft.aulast=Libbrecht&rft.aufirst=Ulrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1981" class="citation journal cs1">Apostol, Tom M. (1981). "An Introduction to the Theory of Numbers (Review of Hardy & Wright.)". <i>Mathematical Reviews (MathSciNet)</i>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</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=0568909">0568909</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Reviews+%28MathSciNet%29&rft.atitle=An+Introduction+to+the+Theory+of+Numbers+%28Review+of+Hardy+%26+Wright.%29&rft.date=1981&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0568909%23id-name%3DMR&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span> (Subscription needed)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker1936" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Oskar_Becker" title="Oskar Becker">Becker, Oskar</a> (1936). "Die Lehre von Geraden und Ungeraden im neunten Buch der euklidischen Elemente". <i>Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik</i>. 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New York: <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+Benjamin&rft.au=Merzbach%2C+Uta+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/details/AHistoryOfMathematics">1968 edition</a> at archive.org</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClark1930" class="citation book cs1">Clark, Walter Eugene (trans.) 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Retrieved <span class="nowrap">7 February</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thales+of+Miletus&rft.pub=The+Internet+Encyclopaedia+of+Philosophy&rft.date=2004-09&rft.aulast=O%27Grady&rft.aufirst=Patricia&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fthales%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPingree1968" class="citation journal cs1"><a href="/wiki/David_Pingree" title="David Pingree">Pingree, David</a>; <a href="/wiki/Ya%CA%BFq%C5%ABb_ibn_%E1%B9%AC%C4%81riq" class="mw-redirect" title="Yaʿqūb ibn Ṭāriq">Ya'qub, ibn Tariq</a> (1968). "The Fragments of the Works of Ya'qub ibn Tariq". <i>Journal of Near Eastern Studies</i>. <b>26</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Near+Eastern+Studies&rft.atitle=The+Fragments+of+the+Works+of+Ya%27qub+ibn+Tariq&rft.volume=26&rft.date=1968&rft.aulast=Pingree&rft.aufirst=David&rft.au=Ya%27qub%2C+ibn+Tariq&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPingree1970" class="citation journal cs1"><a href="/wiki/David_Pingree" title="David Pingree">Pingree, D.</a>; <a href="/wiki/Al-Fazari" title="Al-Fazari">al-Fazari</a> (1970). "The Fragments of the Works of al-Fazari". <i>Journal of Near Eastern Studies</i>. <b>28</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Near+Eastern+Studies&rft.atitle=The+Fragments+of+the+Works+of+al-Fazari&rft.volume=28&rft.date=1970&rft.aulast=Pingree&rft.aufirst=D.&rft.au=al-Fazari&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlofker2008" class="citation book cs1"><a href="/wiki/Kim_Plofker" title="Kim Plofker">Plofker, Kim</a> (2008). <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)"><i>Mathematics in India</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12067-6" title="Special:BookSources/978-0-691-12067-6"><bdi>978-0-691-12067-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+India&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-12067-6&rft.aulast=Plofker&rft.aufirst=Kim&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQian1963" class="citation book cs1 cs1-prop-foreign-lang-source">Qian, Baocong, ed. (1963). <a rel="nofollow" class="external text" href="https://www.scribd.com/doc/53797787/Jigu-Suanjing%E3%80%80%E7%B7%9D%E5%8F%A4%E7%AE%97%E7%B6%93-Qian-Baocong-%E9%8C%A2%E5%AF%B6%E7%90%AE"><i>Suanjing shi shu (Ten Mathematical Classics)</i></a> (in Chinese). Beijing: Zhonghua shuju. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131102154812/http://www.scribd.com/doc/53797787/Jigu-Suanjing%E3%80%80%E7%B7%9D%E5%8F%A4%E7%AE%97%E7%B6%93-Qian-Baocong-%E9%8C%A2%E5%AF%B6%E7%90%AE">Archived</a> from the original on 2013-11-02<span class="reference-accessdate">. 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"Ibn al-Haytham et le théorème de Wilson". <i>Archive for History of Exact Sciences</i>. <b>22</b> (4): 305–321. <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%2FBF00717654">10.1007/BF00717654</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:120885025">120885025</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Ibn+al-Haytham+et+le+th%C3%A9or%C3%A8me+de+Wilson&rft.volume=22&rft.issue=4&rft.pages=305-321&rft.date=1980&rft_id=info%3Adoi%2F10.1007%2FBF00717654&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120885025%23id-name%3DS2CID&rft.aulast=Rashed&rft.aufirst=Roshdi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobson2001" class="citation journal cs1"><a href="/wiki/Eleanor_Robson" title="Eleanor Robson">Robson, Eleanor</a> (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141021070742/http://www.hps.cam.ac.uk/people/robson/neither-sherlock.pdf">"Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322"</a> <span class="cs1-format">(PDF)</span>. <i>Historia Mathematica</i>. <b>28</b> (3): 167–206. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.2001.2317">10.1006/hmat.2001.2317</a>. Archived from <a rel="nofollow" class="external text" href="http://www.hps.cam.ac.uk/people/robson/neither-sherlock.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2014-10-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Neither+Sherlock+Holmes+nor+Babylon%3A+a+Reassessment+of+Plimpton+322&rft.volume=28&rft.issue=3&rft.pages=167-206&rft.date=2001&rft_id=info%3Adoi%2F10.1006%2Fhmat.2001.2317&rft.aulast=Robson&rft.aufirst=Eleanor&rft_id=http%3A%2F%2Fwww.hps.cam.ac.uk%2Fpeople%2Frobson%2Fneither-sherlock.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSachauBīrūni1888" class="citation book cs1"><a href="/wiki/Eduard_Sachau" title="Eduard Sachau">Sachau, Eduard</a>; <a href="/wiki/Ab%C5%AB_Ray%E1%B8%A5%C4%81n_al-B%C4%ABr%C5%ABn%C4%AB" class="mw-redirect" title="Abū Rayḥān al-Bīrūnī">Bīrūni, ̄Muḥammad ibn Aḥmad</a> (1888). <a rel="nofollow" class="external text" href="http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Sachau%2C%20Eduard%2C%201845-1930"><i>Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1</i></a>. London: Kegan, Paul, Trench, Trübner & Co. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303235035/http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Sachau,%20Eduard,%201845-1930">Archived</a> from the original on 2016-03-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alberuni%27s+India%3A+An+Account+of+the+Religion%2C+Philosophy%2C+Literature%2C+Geography%2C+Chronology%2C+Astronomy+and+Astrology+of+India%2C+Vol.+1&rft.place=London&rft.pub=Kegan%2C+Paul%2C+Trench%2C+Tr%C3%BCbner+%26+Co.&rft.date=1888&rft.aulast=Sachau&rft.aufirst=Eduard&rft.au=B%C4%ABr%C5%ABni%2C+%CC%84Mu%E1%B8%A5ammad+ibn+A%E1%B8%A5mad&rft_id=http%3A%2F%2Fonlinebooks.library.upenn.edu%2Fwebbin%2Fbook%2Flookupname%3Fkey%3DSachau%252C%2520Eduard%252C%25201845-1930&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1996" class="citation book cs1"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1996) [1973]. <a rel="nofollow" class="external text" href="https://archive.org/details/courseinarithmet00serr"><i>A Course in Arithmetic</i></a>. Graduate Texts in Mathematics. Vol. 7. <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90040-7" title="Special:BookSources/978-0-387-90040-7"><bdi>978-0-387-90040-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Arithmetic&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-90040-7&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcourseinarithmet00serr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1958" class="citation book cs1">Smith, D.E. (1958). <i>History of Mathematics, Vol I</i>. New York: Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics%2C+Vol+I&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1958&rft.aulast=Smith&rft.aufirst=D.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTanneryHenry1891" class="citation book cs1 cs1-prop-foreign-lang-source cs1-prop-foreign-lang-source"><a href="/wiki/Paul_Tannery" title="Paul Tannery">Tannery, Paul</a>; <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat, Pierre de</a> (1891). <a href="/wiki/Charles_Henry_(librarian)" title="Charles Henry (librarian)">Charles Henry</a> (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdefermat01ferm"><i>Oeuvres de Fermat</i></a>. (4 Vols.) (in French and Latin). Paris: Imprimerie Gauthier-Villars et Fils.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oeuvres+de+Fermat&rft.place=Paris&rft.series=%284+Vols.%29&rft.pub=Imprimerie+Gauthier-Villars+et+Fils&rft.date=1891&rft.aulast=Tannery&rft.aufirst=Paul&rft.au=Fermat%2C+Pierre+de&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foeuvresdefermat01ferm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdefermat01ferm">Volume 1</a> <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdefermat02ferm">Volume 2</a> <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdefermat03ferm">Volume 3</a> <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdefermat04ferm">Volume 4 (1912)</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1818" class="citation book cs1"><a href="/wiki/Iamblichus" title="Iamblichus">Iamblichus</a>; <a href="/wiki/Thomas_Taylor_(neoplatonist)" title="Thomas Taylor (neoplatonist)">Taylor, Thomas (trans.)</a> (1818). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110721184914/http://www.aurumsolis.info/index.php?option=com_phocadownload&view=category&download=1%3Aiamblichus-the-pythagorean-life&id=19%3Awritings-from-the-founders&Itemid=143&lang=en"><i>Life of Pythagoras or, Pythagoric Life</i></a>. London: J.M. Watkins. Archived from the original on 2011-07-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Life+of+Pythagoras+or%2C+Pythagoric+Life&rft.place=London&rft.pub=J.M.+Watkins&rft.date=1818&rft.au=Iamblichus&rft.au=Taylor%2C+Thomas+%28trans.%29&rft_id=http%3A%2F%2Fwww.aurumsolis.info%2Findex.php%3Foption%3Dcom_phocadownload%26view%3Dcategory%26download%3D1%253Aiamblichus-the-pythagorean-life%26id%3D19%253Awritings-from-the-founders%26Itemid%3D143%26lang%3Den&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</span> For other editions, see <a href="/wiki/Iamblichus#List_of_editions_and_translations" title="Iamblichus">Iamblichus#List of editions and translations</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTruesdell1984" class="citation book cs1"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.A.</a> (1984). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mkOhy6v7kIsC">"Leonard Euler, Supreme Geometer"</a>. In Hewlett, John (trans.) (ed.). <i>Leonard Euler, Elements of Algebra</i> (reprint of 1840 5th ed.). New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96014-2" title="Special:BookSources/978-0-387-96014-2"><bdi>978-0-387-96014-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Leonard+Euler%2C+Supreme+Geometer&rft.btitle=Leonard+Euler%2C+Elements+of+Algebra&rft.place=New+York&rft.edition=reprint+of+1840+5th&rft.pub=Springer-Verlag&rft.date=1984&rft.isbn=978-0-387-96014-2&rft.aulast=Truesdell&rft.aufirst=C.A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmkOhy6v7kIsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span> This Google books preview of <i>Elements of algebra</i> lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book:</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTruesdell2007" class="citation book cs1"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.A.</a> (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M4-zUnrSxNoC">"Leonard Euler, Supreme Geometer"</a>. In Dunham, William (ed.). <i>The Genius of Euler: reflections on his life and work</i>. Volume 2 of MAA tercentenary Euler celebration. New York: <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-558-4" title="Special:BookSources/978-0-88385-558-4"><bdi>978-0-88385-558-4</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Leonard+Euler%2C+Supreme+Geometer&rft.btitle=The+Genius+of+Euler%3A+reflections+on+his+life+and+work&rft.place=New+York&rft.series=Volume+2+of+MAA+tercentenary+Euler+celebration&rft.pub=Mathematical+Association+of+America&rft.date=2007&rft.isbn=978-0-88385-558-4&rft.aulast=Truesdell&rft.aufirst=C.A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM4-zUnrSxNoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaradarajan2006" class="citation book cs1">Varadarajan, V.S. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CYyKTREGYd0C"><i>Euler Through Time: A New Look at Old Themes</i></a>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3580-7" title="Special:BookSources/978-0-8218-3580-7"><bdi>978-0-8218-3580-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler+Through+Time%3A+A+New+Look+at+Old+Themes&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-0-8218-3580-7&rft.aulast=Varadarajan&rft.aufirst=V.S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCYyKTREGYd0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVardi1998" class="citation journal cs1">Vardi, Ilan (April 1998). <a rel="nofollow" class="external text" href="https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf">"Archimedes' Cattle Problem"</a> <span class="cs1-format">(PDF)</span>. <i>American Mathematical Monthly</i>. <b>105</b> (4): 305–319. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.383.545">10.1.1.383.545</a></span>. <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%2F2589706">10.2307/2589706</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/2589706">2589706</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120715031904/https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2012-07-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-04-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Archimedes%27+Cattle+Problem&rft.volume=105&rft.issue=4&rft.pages=305-319&rft.date=1998-04&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.383.545%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589706%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2589706&rft.aulast=Vardi&rft.aufirst=Ilan&rft_id=https%3A%2F%2Fwww.cs.drexel.edu%2F~crorres%2FArchimedes%2FCattle%2Fcattle_vardi.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Waerden1961" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, Bartel L.</a>; Dresden, Arnold (trans) (1961). <i>Science Awakening</i>. Vol. 1 or 2. New York: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+Awakening&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1961&rft.aulast=van+der+Waerden&rft.aufirst=Bartel+L.&rft.au=Dresden%2C+Arnold+%28trans%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeil1984" class="citation book cs1"><a href="/wiki/Andr%C3%A9_Weil" title="André Weil">Weil, André</a> (1984). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XSV0hDFj3loC"><i>Number Theory: an Approach Through History – from Hammurapi to Legendre</i></a>. Boston: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3141-3" title="Special:BookSources/978-0-8176-3141-3"><bdi>978-0-8176-3141-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory%3A+an+Approach+Through+History+%E2%80%93+from+Hammurapi+to+Legendre&rft.place=Boston&rft.pub=Birkh%C3%A4user&rft.date=1984&rft.isbn=978-0-8176-3141-3&rft.aulast=Weil&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXSV0hDFj3loC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li></ul> </div> <ul><li><i>This article incorporates material from the <a href="/wiki/Citizendium" title="Citizendium">Citizendium</a> article "<a href="https://en.citizendium.org/wiki/Number_theory" class="extiw" title="citizendium:Number theory">Number theory</a>", which is licensed under the <a href="/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License" class="mw-redirect" title="Wikipedia:Text of Creative Commons Attribution-ShareAlike 3.0 Unported License">Creative Commons Attribution-ShareAlike 3.0 Unported License</a> but not under the <a href="/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License" title="Wikipedia:Text of the GNU Free Documentation License">GFDL</a>.</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Number_theory&action=edit&section=29" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two of the most popular introductions to the subject are: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFG.H._HardyE.M._Wright2008" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">G.H. Hardy</a>; E.M. Wright (2008) [1938]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rey9wfSaJ9EC"><i>An introduction to the theory of numbers</i></a> (rev. by D.R. Heath-Brown and J.H. Silverman, 6th ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921986-5" title="Special:BookSources/978-0-19-921986-5"><bdi>978-0-19-921986-5</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+numbers&rft.edition=rev.+by+D.R.+Heath-Brown+and+J.H.+Silverman%2C+6th&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-921986-5&rft.au=G.H.+Hardy&rft.au=E.M.+Wright&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Drey9wfSaJ9EC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinogradov2003" class="citation book cs1"><a href="/wiki/Ivan_Matveyevich_Vinogradov" class="mw-redirect" title="Ivan Matveyevich Vinogradov">Vinogradov, I.M.</a> (2003) [1954]. <i>Elements of Number Theory</i> (reprint of the 1954 ed.). Mineola, NY: Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Number+Theory&rft.place=Mineola%2C+NY&rft.edition=reprint+of+the+1954&rft.pub=Dover+Publications&rft.date=2003&rft.aulast=Vinogradov&rft.aufirst=I.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li></ul> <p>Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (<a href="#CITEREFApostol1981">Apostol 1981</a>). Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvan_M._NivenHerbert_S._ZuckermanHugh_L._Montgomery2008" class="citation book cs1"><a href="/wiki/Ivan_M._Niven" title="Ivan M. Niven">Ivan M. Niven</a>; Herbert S. Zuckerman; <a href="/wiki/Hugh_L._Montgomery" class="mw-redirect" title="Hugh L. Montgomery">Hugh L. Montgomery</a> (2008) [1960]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V52HIcKguJ4C"><i>An introduction to the theory of numbers</i></a> (reprint of the 5th 1991 ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-265-1811-1" title="Special:BookSources/978-81-265-1811-1"><bdi>978-81-265-1811-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+numbers&rft.edition=reprint+of+the+5th+1991&rft.pub=John+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-81-265-1811-1&rft.au=Ivan+M.+Niven&rft.au=Herbert+S.+Zuckerman&rft.au=Hugh+L.+Montgomery&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DV52HIcKguJ4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenneth_H._Rosen2010" class="citation book cs1">Kenneth H. Rosen (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JqycRAAACAAJ"><i>Elementary Number Theory</i></a> (6th ed.). <a href="/wiki/Pearson_Education" title="Pearson Education">Pearson Education</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-71775-7" title="Special:BookSources/978-0-321-71775-7"><bdi>978-0-321-71775-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory&rft.edition=6th&rft.pub=Pearson+Education&rft.date=2010&rft.isbn=978-0-321-71775-7&rft.au=Kenneth+H.+Rosen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJqycRAAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li></ul> <p>Popular choices for a second textbook include: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorevichShafarevich1966" class="citation book cs1"><a href="/wiki/Borevich" class="mw-redirect" title="Borevich">Borevich, A. I.</a>; <a href="/wiki/Igor_Shafarevich" title="Igor Shafarevich">Shafarevich, Igor R.</a> (1966). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=njgVUjjO-EAC"><i>Number theory</i></a>. Pure and Applied Mathematics. Vol. 20. Boston, MA: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-117850-5" title="Special:BookSources/978-0-12-117850-5"><bdi>978-0-12-117850-5</bdi></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=0195803">0195803</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&rft.place=Boston%2C+MA&rft.series=Pure+and+Applied+Mathematics&rft.pub=Academic+Press&rft.date=1966&rft.isbn=978-0-12-117850-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0195803%23id-name%3DMR&rft.aulast=Borevich&rft.aufirst=A.+I.&rft.au=Shafarevich%2C+Igor+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnjgVUjjO-EAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1996) [1973]. <a rel="nofollow" class="external text" href="https://archive.org/details/courseinarithmet00serr"><i>A course in arithmetic</i></a>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 7. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90040-7" title="Special:BookSources/978-0-387-90040-7"><bdi>978-0-387-90040-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+course+in+arithmetic&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-90040-7&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcourseinarithmet00serr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber+theory" 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=Number_theory&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> 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template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_theory" title="Template talk:Number theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_theory" title="Special:EditPage/Template:Number theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_theory" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> (<a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>, <a href="/wiki/Non-abelian_class_field_theory" title="Non-abelian class field theory">non-abelian class field theory</a>, <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>, <a href="/wiki/Iwasawa%E2%80%93Tate_theory" class="mw-redirect" title="Iwasawa–Tate theory">Iwasawa–Tate theory</a>, <a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a>)</li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a> (<a href="/wiki/L-function" title="L-function">analytic theory of L-functions</a>, <a href="/wiki/Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a>)</li> <li><a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">Geometric number theory</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a> (<a href="/wiki/Arakelov_theory" title="Arakelov theory">Arakelov theory</a>, <a href="/wiki/Hodge%E2%80%93Arakelov_theory" title="Hodge–Arakelov theory">Hodge–Arakelov theory</a>)</li> <li><a href="/wiki/Arithmetic_combinatorics" title="Arithmetic combinatorics">Arithmetic combinatorics</a> (<a href="/wiki/Additive_number_theory" title="Additive number theory">additive number theory</a>)</li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic geometry</a> (<a href="/wiki/Anabelian_geometry" title="Anabelian geometry">anabelian geometry</a>, <a href="/wiki/P-adic_Hodge_theory" title="P-adic Hodge theory">P-adic Hodge theory</a>)</li> <li><a href="/wiki/Arithmetic_topology" title="Arithmetic topology">Arithmetic topology</a></li> <li><a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Number" title="Number">Numbers</a></li> <li><a href="/wiki/0" title="0">0</a></li> <li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></li> <li><a href="/wiki/1" title="1">Unity</a></li> <li><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number">P-adic numbers</a> (<a href="/wiki/P-adic_analysis" title="P-adic analysis">P-adic analysis</a>)</li> <li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></li> <li><a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li> <li><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Advanced concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quadratic_form" title="Quadratic form">Quadratic forms</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular forms</a></li> <li><a href="/wiki/L-function" title="L-function">L-functions</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a></li> <li><a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a></li> <li><a href="/wiki/Irrationality_measure" title="Irrationality measure">Irrationality measure</a></li> <li><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, 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src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_recreational_number_theory_topics" title="List of recreational number theory topics">List of recreational topics</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo.svg" class="mw-file-description" title="Wikibooks page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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data-file-height="512" /></a></span> <a href="https://en.wikiversity.org/wiki/Number_Theory" class="extiw" title="wikiversity:Number Theory">Wikiversity</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Number_theory_tables" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_theory_tables" 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href="/wiki/List_of_Mersenne_primes_and_perfect_numbers" title="List of Mersenne primes and perfect numbers">Mersenne primes and perfect numbers</a></li> <li><a href="/wiki/Table_of_Gaussian_integer_factorizations" title="Table of Gaussian integer factorizations">Gaussian integer factorizations</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td 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href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Network_architecture" title="Network architecture">Network architecture</a></li> <li><a href="/wiki/Network_protocol" class="mw-redirect" title="Network protocol">Network protocol</a></li> <li><a href="/wiki/Networking_hardware" title="Networking hardware">Network components</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Network_performance" title="Network performance">Network performance evaluation</a></li> <li><a href="/wiki/Network_service" title="Network service">Network service</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a href="/wiki/Integrated_development_environment" title="Integrated development environment">Integrated development environment</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration management">Software configuration management</a></li> <li><a href="/wiki/Library_(computing)" title="Library (computing)">Software library</a></li> <li><a href="/wiki/Software_repository" title="Software repository">Software repository</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software_development" title="Software development">Software development</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Control_variable_(programming)" class="mw-redirect" title="Control variable (programming)">Control variable</a></li> <li><a href="/wiki/Software_development_process" title="Software development process">Software development process</a></li> <li><a href="/wiki/Requirements_analysis" title="Requirements analysis">Requirements analysis</a></li> <li><a href="/wiki/Software_design" title="Software design">Software design</a></li> <li><a href="/wiki/Software_construction" title="Software construction">Software construction</a></li> <li><a href="/wiki/Software_deployment" title="Software deployment">Software deployment</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">Software maintenance</a></li> <li><a href="/wiki/Programming_team" title="Programming team">Programming team</a></li> <li><a href="/wiki/Open-source_software" title="Open-source software">Open-source model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Model_of_computation" title="Model of computation">Model of computation</a> <ul><li><a href="/wiki/Stochastic_computing" title="Stochastic computing">Stochastic</a></li></ul></li> <li><a href="/wiki/Formal_language" title="Formal language">Formal language</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic</a></li> <li><a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">Semantics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algorithm" title="Algorithm">Algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed reality</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual 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