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class="vector-toc-list"> <li id="toc-Primality_of_one" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primality_of_one"> <div class="vector-toc-text"> 2.1 Primality of one </div> </a> <ul id="toc-Primality_of_one-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elementary_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elementary_properties"> <div class="vector-toc-text"> 3 Elementary properties </div> </a> <button aria-controls="toc-Elementary_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Elementary properties subsection </button> <ul id="toc-Elementary_properties-sublist" class="vector-toc-list"> <li id="toc-Unique_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unique_factorization"> <div class="vector-toc-text"> 3.1 Unique factorization </div> </a> <ul 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href="#Analytic_properties"> <div class="vector-toc-text"> 4 Analytic properties </div> </a> <button aria-controls="toc-Analytic_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Analytic properties subsection </button> <ul id="toc-Analytic_properties-sublist" class="vector-toc-list"> <li id="toc-Analytical_proof_of_Euclid's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytical_proof_of_Euclid's_theorem"> <div class="vector-toc-text"> 4.1 Analytical proof of Euclid's theorem </div> </a> <ul id="toc-Analytical_proof_of_Euclid's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_of_primes_below_a_given_bound" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_of_primes_below_a_given_bound"> <div class="vector-toc-text"> 4.2 Number of primes below a given bound </div> </a> <ul id="toc-Number_of_primes_below_a_given_bound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_progressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_progressions"> <div class="vector-toc-text"> 4.3 Arithmetic progressions </div> </a> <ul id="toc-Arithmetic_progressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_values_of_quadratic_polynomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_values_of_quadratic_polynomials"> <div class="vector-toc-text"> 4.4 Prime values of quadratic polynomials </div> </a> <ul id="toc-Prime_values_of_quadratic_polynomials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zeta_function_and_the_Riemann_hypothesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zeta_function_and_the_Riemann_hypothesis"> <div class="vector-toc-text"> 4.5 Zeta function and the Riemann hypothesis </div> </a> <ul id="toc-Zeta_function_and_the_Riemann_hypothesis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Abstract_algebra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Abstract_algebra"> <div class="vector-toc-text"> 5 Abstract algebra </div> </a> <button aria-controls="toc-Abstract_algebra-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Abstract algebra subsection </button> <ul id="toc-Abstract_algebra-sublist" class="vector-toc-list"> <li id="toc-Modular_arithmetic_and_finite_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modular_arithmetic_and_finite_fields"> <div class="vector-toc-text"> 5.1 Modular arithmetic and finite fields </div> </a> <ul id="toc-Modular_arithmetic_and_finite_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p-adic_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#p-adic_numbers"> <div class="vector-toc-text"> 5.2 p-adic numbers </div> </a> <ul id="toc-p-adic_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_elements_in_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_elements_in_rings"> <div class="vector-toc-text"> 5.3 Prime elements in rings </div> </a> <ul id="toc-Prime_elements_in_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_ideals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_ideals"> <div class="vector-toc-text"> 5.4 Prime ideals </div> </a> <ul id="toc-Prime_ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_theory"> <div class="vector-toc-text"> 5.5 Group theory </div> </a> <ul id="toc-Group_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computational_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computational_methods"> <div class="vector-toc-text"> 6 Computational methods </div> </a> <button aria-controls="toc-Computational_methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Computational methods subsection </button> <ul id="toc-Computational_methods-sublist" class="vector-toc-list"> <li id="toc-Trial_division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trial_division"> <div class="vector-toc-text"> 6.1 Trial division </div> </a> <ul id="toc-Trial_division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sieves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sieves"> <div class="vector-toc-text"> 6.2 Sieves </div> </a> <ul id="toc-Sieves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primality_testing_versus_primality_proving" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primality_testing_versus_primality_proving"> <div class="vector-toc-text"> 6.3 Primality testing versus primality proving </div> </a> <ul id="toc-Primality_testing_versus_primality_proving-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special-purpose_algorithms_and_the_largest_known_prime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special-purpose_algorithms_and_the_largest_known_prime"> <div class="vector-toc-text"> 6.4 Special-purpose algorithms and the largest known prime </div> </a> <ul id="toc-Special-purpose_algorithms_and_the_largest_known_prime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integer_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integer_factorization"> <div class="vector-toc-text"> 6.5 Integer factorization </div> </a> <ul id="toc-Integer_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_computational_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_computational_applications"> <div class="vector-toc-text"> 6.6 Other computational applications </div> </a> <ul id="toc-Other_computational_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_applications"> <div class="vector-toc-text"> 7 Other applications </div> </a> <button aria-controls="toc-Other_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other applications subsection </button> <ul id="toc-Other_applications-sublist" class="vector-toc-list"> <li id="toc-Constructible_polygons_and_polygon_partitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constructible_polygons_and_polygon_partitions"> <div class="vector-toc-text"> 7.1 Constructible polygons and polygon partitions </div> </a> <ul id="toc-Constructible_polygons_and_polygon_partitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 7.2 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> 7.3 Biology </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arts_and_literature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arts_and_literature"> <div class="vector-toc-text"> 7.4 Arts and literature </div> </a> <ul id="toc-Arts_and_literature-sublist" class="vector-toc-list"> </ul> </li> </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"> 8 Notes </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"> 9 References </div> </a> <ul id="toc-References-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"> 10 External links </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle External links subsection </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Generators_and_calculators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generators_and_calculators"> <div class="vector-toc-text"> 10.1 Generators and calculators </div> </a> <ul id="toc-Generators_and_calculators-sublist" class="vector-toc-list"> </ul> </li> </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" > Toggle the table of contents </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">Prime number</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 138 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-138" aria-hidden="true" > 138 languages </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/Priemgetal" title="Priemgetal – Afrikaans" lang="af" hreflang="af" data-title="Priemgetal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Primzahl" title="Primzahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Primzahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/Frumt%C3%A6l" title="Frumtæl – Old English" lang="ang" hreflang="ang" data-title="Frumtæl" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A3%D9%88%D9%84%D9%8A" title="عدد أولي – Arabic" lang="ar" hreflang="ar" data-title="عدد أولي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_primero" title="Numero primero – Aragonese" lang="an" hreflang="an" data-title="Numero primero" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%8A%D5%A1%D6%80%D5%A6_%D5%A9%D5%AB%D6%82" title="Պարզ թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Պարզ թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="মৌলিক সংখ্যা – Assamese" lang="as" hreflang="as" data-title="মৌলিক সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_primu" title="Númberu primu – Asturian" lang="ast" hreflang="ast" data-title="Númberu primu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sad%C9%99_%C9%99d%C9%99d" title="Sadə ədəd – Azerbaijani" lang="az" hreflang="az" data-title="Sadə ədəd" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B3%D8%A7%D8%AF%D9%87_%D8%B9%D8%AF%D8%AF" title="ساده عدد – South Azerbaijani" lang="azb" hreflang="azb" data-title="ساده عدد" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="মৌলিক সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="মৌলিক সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</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-s%C3%B2%CD%98" title="Sò͘-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AF%D0%B1%D0%B0%D0%B9_%D2%BB%D0%B0%D0%BD" title="Ябай һан – Bashkir" lang="ba" hreflang="ba" data-title="Ябай һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%8B_%D0%BB%D1%96%D0%BA" title="Просты лік – Belarusian" lang="be" hreflang="be" data-title="Просты лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%8B_%D0%BB%D1%96%D0%BA" title="Просты лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Просты лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Просто число – Bulgarian" lang="bg" hreflang="bg" data-title="Просто число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prost_broj" title="Prost broj – Bosnian" lang="bs" hreflang="bs" data-title="Prost broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_kentael" title="Niver kentael – Breton" lang="br" hreflang="br" data-title="Niver kentael" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_primer" title="Nombre primer – Catalan" lang="ca" hreflang="ca" data-title="Nombre primer" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%BD%D1%81%D0%B0%D1%82_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Ансат хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Ансат хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo" title="Prvočíslo – Czech" lang="cs" hreflang="cs" data-title="Prvočíslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cysefin" title="Rhif cysefin – Welsh" lang="cy" hreflang="cy" data-title="Rhif cysefin" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Primtal" title="Primtal – Danish" lang="da" hreflang="da" data-title="Primtal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%B9%D8%A7%D8%AF%D8%A7%D8%AF_%D9%84%D9%88%D9%84%D9%8A" title="عاداد لولي – Moroccan Arabic" lang="ary" hreflang="ary" data-title="عاداد لولي" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Primzahl" title="Primzahl – German" lang="de" hreflang="de" data-title="Primzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Algarv" title="Algarv – Estonian" lang="et" hreflang="et" data-title="Algarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CF%8E%CF%84%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πρώτος αριθμός – Greek" lang="el" hreflang="el" data-title="Πρώτος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_prim" title="Nùmer prim – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer prim" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_primo" title="Número primo – Spanish" lang="es" hreflang="es" data-title="Número primo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Primo" title="Primo – Esperanto" lang="eo" hreflang="eo" data-title="Primo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_lehen" title="Zenbaki lehen – Basque" lang="eu" hreflang="eu" data-title="Zenbaki lehen" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A7%D9%88%D9%84" title="عدد اول – Persian" lang="fa" hreflang="fa" data-title="عدد اول" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Primtal" title="Primtal – Faroese" lang="fo" hreflang="fo" data-title="Primtal" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_premier" title="Nombre premier – French" lang="fr" hreflang="fr" data-title="Nombre premier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_phr%C3%ADomha" title="Uimhir phríomha – Irish" lang="ga" hreflang="ga" data-title="Uimhir phríomha" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_primo" title="Número primo – Galician" lang="gl" hreflang="gl" data-title="Número primo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%B3%AA%E6%95%B8" title="質數 – Gan" lang="gan" hreflang="gan" data-title="質數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%85%E0%AA%B5%E0%AA%BF%E0%AA%AD%E0%AA%BE%E0%AA%9C%E0%AB%8D%E0%AA%AF_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE" title="અવિભાજ્ય સંખ્યા – Gujarati" lang="gu" hreflang="gu" data-title="અવિભાજ્ય સંખ્યા" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%AD%D0%BA%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Экн тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Экн тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론) – Korean" lang="ko" hreflang="ko" data-title="소수 (수론)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Helu_kumu" title="Helu kumu – Hawaiian" lang="haw" hreflang="haw" data-title="Helu kumu" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%A1%D6%80%D5%A6_%D5%A9%D5%AB%D5%BE" title="Պարզ թիվ – Armenian" lang="hy" hreflang="hy" data-title="Պարզ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BE%E0%A4%9C%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अभाज्य संख्या – Hindi" lang="hi" hreflang="hi" data-title="अभाज्य संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Primowa_li%C4%8Dba" title="Primowa ličba – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Primowa ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prosti_broj" title="Prosti broj – Croatian" lang="hr" hreflang="hr" data-title="Prosti broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_prima" title="Bilangan prima – Indonesian" lang="id" hreflang="id" data-title="Bilangan prima" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_prime" title="Numero prime – Interlingua" lang="ia" hreflang="ia" data-title="Numero prime" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Frumtala" title="Frumtala – Icelandic" lang="is" hreflang="is" data-title="Frumtala" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://it.wikipedia.org/wiki/Numero_primo" title="Numero primo – Italian" lang="it" hreflang="it" data-title="Numero primo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99" title="מספר ראשוני – Hebrew" lang="he" hreflang="he" data-title="מספר ראשוני" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Wilangan_prima" title="Wilangan prima – Javanese" lang="jv" hreflang="jv" data-title="Wilangan prima" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%B5%E0%B2%BF%E0%B2%AD%E0%B2%BE%E0%B2%9C%E0%B3%8D%E0%B2%AF_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಅವಿಭಾಜ್ಯ ಸಂಖ್ಯೆ – Kannada" lang="kn" hreflang="kn" data-title="ಅವಿಭಾಜ್ಯ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A0%E1%83%A2%E1%83%98%E1%83%95%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="მარტივი რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="მარტივი რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%B9_%D1%81%D0%B0%D0%BD" title="Жай сан – Kazakh" lang="kk" hreflang="kk" data-title="Жай сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Pennriv" title="Pennriv – Cornish" lang="kw" hreflang="kw" data-title="Pennriv" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_tasa" title="Namba tasa – Swahili" lang="sw" hreflang="sw" data-title="Namba tasa" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Nonm_premye" title="Nonm premye – Haitian Creole" lang="ht" hreflang="ht" data-title="Nonm premye" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_pr%C3%A9my%C3%A9" title="Nonm prémyé – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm prémyé" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_h%C3%AEm%C3%AE" title="Hejmarên hîmî – Kurdish" lang="ku" hreflang="ku" data-title="Hejmarên hîmî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%96%D3%A9%D0%BD%D3%A9%D0%BA%D3%A9%D0%B9_%D1%81%D0%B0%D0%BD" title="Жөнөкөй сан – Kyrgyz" lang="ky" hreflang="ky" data-title="Жөнөкөй сан" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%95%E0%BA%BB%E0%BA%A7%E0%BB%80%E0%BA%A5%E0%BA%81%E0%BA%AB%E0%BA%BC%E0%BA%B1%E0%BA%81" title="ຕົວເລກຫຼັກ – Lao" lang="lo" hreflang="lo" data-title="ຕົວເລກຫຼັກ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_primus" title="Numerus primus – Latin" lang="la" hreflang="la" data-title="Numerus primus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Pirmskaitlis" title="Pirmskaitlis – Latvian" lang="lv" hreflang="lv" data-title="Pirmskaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Primzuel" title="Primzuel – Luxembourgish" lang="lb" hreflang="lb" data-title="Primzuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pirminis_skai%C4%8Dius" title="Pirminis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Pirminis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Priemgetaal" title="Priemgetaal – Limburgish" lang="li" hreflang="li" data-title="Priemgetaal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/nalfendi_kacna%27u" title="nalfendi kacna'u – Lojban" lang="jbo" hreflang="jbo" data-title="nalfendi kacna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://lmo.wikipedia.org/wiki/Numer_primm" title="Numer primm – Lombard" lang="lmo" hreflang="lmo" data-title="Numer primm" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pr%C3%ADmsz%C3%A1mok" title="Prímszámok – Hungarian" lang="hu" hreflang="hu" data-title="Prímszámok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82_%D0%B1%D1%80%D0%BE%D1%98" title="Прост број – Macedonian" lang="mk" hreflang="mk" data-title="Прост број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsy_azo_tsinjaraina" title="Isa tsy azo tsinjaraina – Malagasy" lang="mg" hreflang="mg" data-title="Isa tsy azo tsinjaraina" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%AD%E0%B4%BE%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="അഭാജ്യസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="അഭാജ്യസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Numru_l-ewwel" title="Numru l-ewwel – Maltese" lang="mt" hreflang="mt" data-title="Numru l-ewwel" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AE%E0%A5%82%E0%A4%B3_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="मूळ संख्या – Marathi" lang="mr" hreflang="mr" data-title="मूळ संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A7%D9%88%D9%84%D9%89" title="عدد اولى – Egyptian Arabic" lang="arz" hreflang="arz" data-title="عدد اولى" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_perdana" title="Nombor perdana – Malay" lang="ms" hreflang="ms" data-title="Nombor perdana" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BD%D1%85%D0%BD%D1%8B_%D1%82%D0%BE%D0%BE" title="Анхны тоо – Mongolian" lang="mn" hreflang="mn" data-title="Анхны тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%AF%E1%80%92%E1%80%B9%E1%80%93%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="သုဒ္ဓကိန်း – Burmese" lang="my" hreflang="my" data-title="သုဒ္ဓကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Naba_taumada" title="Naba taumada – Fijian" lang="fj" hreflang="fj" data-title="Naba taumada" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Priemgetal" title="Priemgetal – Dutch" lang="nl" hreflang="nl" data-title="Priemgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%87%E0%A4%AE_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्राइम सङ्ख्या – Nepali" lang="ne" hreflang="ne" data-title="प्राइम सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B4%A0%E6%95%B0" title="素数 – Japanese" lang="ja" hreflang="ja" data-title="素数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Primtaal" title="Primtaal – Northern Frisian" lang="frr" hreflang="frr" data-title="Primtaal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Primtall" title="Primtall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Primtall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Primtal" title="Primtal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Primtal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_primi%C3%A8r" title="Nombre primièr – Occitan" lang="oc" hreflang="oc" data-title="Nombre primièr" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AE%E0%AD%8C%E0%AC%B3%E0%AC%BF%E0%AC%95_%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE" title="ମୌଳିକ ସଂଖ୍ୟା – Odia" lang="or" hreflang="or" data-title="ମୌଳିକ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tub_son" title="Tub son – Uzbek" lang="uz" hreflang="uz" data-title="Tub son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%AD%E0%A8%BE%E0%A8%9C_%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86" title="ਅਭਾਜ ਸੰਖਿਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਭਾਜ ਸੰਖਿਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%BE%D8%B1%D8%A7%D8%A6%D9%85_%D9%86%D9%85%D8%A8%D8%B1" title="پرائم نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="پرائم نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Praim_nomba" title="Praim nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Praim nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%94%E1%9E%8B%E1%9E%98" title="ចំនួនបឋម – Khmer" lang="km" hreflang="km" data-title="ចំនួនបឋម" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_prim" title="Nùmer prim – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer prim" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Primtall" title="Primtall – Low German" lang="nds" hreflang="nds" data-title="Primtall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_pierwsze" title="Liczby pierwsze – Polish" lang="pl" hreflang="pl" data-title="Liczby pierwsze" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_primo" title="Número primo – Portuguese" lang="pt" hreflang="pt" data-title="Número primo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_prim" title="Număr prim – Romanian" lang="ro" hreflang="ro" data-title="Număr prim" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Простое число – Russian" lang="ru" hreflang="ru" data-title="Простое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Простое число – Yakut" lang="sah" hreflang="sah" data-title="Простое число" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri_i_thjesht%C3%AB" title="Numri i thjeshtë – Albanian" lang="sq" hreflang="sq" data-title="Numri i thjeshtë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_primu" title="Nùmmuru primu – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru primu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%AE%E0%B6%B8%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="ප්‍රථමක සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="ප්‍රථමක සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Prime_number" title="Prime number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Prime number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo" title="Prvočíslo – Slovak" lang="sk" hreflang="sk" data-title="Prvočíslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pra%C5%A1tevilo" title="Praštevilo – Slovenian" lang="sl" hreflang="sl" data-title="Praštevilo" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Pjyrszo_n%C5%AFmera" title="Pjyrszo nůmera – Silesian" lang="szl" hreflang="szl" data-title="Pjyrszo nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_mutuxan" title="Thiin mutuxan – Somali" lang="so" hreflang="so" data-title="Thiin mutuxan" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%B3%DB%95%D8%B1%DB%95%D8%AA%D8%A7%DB%8C%DB%8C" title="ژمارەی سەرەتایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی سەرەتایی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82_%D0%B1%D1%80%D0%BE%D1%98" title="Прост број – Serbian" lang="sr" hreflang="sr" data-title="Прост број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prost_broj" title="Prost broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Prost broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Alkuluku" title="Alkuluku – Finnish" lang="fi" hreflang="fi" data-title="Alkuluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Primtal" title="Primtal – Swedish" lang="sv" hreflang="sv" data-title="Primtal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pangunahing_bilang" title="Pangunahing bilang – Tagalog" lang="tl" hreflang="tl" data-title="Pangunahing bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="பகா எண் – Tamil" lang="ta" hreflang="ta" data-title="பகா எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Am%E1%B8%8Dan_amnzu" title="Amḍan amnzu – Tachelhit" lang="shi" hreflang="shi" data-title="Amḍan amnzu" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B1%8D%E0%B0%B0%E0%B0%A7%E0%B0%BE%E0%B0%A8_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="ప్రధాన సంఖ్య – Telugu" lang="te" hreflang="te" data-title="ప్రధాన సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="จำนวนเฉพาะ – Thai" lang="th" hreflang="th" data-title="จำนวนเฉพาะ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D2%B3%D0%BE%D0%B8_%D1%81%D0%BE%D0%B4%D0%B0" title="Ададҳои сода – Tajik" lang="tg" hreflang="tg" data-title="Ададҳои сода" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Asal_say%C4%B1" title="Asal sayı – Turkish" lang="tr" hreflang="tr" data-title="Asal sayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Просте число – Ukrainian" lang="uk" hreflang="uk" data-title="Просте число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D9%81%D8%B1%D8%AF_%D8%B9%D8%AF%D8%AF" title="مفرد عدد – Urdu" lang="ur" hreflang="ur" data-title="مفرد عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%AA%DB%88%D9%BE_%D8%B3%D8%A7%D9%86" title="تۈپ سان – Uyghur" lang="ug" hreflang="ug" data-title="تۈپ سان" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target">ئۇيغۇرچە / Uyghurche</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/N%C3%B9maro_primo" title="Nùmaro primo – Venetian" lang="vec" hreflang="vec" data-title="Nùmaro primo" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Palatoi_lugu" title="Palatoi lugu – Veps" lang="vep" hreflang="vep" data-title="Palatoi lugu" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91" title="Số nguyên tố – Vietnamese" lang="vi" hreflang="vi" data-title="Số nguyên tố" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Algarv" title="Algarv – Võro" lang="vro" hreflang="vro" data-title="Algarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Nombe_prum%C3%AE" title="Nombe prumî – Walloon" lang="wa" hreflang="wa" data-title="Nombe prumî" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%B3%AA%E6%95%B8" title="質數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="質數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Priemgetal" title="Priemgetal – West Flemish" lang="vls" hreflang="vls" data-title="Priemgetal" data-language-autonym="West-Vlams" data-language-local-name="West 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Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> <div id="mw-indicator-pp-default" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected due to vandalism"><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Number divisible only by 1 or itself</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">"Prime" redirects here. For other uses, see <a href="/wiki/Prime_(disambiguation)" class="mw-disambig" title="Prime (disambiguation)">Prime (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Primes-vs-composites.svg" class="mw-file-description"><img alt="Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Primes-vs-composites.svg/220px-Primes-vs-composites.svg.png" decoding="async" width="220" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Primes-vs-composites.svg/330px-Primes-vs-composites.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Primes-vs-composites.svg/440px-Primes-vs-composites.svg.png 2x" data-file-width="468" data-file-height="658" /></a><figcaption><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> can be arranged into <a href="/wiki/Rectangle" title="Rectangle">rectangles</a> but prime numbers cannot.</figcaption></figure> A prime number (or a prime) is a <a href="/wiki/Natural_number" title="Natural number">natural number</a> greater than 1 that is not a <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of two smaller natural numbers. A natural number greater than 1 that is not prime is called a <a href="/wiki/Composite_number" title="Composite number">composite number</a>. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in <a href="/wiki/Number_theory" title="Number theory">number theory</a> because of the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>: every natural number greater than 1 is either a prime itself or can be <a href="/wiki/Factorization" title="Factorization">factorized</a> as a product of primes that is unique <a href="/wiki/Up_to" title="Up to">up to</a> their order. The property of being prime is called primality. A simple but slow <a href="/wiki/Primality_test" title="Primality test">method of checking the primality</a> of a given number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, called <a href="/wiki/Trial_division" title="Trial division">trial division</a>, tests whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is a multiple of any integer between 2 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}">. Faster algorithms include the <a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin primality test</a>, which is fast but has a small chance of error, and the <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a>, which always produces the correct answer in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne numbers</a>. As of October 2024<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Prime_number&action=edit">[update]</a> the <a href="/wiki/Largest_known_prime_number" title="Largest known prime number">largest known prime number</a> is a Mersenne prime with 41,024,320 <a href="/wiki/Numerical_digit" title="Numerical digit">decimal digits</a>.<a href="#cite_note-GIMPS-2024-1">[1]</a><a href="#cite_note-2">[2]</a> There are <a href="/wiki/Infinite_set" title="Infinite set">infinitely many</a> primes, as <a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">demonstrated by Euclid</a> around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>, proven at the end of the 19th century, which says that the <a href="/wiki/Probability" title="Probability">probability</a> of a randomly chosen large number being prime is inversely <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to its number of digits, that is, to its <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>. Several historical questions regarding prime numbers are still unsolved. These include <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a>, that every even integer greater than 2 can be expressed as the sum of two primes, and the <a href="/wiki/Twin_prime" title="Twin prime">twin prime</a> conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic</a> or <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic</a> aspects of numbers. Primes are used in several routines in <a href="/wiki/Information_technology" title="Information technology">information technology</a>, such as <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>, which relies on the difficulty of <a href="/wiki/Integer_factorization" title="Integer factorization">factoring</a> large numbers into their prime factors. In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, objects that behave in a generalized way like prime numbers include <a href="/wiki/Prime_element" title="Prime element">prime elements</a> and <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_examples">Definition and examples</h2></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/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div> A <a href="/wiki/Natural_number" title="Natural number">natural number</a> (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called <a href="/wiki/Composite_number" title="Composite number">composite numbers</a>.<a href="#cite_note-3">[3]</a> In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is prime if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> items cannot be divided up into smaller equal-size groups of more than one item,<a href="#cite_note-4">[4]</a> or if it is not possible to arrange <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> dots into a rectangular grid that is more than one dot wide and more than one dot high.<a href="#cite_note-5">[5]</a> For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,<a href="#cite_note-6">[6]</a> as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Prime_number_Cuisenaire_rods_7.png" class="mw-file-description"><img alt="refer to caption" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Prime_number_Cuisenaire_rods_7.png/260px-Prime_number_Cuisenaire_rods_7.png" decoding="async" width="260" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Prime_number_Cuisenaire_rods_7.png/390px-Prime_number_Cuisenaire_rods_7.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Prime_number_Cuisenaire_rods_7.png/520px-Prime_number_Cuisenaire_rods_7.png 2x" data-file-width="580" data-file-height="388" /></a><figcaption>Demonstration, with <a href="/wiki/Cuisenaire_rods" title="Cuisenaire rods">Cuisenaire rods</a>, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly</figcaption></figure> The <a href="/wiki/Divisor" title="Divisor">divisors</a> of a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> are the natural numbers that divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive <a href="/wiki/Divisor" title="Divisor">divisors</a>. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition.<a href="#cite_note-7">[7]</a> Yet another way to express the same thing is that a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is prime if it is greater than one and if none of the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,3,\dots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,3,\dots ,n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc661ca19f3a62fb3fcf54e9e2ba22bfae50edda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.935ex; height:2.509ex;" alt="{\displaystyle 2,3,\dots ,n-1}"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> evenly.<a href="#cite_note-8">[8]</a> The first 25 prime numbers (all the prime numbers less than 100) are:<a href="#cite_note-ziegler-9">[9]</a> <dl><dd><a href="/wiki/2" title="2">2</a>, <a href="/wiki/3" title="3">3</a>, <a href="/wiki/5" title="5">5</a>, <a href="/wiki/7" title="7">7</a>, <a href="/wiki/11_(number)" title="11 (number)">11</a>, <a href="/wiki/13_(number)" title="13 (number)">13</a>, <a href="/wiki/17_(number)" title="17 (number)">17</a>, <a href="/wiki/19_(number)" title="19 (number)">19</a>, <a href="/wiki/23_(number)" title="23 (number)">23</a>, <a href="/wiki/29_(number)" title="29 (number)">29</a>, <a href="/wiki/31_(number)" title="31 (number)">31</a>, <a href="/wiki/37_(number)" title="37 (number)">37</a>, <a href="/wiki/41_(number)" title="41 (number)">41</a>, <a href="/wiki/43_(number)" title="43 (number)">43</a>, <a href="/wiki/47_(number)" title="47 (number)">47</a>, <a href="/wiki/53_(number)" title="53 (number)">53</a>, <a href="/wiki/59_(number)" title="59 (number)">59</a>, <a href="/wiki/61_(number)" title="61 (number)">61</a>, <a href="/wiki/67_(number)" title="67 (number)">67</a>, <a href="/wiki/71_(number)" title="71 (number)">71</a>, <a href="/wiki/73_(number)" title="73 (number)">73</a>, <a href="/wiki/79_(number)" title="79 (number)">79</a>, <a href="/wiki/83_(number)" title="83 (number)">83</a>, <a href="/wiki/89_(number)" title="89 (number)">89</a>, <a href="/wiki/97_(number)" title="97 (number)">97</a> (sequence <a href="//oeis.org/A000040" class="extiw" title="oeis:A000040">A000040</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> No <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> greater than 2 is prime because any such number can be expressed as the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times n/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times n/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b3d623bd1e97df55c447f8b187c7a62418064c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.722ex; height:2.843ex;" alt="{\displaystyle 2\times n/2}">. Therefore, every prime number other than 2 is an <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd number</a>, and is called an odd prime.<a href="#cite_note-10">[10]</a> Similarly, when written in the usual <a href="/wiki/Decimal" title="Decimal">decimal</a> system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.<a href="#cite_note-11">[11]</a> The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all primes is sometimes denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"> (a <a href="/wiki/Boldface" class="mw-redirect" title="Boldface">boldface</a> capital P)<a href="#cite_note-12">[12]</a> or by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"> (a <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> capital P).<a href="#cite_note-13">[13]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rhind_Mathematical_Papyrus.jpg" class="mw-file-description"><img alt="The Rhind Mathematical Papyrus" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/220px-Rhind_Mathematical_Papyrus.jpg" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/330px-Rhind_Mathematical_Papyrus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/440px-Rhind_Mathematical_Papyrus.jpg 2x" data-file-width="750" data-file-height="449" /></a><figcaption>The <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a></figcaption></figure> The <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>, from around 1550 BC, has <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</a> expansions of different forms for prime and composite numbers.<a href="#cite_note-14">[14]</a> However, the earliest surviving records of the study of prime numbers come from the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematicians</a>, who called them prōtos arithmòs (πρῶτος ἀριθμὸς). <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a> (c. 300 BC) proves the <a href="/wiki/Infinitude_of_primes" class="mw-redirect" title="Infinitude of primes">infinitude of primes</a> and the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, and shows how to construct a <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a> from a <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne prime</a>.<a href="#cite_note-stillwell-2010-p40-15">[15]</a> Another Greek invention, the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a>, is still used to construct lists of primes.<a href="#cite_note-pomerance-sciam-16">[16]</a><a href="#cite_note-mollin-17">[17]</a> Around 1000 AD, the <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Islamic</a> mathematician <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> (Alhazen) found <a href="/wiki/Wilson%27s_theorem" title="Wilson's theorem">Wilson's theorem</a>, characterizing the prime numbers as the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> that evenly divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)!+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5241d2022a4a3e94d24081a5cfb9b7b5fbf98b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.857ex; height:2.843ex;" alt="{\displaystyle (n-1)!+1}">. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.<a href="#cite_note-18">[18]</a> Another Islamic mathematician, <a href="/wiki/Ibn_al-Banna%27_al-Marrakushi" title="Ibn al-Banna' al-Marrakushi">Ibn al-Banna' al-Marrakushi</a>, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit.<a href="#cite_note-mollin-17">[17]</a> <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> took the innovations from Islamic mathematics to Europe. His book <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a> (1202) was the first to describe <a href="/wiki/Trial_division" title="Trial division">trial division</a> for testing primality, again using divisors only up to the square root.<a href="#cite_note-mollin-17">[17]</a> In 1640 <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> stated (without proof) <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> (later proved by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>).<a href="#cite_note-19">[19]</a> Fermat also investigated the primality of the <a href="/wiki/Fermat_number" title="Fermat number">Fermat numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27f57a4191be088259902a790ef2fb093ffb812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 2^{2^{n}}+1}">,<a href="#cite_note-20">[20]</a> and <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a> studied the <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne primes</a>, prime numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{p}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c5977dbf385ba719fbb90f67b0a3d91e1da6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.224ex; height:2.509ex;" alt="{\displaystyle 2^{p}-1}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> itself a prime.<a href="#cite_note-21">[21]</a> <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a> formulated <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a>, that every even number is the sum of two primes, in a 1742 letter to Euler.<a href="#cite_note-22">[22]</a> Euler proved Alhazen's conjecture (now the <a href="/wiki/Euclid%E2%80%93Euler_theorem" title="Euclid–Euler theorem">Euclid–Euler theorem</a>) that all even perfect numbers can be constructed from Mersenne primes.<a href="#cite_note-stillwell-2010-p40-15">[15]</a> He introduced methods from <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> to this area in his proofs of the infinitude of the primes and the <a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">divergence of the sum of the reciprocals of the primes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57e94d60efb5ad04a9ddebe807c98fff5d6d2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.038ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots }">.<a href="#cite_note-23">[23]</a> At the start of the 19th century, Legendre and Gauss conjectured that as <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><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}"> tends to infinity, the number of primes up to <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><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}"> is <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x/\log x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x/\log x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dcc3d89d9a592249b29d9ec90ec429423fc3db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.568ex; height:2.843ex;" alt="{\displaystyle x/\log x}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d453de713a8c45f7bf99108752531ed7d6dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.689ex; height:2.509ex;" alt="{\displaystyle \log x}"> is the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of <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><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}">. A weaker consequence of this high density of primes was <a href="/wiki/Bertrand%27s_postulate" title="Bertrand's postulate">Bertrand's postulate</a>, that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" 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>1}"> there is a prime between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}">, proved in 1852 by <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>.<a href="#cite_note-24">[24]</a> Ideas of <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> in his <a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">1859 paper on the zeta-function</a> sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> remains unproven, Riemann's outline was completed in 1896 by <a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Hadamard</a> and <a href="/wiki/Charles_Jean_de_la_Vall%C3%A9e-Poussin" class="mw-redirect" title="Charles Jean de la Vallée-Poussin">de la Vallée Poussin</a>, and the result is now known as the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>.<a href="#cite_note-25">[25]</a> Another important 19th century result was <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a>, that certain <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progressions</a> contain infinitely many primes.<a href="#cite_note-26">[26]</a> Many mathematicians have worked on <a href="/wiki/Primality_test" title="Primality test">primality tests</a> for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include <a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's test</a> for Fermat numbers (1877),<a href="#cite_note-27">[27]</a> <a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a> (c. 1878),<a href="#cite_note-28">[28]</a> the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer primality test</a> (originated 1856), and the generalized <a href="/wiki/Lucas_primality_test" title="Lucas primality test">Lucas primality test</a>.<a href="#cite_note-mollin-17">[17]</a> Since 1951 all the <a href="/wiki/Largest_known_prime" class="mw-redirect" title="Largest known prime">largest known primes</a> have been found using these tests on <a href="/wiki/Computer" title="Computer">computers</a>.<a href="#cite_note-30">[a]</a> The search for ever larger primes has generated interest outside mathematical circles, through the <a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a> and other <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> projects.<a href="#cite_note-ziegler-9">[9]</a><a href="#cite_note-31">[30]</a> The idea that prime numbers had few applications outside of <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a><a href="#cite_note-pure-34">[b]</a> was shattered in the 1970s when <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a> and the <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA</a> cryptosystem were invented, using prime numbers as their basis.<a href="#cite_note-ent-7-35">[33]</a> The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form.<a href="#cite_note-pomerance-sciam-16">[16]</a><a href="#cite_note-36">[34]</a><a href="#cite_note-37">[35]</a> The mathematical theory of prime numbers also moved forward with the <a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a> (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and <a href="/wiki/Yitang_Zhang" title="Yitang Zhang">Yitang Zhang</a>'s 2013 proof that there exist infinitely many <a href="/wiki/Prime_gap" title="Prime gap">prime gaps</a> of bounded size.<a href="#cite_note-neale-18-47-38">[36]</a> <div class="mw-heading mw-heading3"><h3 id="Primality_of_one">Primality of one</h3></div> Most early Greeks did not even consider 1 to be a number,<a href="#cite_note-crxk-34-39">[37]</a><a href="#cite_note-40">[38]</a> so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including <a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a>, <a href="/wiki/Iamblichus" title="Iamblichus">Iamblichus</a>, <a href="/wiki/Boethius" title="Boethius">Boethius</a>, and <a href="/wiki/Cassiodorus" title="Cassiodorus">Cassiodorus</a>, also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">medieval Islamic mathematicians</a> largely followed the Greeks in viewing 1 as not being a number.<a href="#cite_note-crxk-34-39">[37]</a> By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some of them included it as the first prime number.<a href="#cite_note-41">[39]</a> In the mid-18th century, <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a> listed 1 as prime in his correspondence with <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>;<a href="#cite_note-42">[40]</a> however, Euler himself did not consider 1 to be prime.<a href="#cite_note-43">[41]</a> Many 19th century mathematicians still considered 1 to be prime,<a href="#cite_note-cx-44">[42]</a> and <a href="/wiki/Derrick_Norman_Lehmer" title="Derrick Norman Lehmer">Derrick Norman Lehmer</a> included 1 in his list of primes less than ten million published in 1914.<a href="#cite_note-FOOTNOTEConwayGuy1996130-45">[43]</a> Lists of primes that included 1 continued to be published as recently as 1956.<a href="#cite_note-46">[44]</a><a href="#cite_note-cg-bon-129-130-47">[45]</a> However, around this time, by the early 20th century, mathematicians started to agree that 1 should not be classified as a prime number.<a href="#cite_note-cx-44">[42]</a> If 1 were to be considered a prime, many statements involving primes would need to be awkwardly reworded. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1.<a href="#cite_note-cx-44">[42]</a> Similarly, the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1.<a href="#cite_note-cg-bon-129-130-47">[45]</a> Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a> or for the <a href="/wiki/Sum-of-divisors_function" class="mw-redirect" title="Sum-of-divisors function">sum of divisors function</a> are different for prime numbers than they are for 1.<a href="#cite_note-48">[46]</a> By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "<a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>".<a href="#cite_note-cx-44">[42]</a> <div class="mw-heading mw-heading2"><h2 id="Elementary_properties">Elementary properties</h2></div> <div class="mw-heading mw-heading3"><h3 id="Unique_factorization">Unique factorization</h3></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/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></div> Writing a number as a product of prime numbers is called a prime factorization of the number. For example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}50&=2\times 5\times 5\\&=2\times 5^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>50</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}50&=2\times 5\times 5\\&=2\times 5^{2}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027ed6ef8162c47c25e9dc13c0aefa473d153a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.343ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}50&=2\times 5\times 5\\&=2\times 5^{2}.\end{aligned}}}"></dd></dl> The terms in the product are called prime factors. The same prime factor may occur more than once; this example has two copies of the prime factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62cfda82378d02d6ff65a09e66873314c7013888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 5.}"> When a prime occurs multiple times, <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac42c99ce3b9abb17b955f9995dd37ba270905f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 5^{2}}"> denotes the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> or second power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62cfda82378d02d6ff65a09e66873314c7013888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 5.}"> The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.<a href="#cite_note-49">[47]</a> This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ.<a href="#cite_note-50">[48]</a> So, although there are many different ways of finding a factorization using an <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers.<a href="#cite_note-51">[49]</a> Some proofs of the uniqueness of prime factorizations are based on <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is a prime number and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> divides a product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"> of integers <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><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}"> and <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> divides <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><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}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> divides <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><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}"> (or both).<a href="#cite_note-52">[50]</a> Conversely, if a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> has the property that when it divides a product it always divides at least one factor of the product, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> must be prime.<a href="#cite_note-53">[51]</a> <div class="mw-heading mw-heading3"><h3 id="Infinitude">Infinitude</h3></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/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a></div> There are <a href="/wiki/Infinitely" class="mw-redirect" title="Infinitely">infinitely</a> many prime numbers. Another way of saying this is that the sequence <dl><dd>2, 3, 5, 7, 11, 13, ...</dd></dl> of prime numbers never ends. This statement is referred to as Euclid's theorem in honor of the ancient Greek mathematician <a href="/wiki/Euclid" title="Euclid">Euclid</a>, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analytical</a> proof by <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>, <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Goldbach's</a> <a href="/wiki/Fermat_number#Basic_properties" title="Fermat number">proof</a> based on <a href="/wiki/Fermat_number" title="Fermat number">Fermat numbers</a>,<a href="#cite_note-54">[52]</a> <a href="/wiki/Hillel_Furstenberg" title="Hillel Furstenberg">Furstenberg's</a> <a href="/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes" title="Furstenberg's proof of the infinitude of primes">proof using general topology</a>,<a href="#cite_note-55">[53]</a> and <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer's</a> elegant proof.<a href="#cite_note-56">[54]</a> <a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's proof</a><a href="#cite_note-57">[55]</a> shows that every <a href="/wiki/Finite_set" title="Finite set">finite list</a> of primes is incomplete. The key idea is to multiply together the primes in any given list and add <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}"> If the list consists of the primes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},\ldots ,p_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},\ldots ,p_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1e3bbaf2b32c0dc5e79c1957b04b108bd09d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.784ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},\ldots ,p_{n},}"> this gives the number <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1+p_{1}\cdot p_{2}\cdots p_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1+p_{1}\cdot p_{2}\cdots p_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1d057bffd1dbbc94420d3a0b32703cc3205f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.824ex; height:2.509ex;" alt="{\displaystyle N=1+p_{1}\cdot p_{2}\cdots p_{n}.}"></dd></dl> By the fundamental theorem, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> has a prime factorization <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=p'_{1}\cdot p'_{2}\cdots p'_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>⋅</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=p'_{1}\cdot p'_{2}\cdots p'_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606c652811e0772e18cccc88c216bb3647958d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.63ex; height:2.843ex;" alt="{\displaystyle N=p'_{1}\cdot p'_{2}\cdots p'_{m}}"></dd></dl> with one or more prime factors. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is evenly divisible by each of these factors, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes. The numbers formed by adding one to the products of the smallest primes are called <a href="/wiki/Euclid_number" title="Euclid number">Euclid numbers</a>.<a href="#cite_note-58">[56]</a> The first five of them are prime, but the sixth, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\big (}2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13{\big )}=30031=59\cdot 509,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>7</mn> <mo>⋅</mo> <mn>11</mn> <mo>⋅</mo> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>30031</mn> <mo>=</mo> <mn>59</mn> <mo>⋅</mo> <mn>509</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\big (}2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13{\big )}=30031=59\cdot 509,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24968a289b6e1411be6a5825a10543603d4613b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.975ex; height:3.176ex;" alt="{\displaystyle 1+{\big (}2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13{\big )}=30031=59\cdot 509,}"></dd></dl> is a composite number. <div class="mw-heading mw-heading3"><h3 id="Formulas_for_primes">Formulas for primes</h3></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/Formula_for_primes" title="Formula for primes">Formula for primes</a></div> There is no known efficient formula for primes. For example, there is no non-constant <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, even in several variables, that takes only prime values.<a href="#cite_note-matiyasevich-59">[57]</a> However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on <a href="/wiki/Wilson%27s_theorem" title="Wilson's theorem">Wilson's theorem</a> and generates the number 2 many times and all other primes exactly once.<a href="#cite_note-60">[58]</a> There is also a set of <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a> in nine variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its positive values are prime.<a href="#cite_note-matiyasevich-59">[57]</a> Other examples of prime-generating formulas come from <a href="/wiki/Mills%27_theorem" class="mw-redirect" title="Mills' theorem">Mills' theorem</a> and a theorem of <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright</a>. These assert that there are real constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f763115cc85f7fd1d4ffd590bdc135c74a58afec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A>1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\lfloor A^{3^{n}}\right\rfloor {\text{ and }}\left\lfloor 2^{\cdots ^{2^{2^{\mu }}}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>⌋</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow> <mo>⌊</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor A^{3^{n}}\right\rfloor {\text{ and }}\left\lfloor 2^{\cdots ^{2^{2^{\mu }}}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6695fd076eb01dd676dbadefc5375fe802d7355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.501ex; height:6.176ex;" alt="{\displaystyle \left\lfloor A^{3^{n}}\right\rfloor {\text{ and }}\left\lfloor 2^{\cdots ^{2^{2^{\mu }}}}\right\rfloor }"></dd></dl> are prime for any natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> in the first formula, and any number of exponents in the second formula.<a href="#cite_note-61">[59]</a> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor {}\cdot {}\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor {}\cdot {}\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a17bb92075611951d0ca47b9ab7876515c2507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.744ex; height:2.843ex;" alt="{\displaystyle \lfloor {}\cdot {}\rfloor }"> represents the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a>, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of <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><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}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ef6db045c1f6193799bd25a4b68ba9f78646d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.049ex; height:2.176ex;" alt="{\displaystyle \mu .}"><a href="#cite_note-matiyasevich-59">[57]</a> <div class="mw-heading mw-heading3"><h3 id="Open_questions">Open questions</h3></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/Category:Conjectures_about_prime_numbers" title="Category:Conjectures about prime numbers">Category:Conjectures about prime numbers</a></div> Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of <a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's problems</a> from 1912 are still unsolved.<a href="#cite_note-62">[60]</a> One of them is <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a>, which asserts that every even integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> greater than 2 can be written as a sum of two primes.<a href="#cite_note-63">[61]</a> As of 2014<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Prime_number&action=edit">[update]</a>, this conjecture has been verified for all numbers up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4\cdot 10^{18}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4\cdot 10^{18}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c89cde8425a207680113a97590e763bdbf5126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.183ex; height:2.676ex;" alt="{\displaystyle n=4\cdot 10^{18}.}"><a href="#cite_note-64">[62]</a> Weaker statements than this have been proven; for example, <a href="/wiki/Vinogradov%27s_theorem" title="Vinogradov's theorem">Vinogradov's theorem</a> says that every sufficiently large odd integer can be written as a sum of three primes.<a href="#cite_note-65">[63]</a> <a href="/wiki/Chen%27s_theorem" title="Chen's theorem">Chen's theorem</a> says that every sufficiently large even number can be expressed as the sum of a prime and a <a href="/wiki/Semiprime" title="Semiprime">semiprime</a> (the product of two primes).<a href="#cite_note-66">[64]</a> Also, any even integer greater than 10 can be written as the sum of six primes.<a href="#cite_note-67">[65]</a> The branch of number theory studying such questions is called <a href="/wiki/Additive_number_theory" title="Additive number theory">additive number theory</a>.<a href="#cite_note-68">[66]</a> Another type of problem concerns <a href="/wiki/Prime_gap" title="Prime gap">prime gaps</a>, the differences between consecutive primes. The existence of arbitrarily large prime gaps can be seen by noting that the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!+2,n!+3,\dots ,n!+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>!</mo> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>!</mo> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!+2,n!+3,\dots ,n!+n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf39c77f9ee05de53b4ae8bcf69ab8e5bbc0d119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.578ex; height:2.509ex;" alt="{\displaystyle n!+2,n!+3,\dots ,n!+n}"> consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> composite numbers, for any natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"><a href="#cite_note-69">[67]</a> However, large prime gaps occur much earlier than this argument shows.<a href="#cite_note-riesel-gaps-70">[68]</a> For example, the first prime gap of length 8 is between the primes 89 and 97,<a href="#cite_note-71">[69]</a> much smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8!=40320.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>!</mo> <mo>=</mo> <mn>40320.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8!=40320.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76794af19af9e0c06865ed95d9a57e594a2e9bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.367ex; height:2.176ex;" alt="{\displaystyle 8!=40320.}"> It is conjectured that there are infinitely many <a href="/wiki/Twin_prime" title="Twin prime">twin primes</a>, pairs of primes with difference 2; this is the <a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">twin prime conjecture</a>. <a href="/wiki/Polignac%27s_conjecture" title="Polignac's conjecture">Polignac's conjecture</a> states more generally that for every positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e185ab9c990830d5055fa3ae698a4225ce67e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle k,}"> there are infinitely many pairs of consecutive primes that differ by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2k.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f52ba9e75bf2fb39818de19d98af82aafc4d67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.021ex; height:2.176ex;" alt="{\displaystyle 2k.}"><a href="#cite_note-rib-gaps-72">[70]</a> <a href="/wiki/Andrica%27s_conjecture" title="Andrica's conjecture">Andrica's conjecture</a>,<a href="#cite_note-rib-gaps-72">[70]</a> <a href="/wiki/Brocard%27s_conjecture" title="Brocard's conjecture">Brocard's conjecture</a>,<a href="#cite_note-rib-183-73">[71]</a> <a href="/wiki/Legendre%27s_conjecture" title="Legendre's conjecture">Legendre's conjecture</a>,<a href="#cite_note-chan-74">[72]</a> and <a href="/wiki/Oppermann%27s_conjecture" title="Oppermann's conjecture">Oppermann's conjecture</a><a href="#cite_note-rib-183-73">[71]</a> all suggest that the largest gaps between primes from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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 1}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> should be at most approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7173717a40b3c0819656acd987cacbc2feaf533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.977ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}},}"> a result that is known to follow from the Riemann hypothesis, while the much stronger <a href="/wiki/Cram%C3%A9r_conjecture" class="mw-redirect" title="Cramér conjecture">Cramér conjecture</a> sets the largest gap size at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((\log n)^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O((\log n)^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6248bf94a4ffabcefd6bcd1c7e2dc01d3d05f0e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.847ex; height:3.176ex;" alt="{\displaystyle O((\log n)^{2}).}"><a href="#cite_note-rib-gaps-72">[70]</a> Prime gaps can be generalized to <a href="/wiki/Prime_k-tuple" title="Prime k-tuple">prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-tuples</a>, patterns in the differences among more than two prime numbers. Their infinitude and density are the subject of the <a href="/wiki/First_Hardy%E2%80%93Littlewood_conjecture" title="First Hardy–Littlewood conjecture">first Hardy–Littlewood conjecture</a>, which can be motivated by the <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.<a href="#cite_note-75">[73]</a> <div class="mw-heading mw-heading2"><h2 id="Analytic_properties">Analytic properties</h2></div> <a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a> studies number theory through the lens of <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>, <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>, <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>, and the related mathematics of the infinite and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a>. This area of study began with <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> and his first major result, the solution to the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a>. The problem asked for the value of the infinite sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/879448d9d12ad037b8fde0be5082844a3da6ae09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.078ex; height:3.676ex;" alt="{\displaystyle 1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,}"> which today can be recognized as the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff246e5aba5259593186618c576a3b7e14bc3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.067ex; height:2.843ex;" alt="{\displaystyle \zeta (2)}"> of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>. Euler showed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (2)=\pi ^{2}/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2)=\pi ^{2}/6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780b5b024e5d868b7e89ce95dd4cb682919623f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.879ex; height:3.176ex;" alt="{\displaystyle \zeta (2)=\pi ^{2}/6}">.<a href="#cite_note-76">[74]</a> The reciprocal of this number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6/\pi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6/\pi ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a29a93991c7d0dfd47e0c5b4ea3468fc2e41f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:3.176ex;" alt="{\displaystyle 6/\pi ^{2}}">, is the limiting probability that two random numbers selected uniformly from a large range are <a href="/wiki/Coprime_integers" title="Coprime integers">relatively prime</a> (have no factors in common).<a href="#cite_note-77">[75]</a> The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>, but no efficient <a href="/wiki/Formula_for_primes" title="Formula for primes">formula for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-th prime</a> is known. <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a>, in its basic form, asserts that linear polynomials <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)=a+bn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)=a+bn}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ec0a7d51df349db548143b777ac3a56f88d8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.024ex; height:2.843ex;" alt="{\displaystyle p(n)=a+bn}"></dd></dl> with relatively prime integers <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><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}"> and <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><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}"> take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same <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><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}"> have approximately the same proportions of primes. Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often. <div class="mw-heading mw-heading3"><h3 id="Analytical_proof_of_Euclid's_theorem">Analytical proof of Euclid's theorem</h3></div> <a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">Euler's proof that there are infinitely many primes</a> considers the sums of <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of primes, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots +{\frac {1}{p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots +{\frac {1}{p}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398379c591d6687ff6a128f25a7cd2fd3b20cfc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.572ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots +{\frac {1}{p}}.}"></dd></dl> Euler showed that, for any arbitrary <a href="/wiki/Real_number" title="Real number">real number</a> <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><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}">, there exists a prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> for which this sum is bigger than <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><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}">.<a href="#cite_note-78">[76]</a> This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every <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><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}">. The growth rate of this sum is described more precisely by <a href="/wiki/Mertens%27_theorems" title="Mertens' theorems">Mertens' second theorem</a>.<a href="#cite_note-79">[77]</a> For comparison, the sum <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7be4306447ca3c94aa41f5683e7372ef912b72a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.528ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}}"></dd></dl> does not grow to infinity as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> goes to infinity (see the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a>). In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite.<a href="#cite_note-mtb-invitation-80">[78]</a> <a href="/wiki/Brun%27s_theorem" title="Brun's theorem">Brun's theorem</a> states that the sum of the reciprocals of <a href="/wiki/Twin_prime" title="Twin prime">twin primes</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>13</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dd979f913d0f2173e98f7d4095909780bd6977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.38ex; height:6.176ex;" alt="{\displaystyle \left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots ,}"></dd></dl> is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the <a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">twin prime conjecture</a>, that there exist infinitely many twin primes.<a href="#cite_note-mtb-invitation-80">[78]</a> <div class="mw-heading mw-heading3"><h3 id="Number_of_primes_below_a_given_bound">Number of primes below a given bound</h3></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/Prime_number_theorem" title="Prime number theorem">Prime number theorem</a> and <a href="/wiki/Prime-counting_function" title="Prime-counting function">Prime-counting function</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Prime-counting_relative_error.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Prime-counting_relative_error.svg/350px-Prime-counting_relative_error.svg.png" decoding="async" width="350" height="301" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Prime-counting_relative_error.svg/525px-Prime-counting_relative_error.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Prime-counting_relative_error.svg/700px-Prime-counting_relative_error.svg.png 2x" data-file-width="1012" data-file-height="870" /></a><figcaption>The <a href="/wiki/Approximation_error" title="Approximation error">relative error</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{\log n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{\log n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942db377004f15abd6b11d4d5487c436a2249d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.311ex; height:3.676ex;" alt="{\displaystyle {\tfrac {n}{\log n}}}"> and the logarithmic integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Li} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Li</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c711c5eda9013c9c1d65e69f35f05b99f03629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.304ex; height:2.843ex;" alt="{\displaystyle \operatorname {Li} (n)}"> as approximations to the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a>. Both relative errors decrease to zero as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> grows, but the convergence to zero is much more rapid for the logarithmic integral.</figcaption></figure> The <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac42d38c71b368d5fbf1e05753e9c5c038cd671b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.536ex; height:2.843ex;" alt="{\displaystyle \pi (n)}"> is defined as the number of primes not greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-81">[79]</a> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (11)=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mn>11</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (11)=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37a1ec8c22561cae405286228241267bec24e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.727ex; height:2.843ex;" alt="{\displaystyle \pi (11)=5}">, since there are five primes less than or equal to 11. Methods such as the <a href="/wiki/Meissel%E2%80%93Lehmer_algorithm" title="Meissel–Lehmer algorithm">Meissel–Lehmer algorithm</a> can compute exact values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac42d38c71b368d5fbf1e05753e9c5c038cd671b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.536ex; height:2.843ex;" alt="{\displaystyle \pi (n)}"> faster than it would be possible to list each prime up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-82">[80]</a> The <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> states that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac42d38c71b368d5fbf1e05753e9c5c038cd671b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.536ex; height:2.843ex;" alt="{\displaystyle \pi (n)}"> is asymptotic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/\log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/\log n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf375088f1428c429e63be39e7852e8b39c77c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.698ex; height:2.843ex;" alt="{\displaystyle n/\log n}">, which is denoted as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)\sim {\frac {n}{\log n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)\sim {\frac {n}{\log n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408b397046af4ade735aebb8995aa70ac2244838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.871ex; height:5.176ex;" alt="{\displaystyle \pi (n)\sim {\frac {n}{\log n}},}"></dd></dl> and means that the ratio of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac42d38c71b368d5fbf1e05753e9c5c038cd671b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.536ex; height:2.843ex;" alt="{\displaystyle \pi (n)}"> to the right-hand fraction <a href="/wiki/Convergent_sequence" class="mw-redirect" title="Convergent sequence">approaches</a> 1 as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> grows to infinity.<a href="#cite_note-cranpom10-83">[81]</a> This implies that the likelihood that a randomly chosen number less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is prime is (approximately) inversely proportional to the number of digits in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-84">[82]</a> It also implies that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">th prime number is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\log n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560dfdce0353a330e03e4b3e0b7ca6e484bb40fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.535ex; height:2.509ex;" alt="{\displaystyle n\log n}"><a href="#cite_note-85">[83]</a> and therefore that the average size of a prime gap is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317ab5292da7c7935aec01a570461fe0613b21d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.754ex; height:2.509ex;" alt="{\displaystyle \log n}">.<a href="#cite_note-riesel-gaps-70">[68]</a> A more accurate estimate for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac42d38c71b368d5fbf1e05753e9c5c038cd671b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.536ex; height:2.843ex;" alt="{\displaystyle \pi (n)}"> is given by the <a href="/wiki/Offset_logarithmic_integral" class="mw-redirect" title="Offset logarithmic integral">offset logarithmic integral</a><a href="#cite_note-cranpom10-83">[81]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (n)\sim \operatorname {Li} (n)=\int _{2}^{n}{\frac {dt}{\log t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>Li</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (n)\sim \operatorname {Li} (n)=\int _{2}^{n}{\frac {dt}{\log t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b5a9dac3f86b36f631c01ed4aae859cdedf79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.788ex; height:5.843ex;" alt="{\displaystyle \pi (n)\sim \operatorname {Li} (n)=\int _{2}^{n}{\frac {dt}{\log t}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_progressions">Arithmetic progressions</h3></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/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a> and <a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a></div> An <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.<a href="#cite_note-86">[84]</a> This difference is called the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulus</a> of the progression.<a href="#cite_note-87">[85]</a> For example, <dl><dd>3, 12, 21, 30, 39, ...,</dd></dl> is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,a+q,a+2q,a+3q,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>q</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>q</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>q</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,a+q,a+2q,a+3q,\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d16a7879ded2fe4112aae7ab945259fc40eda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.833ex; height:2.509ex;" alt="{\displaystyle a,a+q,a+2q,a+3q,\dots }"></dd></dl> can have more than one prime only when its remainder <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><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}"> and modulus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> are relatively prime. If they are relatively prime, <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a> asserts that the progression contains infinitely many primes.<a href="#cite_note-88">[86]</a> <div class="thumb tnone" style="margin-left:auto;margin-right:auto;overflow:hidden;width:auto;max-width:823px"><div class="thumbinner"><div class="noresize" style="overflow:auto"><a href="/wiki/File:Prime_numbers_in_arithmetic_progression_mod_9_zoom_in.png" class="mw-file-description" title="Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression."><img alt="Prime numbers in arithmetic progression mod 9" src="//upload.wikimedia.org/wikipedia/commons/e/e3/Prime_numbers_in_arithmetic_progression_mod_9_zoom_in.png" decoding="async" width="815" height="385" class="mw-file-element" data-file-width="815" data-file-height="385" /></a></div><div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Prime_numbers_in_arithmetic_progression_mod_9_zoom_in.png" title="File:Prime numbers in arithmetic progression mod 9 zoom in.png"> </a></div>Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.</div></div></div> The <a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a> shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.<a href="#cite_note-neale-18-47-38">[36]</a><a href="#cite_note-89">[87]</a> <div class="mw-heading mw-heading3"><h3 id="Prime_values_of_quadratic_polynomials">Prime values of quadratic polynomials</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ulam_2.png" class="mw-file-description"><img alt="The Ulam spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Ulam_2.png/240px-Ulam_2.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/75/Ulam_2.png 1.5x" data-file-width="345" data-file-height="345" /></a><figcaption>The <a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam spiral</a>. Prime numbers (red) cluster on some diagonals and not others. Prime values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4n^{2}-2n+41}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>41</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4n^{2}-2n+41}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f7b0b039e84f6a35cf179e74583d3217eced75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.174ex; height:2.843ex;" alt="{\displaystyle 4n^{2}-2n+41}"> are shown in blue.</figcaption></figure> Euler noted that the function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}-n+41}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>41</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}-n+41}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c4504253486e5ff537490356c8529390687bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.849ex; height:2.843ex;" alt="{\displaystyle n^{2}-n+41}"></dd></dl> yields prime numbers for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq n\leq 40}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>40</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq n\leq 40}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d90b605eed00af6cd0590bb9591798ea4497f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.079ex; height:2.343ex;" alt="{\displaystyle 1\leq n\leq 40}">, although composite numbers appear among its later values.<a href="#cite_note-90">[88]</a><a href="#cite_note-91">[89]</a> The search for an explanation for this phenomenon led to the deep <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> of <a href="/wiki/Heegner_number" title="Heegner number">Heegner numbers</a> and the <a href="/wiki/Class_number_problem" title="Class number problem">class number problem</a>.<a href="#cite_note-92">[90]</a> The <a href="/wiki/Hardy%E2%80%93Littlewood_conjecture_F" class="mw-redirect" title="Hardy–Littlewood conjecture F">Hardy–Littlewood conjecture F</a> predicts the density of primes among the values of <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomials</a> with integer <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.<a href="#cite_note-guy-a1-93">[91]</a> The <a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam spiral</a> arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.<a href="#cite_note-guy-a1-93">[91]</a> <div class="mw-heading mw-heading3"><h3 id="Zeta_function_and_the_Riemann_hypothesis">Zeta function and the Riemann hypothesis</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Riemann_zeta_function_absolute_value.png" class="mw-file-description"><img alt="Plot of the absolute values of the zeta function" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Riemann_zeta_function_absolute_value.png/330px-Riemann_zeta_function_absolute_value.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Riemann_zeta_function_absolute_value.png/495px-Riemann_zeta_function_absolute_value.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Riemann_zeta_function_absolute_value.png/660px-Riemann_zeta_function_absolute_value.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Plot of the absolute values of the zeta function, showing some of its features</figcaption></figure> One of the most famous unsolved questions in mathematics, dating from 1859, and one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>, is the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, which asks where the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd45922057e4d7a5718ce5ed703ab493c63897a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.995ex; height:2.843ex;" alt="{\displaystyle \zeta (s)}"> are located. This function is an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> on the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. For complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> with real part greater than one it equals both an <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite sum</a> over all integers, and an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> over the prime numbers, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ee4dae49e217914ed83b6d9c27155a6eca1c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.015ex; height:7.176ex;" alt="{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}.}"></dd></dl> This equality between a sum and a product, discovered by Euler, is called an <a href="/wiki/Euler_product" title="Euler product">Euler product</a>.<a href="#cite_note-94">[92]</a> The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers.<a href="#cite_note-95">[93]</a> It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac386d8f227fb823cede9b3e33d706cad3ed306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}">, but the sum would diverge (it is the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391a7209d3509160364ad66fc270afc07d84c0db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.723ex; height:3.676ex;" alt="{\displaystyle 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots }">) while the product would be finite, a contradiction.<a href="#cite_note-96">[94]</a> The Riemann hypothesis states that the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of the zeta-function are all either negative even numbers, or complex numbers with <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> equal to 1/2.<a href="#cite_note-97">[95]</a> The original proof of the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,<a href="#cite_note-98">[96]</a><a href="#cite_note-bcrw18-99">[97]</a> although other more elementary proofs have been found.<a href="#cite_note-100">[98]</a> The prime-counting function can be expressed by <a href="/wiki/Riemann%27s_explicit_formula" class="mw-redirect" title="Riemann's explicit formula">Riemann's explicit formula</a> as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term.<a href="#cite_note-101">[99]</a> In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the <a href="/wiki/Asymptotic_distribution" title="Asymptotic distribution">asymptotic distribution</a> of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of <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><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}"> for intervals near a number <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><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}">).<a href="#cite_note-bcrw18-99">[97]</a> <div class="mw-heading mw-heading2"><h2 id="Abstract_algebra">Abstract algebra</h2></div> <div class="mw-heading mw-heading3"><h3 id="Modular_arithmetic_and_finite_fields">Modular arithmetic and finite fields</h3></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/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></div> Modular arithmetic modifies usual arithmetic by only using the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\dots ,n-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\dots ,n-1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a274c78335c05b113ff9a19b66526a9da869fbfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.456ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\dots ,n-1\}}">, for a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-102">[100]</a> Modular sums, differences and products are calculated by performing the same replacement by the remainder on the result of the usual sum, difference, or product of integers.<a href="#cite_note-103">[101]</a> Equality of integers corresponds to congruence in modular arithmetic: <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> are congruent (written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\equiv y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344d00ea5455ff5454015cc1f00cf534be450a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\equiv y}"> mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">) when they have the same remainder after division by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-104">[102]</a> However, in this system of numbers, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee716ec61382a6b795092c0edd859d12e64cbba8" 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 7}"> as modulus, division by <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><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}"> is possible: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/3\equiv 3{\bmod {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>≡</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2/3\equiv 3{\bmod {7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e6eb9bb6b2d861011df01e930241a2b1338af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.592ex; height:2.843ex;" alt="{\displaystyle 2/3\equiv 3{\bmod {7}}}">, because <a href="/wiki/Clearing_denominators" title="Clearing denominators">clearing denominators</a> by multiplying both sides by <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><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}"> gives the valid formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\equiv 9{\bmod {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>≡</mo> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\equiv 9{\bmod {7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cc49f8e6ba7fbf81201a0a8d5b3ce614135853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.267ex; height:2.176ex;" alt="{\displaystyle 2\equiv 9{\bmod {7}}}">. However, with the composite modulus <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><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}">, division by <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><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}"> is impossible. There is no valid solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/3\equiv x{\bmod {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>≡</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2/3\equiv x{\bmod {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959d2e86e50ec85744347d794e18cc3d107979cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.759ex; height:2.843ex;" alt="{\displaystyle 2/3\equiv x{\bmod {6}}}">: clearing denominators by multiplying by <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><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}"> causes the left-hand side to become <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><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}"> while the right-hand side becomes either <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><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}"> or <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><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}">. In the terminology of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, the ability to perform division means that modular arithmetic modulo a prime number forms a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or, more specifically, a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, while other moduli only give a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> but not a field.<a href="#cite_note-105">[103]</a> Several theorems about primes can be formulated using modular arithmetic. For instance, <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> states that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\not \equiv 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≢</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\not \equiv 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4a234827d107c1dd3ffc19005990ea92f4e4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.492ex; height:2.676ex;" alt="{\displaystyle a\not \equiv 0}"> (mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{p-1}\equiv 1}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{p-1}\equiv 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d0c1282f62b25fc17d3ce40e36e3b2a2f541f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.65ex; height:2.676ex;" alt="{\displaystyle a^{p-1}\equiv 1}"> (mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">).<a href="#cite_note-106">[104]</a> Summing this over all choices of <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><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}"> gives the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{a=1}^{p-1}a^{p-1}\equiv (p-1)\cdot 1\equiv -1{\pmod {p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>1</mn> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a=1}^{p-1}a^{p-1}\equiv (p-1)\cdot 1\equiv -1{\pmod {p}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac4572eadd7182309e72510091be6fe559e024f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.623ex; height:7.343ex;" alt="{\displaystyle \sum _{a=1}^{p-1}a^{p-1}\equiv (p-1)\cdot 1\equiv -1{\pmod {p}},}"></dd></dl> valid whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is prime. <a href="/wiki/Giuga%27s_conjecture" class="mw-redirect" title="Giuga's conjecture">Giuga's conjecture</a> says that this equation is also a sufficient condition for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> to be prime.<a href="#cite_note-107">[105]</a> <a href="/wiki/Wilson%27s_theorem" title="Wilson's theorem">Wilson's theorem</a> says that an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f127e7a5f2449ddf3edb8164c2d2439120641f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>1}"> is prime if and only if the <a href="/wiki/Factorial" title="Factorial">factorial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p-1)!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba68358eefe786c8448a04d3be5fa11279a59c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.628ex; height:2.843ex;" alt="{\displaystyle (p-1)!}"> is congruent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> mod <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">. For a composite number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;n=r\cdot s\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>s</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;n=r\cdot s\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42391cbdb9185ff409ed6c393c297c43f005078a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.602ex; height:1.676ex;" alt="{\displaystyle \;n=r\cdot s\;}"> this cannot hold, since one of its factors divides both n and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb71a5f562b97650728f30493f43e96c15b4287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.854ex; height:2.843ex;" alt="{\displaystyle (n-1)!}">, and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)!\equiv -1{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)!\equiv -1{\pmod {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b29169bc8c822e9d212b319d1f610e0998d3fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.002ex; height:2.843ex;" alt="{\displaystyle (n-1)!\equiv -1{\pmod {n}}}"> is impossible.<a href="#cite_note-108">[106]</a> <div class="mw-heading mw-heading3"><h3 id="p-adic_numbers">p-adic numbers</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/P-adic_number" title="P-adic number">p-adic number</a></div> The <a href="/wiki/P-adic_order" class="mw-redirect" title="P-adic order"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{p}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{p}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a740c63898dac75666e43736eb3ae87eb8c62b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.412ex; height:3.009ex;" alt="{\displaystyle \nu _{p}(n)}"> of an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is the number of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> in the prime factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. The same concept can be extended from integers to rational numbers by defining the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic order of a fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eebcb27a9df80445dbe86eefee5d131d6e0e7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.598ex; height:2.843ex;" alt="{\displaystyle m/n}"> to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{p}(m)-\nu _{p}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{p}(m)-\nu _{p}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d324584195f4488cae3200e33bcb93f605f43ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.309ex; height:3.009ex;" alt="{\displaystyle \nu _{p}(m)-\nu _{p}(n)}">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |q|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |q|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cfd1fc1d9b3543ebf4cb30e9d3a5a73a61bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.422ex; height:3.176ex;" alt="{\displaystyle |q|_{p}}"> of any rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> is then defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |q|_{p}=p^{-\nu _{p}(q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |q|_{p}=p^{-\nu _{p}(q)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121493a70b4bc560c8fe699075a1784122258128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.884ex; height:3.676ex;" alt="{\displaystyle |q|_{p}=p^{-\nu _{p}(q)}}">. Multiplying an integer by its <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic absolute value cancels out the factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic distance, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a <a href="/wiki/Complete_field" title="Complete field">complete field</a>, the rational numbers with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic distance can be extended to a different complete field, the <a href="/wiki/P-adic_number" title="P-adic number"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic numbers</a>.<a href="#cite_note-childress-109">[107]</a><a href="#cite_note-110">[108]</a> This picture of an order, absolute value, and complete field derived from them can be generalized to <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> and their <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuations</a> (certain mappings from the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a> of the field to a <a href="/wiki/Totally_ordered_group" class="mw-redirect" title="Totally ordered group">totally ordered additive group</a>, also called orders), <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute values</a> (certain multiplicative mappings from the field to the real numbers, also called norms),<a href="#cite_note-childress-109">[107]</a> and places (extensions to <a href="/wiki/Complete_field" title="Complete field">complete fields</a> in which the given field is a <a href="/wiki/Dense_set" title="Dense set">dense set</a>, also called completions).<a href="#cite_note-111">[109]</a> The extension from the rational numbers to the <a href="/wiki/Real_number" title="Real number">real numbers</a>, for instance, is a place in which the distance between numbers is the usual <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of their difference. The corresponding mapping to an additive group would be the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the absolute value, although this does not meet all the requirements of a valuation. According to <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a>, up to a natural notion of equivalence, the real numbers and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers.<a href="#cite_note-childress-109">[107]</a> The <a href="/wiki/Local-global_principle" class="mw-redirect" title="Local-global principle">local-global principle</a> allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.<a href="#cite_note-112">[110]</a> <div class="mw-heading mw-heading3"><h3 id="Prime_elements_in_rings">Prime elements in rings</h3></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/Prime_element" title="Prime element">Prime element</a> and <a href="/wiki/Irreducible_element" title="Irreducible element">Irreducible element</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gaussian_primes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Gaussian_primes.png/220px-Gaussian_primes.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Gaussian_primes.png/330px-Gaussian_primes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Gaussian_primes.png/440px-Gaussian_primes.png 2x" data-file-width="1200" data-file-height="1198" /></a><figcaption>The <a href="/wiki/Gaussian_prime" class="mw-redirect" title="Gaussian prime">Gaussian primes</a> with norm less than 500</figcaption></figure> A <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> is an <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, prime elements and irreducible elements. An element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> of a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is called prime if it is nonzero, has no <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> (that is, it is not a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>), and satisfies the following requirement: whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> divides the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> of two elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, it also divides at least one of <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><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}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">. An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>11</mn> <mo>,</mo> <mo>−</mo> <mn>7</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521b9a30f8065b6659540de5e2d79f23b8ae8009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.556ex; height:2.843ex;" alt="{\displaystyle \{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}\,.}"></dd></dl> In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a>.<a href="#cite_note-113">[111]</a> The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i]}">, the ring of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> denotes the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> and <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><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}"> and <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><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}"> are arbitrary integers. Its prime elements are known as <a href="/wiki/Gaussian_prime" class="mw-redirect" title="Gaussian prime">Gaussian primes</a>. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a65e41a5c0369e908cf26a2452046f19bab946d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle 1+i}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/685fed923493e7884e748f0dd6f932311f66bd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle 1-i}">. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.<a href="#cite_note-114">[112]</a> This is a consequence of <a href="/wiki/Fermat%27s_theorem_on_sums_of_two_squares" title="Fermat's theorem on sums of two squares">Fermat's theorem on sums of two squares</a>, which states that an odd prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is expressible as the sum of two squares, <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><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}}">, and therefore factorable as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=(x+iy)(x-iy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(x+iy)(x-iy)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba304bbcb900df2560ef81f194004d2cd365ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.232ex; height:2.843ex;" alt="{\displaystyle p=(x+iy)(x-iy)}">, exactly when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is 1 mod 4.<a href="#cite_note-115">[113]</a> <div class="mw-heading mw-heading3"><h3 id="Prime_ideals">Prime ideals</h3></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/Prime_ideals" class="mw-redirect" title="Prime ideals">Prime ideals</a></div> Not every ring is a unique factorization domain. For instance, in the ring of numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-5}}}"> <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> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5894d5cf1749ce9cb9be987542cabde8b3b8f2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-5}}}"> (for integers <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><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}"> and <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><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}">) the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 21}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ebb9ccf6080ba5c9a6ea8973cb2f26c50211cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 21}"> has two factorizations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21=3\cdot 7=(1+2{\sqrt {-5}})(1-2{\sqrt {-5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>7</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <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> <mn>2</mn> <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 21=3\cdot 7=(1+2{\sqrt {-5}})(1-2{\sqrt {-5}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ef78d9e6efc73f20864dade3df3c2c9ae57f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.288ex; height:3.009ex;" alt="{\displaystyle 21=3\cdot 7=(1+2{\sqrt {-5}})(1-2{\sqrt {-5}})}">, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements. Prime ideals, which generalize prime elements in the sense that the <a href="/wiki/Principal_ideal" title="Principal ideal">principal ideal</a> generated by a prime element is a prime ideal, are an important tool and object of study in <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, <a href="/wiki/Number_theory" title="Number theory">algebraic number theory</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the <a href="/wiki/Lasker%E2%80%93Noether_theorem" class="mw-redirect" title="Lasker–Noether theorem">Lasker–Noether theorem</a>, which expresses every ideal in a <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> as an intersection of <a href="/wiki/Primary_ideal" title="Primary ideal">primary ideals</a>, which are the appropriate generalizations of <a href="/wiki/Prime_power" title="Prime power">prime powers</a>.<a href="#cite_note-116">[114]</a> The <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a> is a geometric space whose points are the prime ideals of the ring.<a href="#cite_note-117">[115]</a> <a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic geometry</a> also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or <a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">ramification</a> of prime ideals when lifted to an <a href="/wiki/Field_extension" title="Field extension">extension field</a>, a basic problem of algebraic number theory, bears some resemblance with <a href="/wiki/Ramified_cover" class="mw-redirect" title="Ramified cover">ramification in geometry</a>. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of <a href="/wiki/Quadratic_number_field" class="mw-redirect" title="Quadratic number field">quadratic number fields</a> can be used in proving <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>, a statement that concerns the existence of square roots modulo integer prime numbers.<a href="#cite_note-118">[116]</a> Early attempts to prove <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> led to <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer</a>'s introduction of <a href="/wiki/Regular_prime" title="Regular prime">regular primes</a>, integer prime numbers connected with the failure of unique factorization in the <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic integers</a>.<a href="#cite_note-119">[117]</a> The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by <a href="/wiki/Chebotarev%27s_density_theorem" title="Chebotarev's density theorem">Chebotarev's density theorem</a>, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.<a href="#cite_note-120">[118]</a> <div class="mw-heading mw-heading3"><h3 id="Group_theory">Group theory</h3></div> In the theory of <a href="/wiki/Finite_group" title="Finite group">finite groups</a> the <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a> imply that, if a power of a prime number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}"> divides the <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order of a group</a>, then the group has a subgroup of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}">. By <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>, any group of prime order is a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a>, and by <a href="/wiki/Burnside%27s_theorem" title="Burnside's theorem">Burnside's theorem</a> any group whose order is divisible by only two primes is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>.<a href="#cite_note-121">[119]</a> <div class="mw-heading mw-heading2"><h2 id="Computational_methods">Computational methods</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gears_large.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gears_large.jpg/220px-Gears_large.jpg" decoding="async" width="220" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gears_large.jpg/330px-Gears_large.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gears_large.jpg/440px-Gears_large.jpg 2x" data-file-width="500" data-file-height="680" /></a><figcaption>The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13.</figcaption></figure> For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics<a href="#cite_note-pure-34">[b]</a> other than the use of prime numbered gear teeth to distribute wear evenly.<a href="#cite_note-122">[120]</a> In particular, number theorists such as <a href="/wiki/United_Kingdom" title="United Kingdom">British</a> mathematician <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> prided themselves on doing work that had absolutely no military significance.<a href="#cite_note-123">[121]</a> This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a> algorithms.<a href="#cite_note-ent-7-35">[33]</a> These applications have led to significant study of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for computing with prime numbers, and in particular of <a href="/wiki/Primality_test" title="Primality test">primality testing</a>, methods for determining whether a given number is prime. The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called <a href="/wiki/Integer_factorization" title="Integer factorization">factorization</a> algorithms. Prime numbers are also used in computing for <a href="/wiki/Checksum" title="Checksum">checksums</a>, <a href="/wiki/Hash_table" title="Hash table">hash tables</a>, and <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a>. <div class="mw-heading mw-heading3"><h3 id="Trial_division">Trial division</h3></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/Trial_division" title="Trial division">Trial division</a></div> The most basic method of checking the primality of a given integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is called <a href="/wiki/Trial_division" title="Trial division">trial division</a>. This method divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> by each integer from 2 up to the <a href="/wiki/Square_root" title="Square root">square root</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. Any such integer dividing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> evenly establishes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/083e9fcf193435c0829806a150b537e26641793c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.176ex;" alt="{\displaystyle n=a\cdot b}">, one of the two factors <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><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}"> and <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><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}"> is less than or equal to the <a href="/wiki/Square_root" title="Square root">square root</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. Another optimization is to check only primes as factors in this range.<a href="#cite_note-124">[122]</a> For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {37}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>37</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {37}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2253dfc4b293b631b7fa755e6e735fc6c40de39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {37}}}">, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs <a href="/wiki/Exponential_growth" title="Exponential growth">grows exponentially</a> as a function of the number of digits of these integers.<a href="#cite_note-125">[123]</a> However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.<a href="#cite_note-p._220-126">[124]</a> <div class="mw-heading mw-heading3"><h3 id="Sieves">Sieves</h3></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Sieve_of_Eratosthenes_animation.gif" class="mw-file-description"><img alt="Animation of the sieve of Eratosthenes" src="//upload.wikimedia.org/wikipedia/commons/b/b9/Sieve_of_Eratosthenes_animation.gif" decoding="async" width="445" height="369" class="mw-file-element" data-file-width="445" data-file-height="369" /></a><figcaption>The <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a></div> Before computers, <a href="/wiki/Mathematical_table" title="Mathematical table">mathematical tables</a> listing all of the primes or prime factorizations up to a given limit were commonly printed.<a href="#cite_note-127">[125]</a> The oldest known method for generating a list of primes is called the sieve of Eratosthenes.<a href="#cite_note-128">[126]</a> The animation shows an optimized variant of this method.<a href="#cite_note-129">[127]</a> Another more asymptotically efficient sieving method for the same problem is the <a href="/wiki/Sieve_of_Atkin" title="Sieve of Atkin">sieve of Atkin</a>.<a href="#cite_note-130">[128]</a> In advanced mathematics, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a> applies similar methods to other problems.<a href="#cite_note-131">[129]</a> <div class="mw-heading mw-heading3"><h3 id="Primality_testing_versus_primality_proving">Primality testing versus primality proving</h3></div> Some of the fastest modern tests for whether an arbitrary given number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is prime are <a href="/wiki/Probabilistic_algorithm" class="mw-redirect" title="Probabilistic algorithm">probabilistic</a> (or <a href="/wiki/Monte_Carlo_algorithm" title="Monte Carlo algorithm">Monte Carlo</a>) algorithms, meaning that they have a small random chance of producing an incorrect answer.<a href="#cite_note-hromkovic-132">[130]</a> For instance the <a href="/wiki/Solovay%E2%80%93Strassen_primality_test" title="Solovay–Strassen primality test">Solovay–Strassen primality test</a> on a given number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> chooses a number <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><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}"> randomly from <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><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}"> through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54de64bfce933aaf8b8317dc54f927ef25c96774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p-2}"> and uses <a href="/wiki/Modular_exponentiation" title="Modular exponentiation">modular exponentiation</a> to check whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{(p-1)/2}\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{(p-1)/2}\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e41c9364089b9540708ee3514f73b1bfa62a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.316ex; height:2.843ex;" alt="{\displaystyle a^{(p-1)/2}\pm 1}"> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">.<a href="#cite_note-133">[c]</a> If so, it answers yes and otherwise it answers no. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> really is prime, it will always answer yes, but if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2.<a href="#cite_note-koblitz-134">[131]</a> If this test is repeated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> times on the same number, the probability that a composite number could pass the test every time is at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcca0eda567be34036ddbb204671dc6aee9975f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\displaystyle 1/2^{n}}">. Because this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite.<a href="#cite_note-135">[132]</a> A composite number that passes such a test is called a <a href="/wiki/Pseudoprime" title="Pseudoprime">pseudoprime</a>.<a href="#cite_note-koblitz-134">[131]</a> In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministic</a> (non-random) algorithms, such as the <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a>,<a href="#cite_note-tao-aks-136">[133]</a> and randomized <a href="/wiki/Las_Vegas_algorithm" title="Las Vegas algorithm">Las Vegas algorithms</a> where the random choices made by the algorithm do not affect its final answer, such as some variations of <a href="/wiki/Elliptic_curve_primality_proving" class="mw-redirect" title="Elliptic curve primality proving">elliptic curve primality proving</a>.<a href="#cite_note-hromkovic-132">[130]</a> When the elliptic curve method concludes that a number is prime, it provides <a href="/wiki/Primality_certificate" title="Primality certificate">primality certificate</a> that can be verified quickly.<a href="#cite_note-atkin-morain-137">[134]</a> The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on <a href="/wiki/Heuristic_argument" title="Heuristic argument">heuristic arguments</a> rather than rigorous proofs. The <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a> has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice.<a href="#cite_note-morain-138">[135]</a> These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime.<a href="#cite_note-139">[d]</a> The following table lists some of these tests. Their running time is given in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the number to be tested and, for probabilistic algorithms, the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> of tests performed. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> is an arbitrarily small positive number, and log is the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> to an unspecified base. The <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a> means that each time bound should be multiplied by a <a href="/wiki/Constant_factor" class="mw-redirect" title="Constant factor">constant factor</a> to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">. <table class="wikitable sortable"> <tbody><tr> <th>Test </th> <th>Developed in </th> <th>Type </th> <th>Running time </th> <th>Notes </th> <th class="unsortable">References </th></tr> <tr> <td><a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a> </td> <td>2002 </td> <td>deterministic </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((\log n)^{6+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O((\log n)^{6+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc16469e3af6c2a0827c49a1bc042109f80f893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.244ex; height:3.176ex;" alt="{\displaystyle O((\log n)^{6+\varepsilon })}"> </td> <td> </td> <td><a href="#cite_note-tao-aks-136">[133]</a><a href="#cite_note-140">[136]</a> </td></tr> <tr> <td><a href="/wiki/Elliptic_curve_primality_proving" class="mw-redirect" title="Elliptic curve primality proving">Elliptic curve primality proving</a> </td> <td>1986 </td> <td>Las Vegas </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((\log n)^{4+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O((\log n)^{4+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfc882e3d4fafd81c7e758315ce91a8100bd432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.244ex; height:3.176ex;" alt="{\displaystyle O((\log n)^{4+\varepsilon })}"> heuristically </td> <td> </td> <td><a href="#cite_note-morain-138">[135]</a> </td></tr> <tr> <td><a href="/wiki/Baillie%E2%80%93PSW_primality_test" title="Baillie–PSW primality test">Baillie–PSW primality test</a> </td> <td>1980 </td> <td>Monte Carlo </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((\log n)^{2+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O((\log n)^{2+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26335494299bcaa20a6732274e174f5eaa8461e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.244ex; height:3.176ex;" alt="{\displaystyle O((\log n)^{2+\varepsilon })}"> </td> <td> </td> <td><a href="#cite_note-PSW-141">[137]</a><a href="#cite_note-lpsp-142">[138]</a> </td></tr> <tr> <td><a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin primality test</a> </td> <td>1980 </td> <td>Monte Carlo </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k(\log n)^{2+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(k(\log n)^{2+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0341577e6a6f01b2c97fb60b0585bcad6ecbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.456ex; height:3.176ex;" alt="{\displaystyle O(k(\log n)^{2+\varepsilon })}"> </td> <td>error probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7b428178b13d0649e586b49bd7d14fd33e42fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle 4^{-k}}"> </td> <td><a href="#cite_note-monier-143">[139]</a> </td></tr> <tr> <td><a href="/wiki/Solovay%E2%80%93Strassen_primality_test" title="Solovay–Strassen primality test">Solovay–Strassen primality test</a> </td> <td>1977 </td> <td>Monte Carlo </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k(\log n)^{2+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(k(\log n)^{2+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0341577e6a6f01b2c97fb60b0585bcad6ecbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.456ex; height:3.176ex;" alt="{\displaystyle O(k(\log n)^{2+\varepsilon })}"> </td> <td>error probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ba5275e798107d7738afeb76e2bdb91cd37e0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle 2^{-k}}"> </td> <td><a href="#cite_note-monier-143">[139]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Special-purpose_algorithms_and_the_largest_known_prime">Special-purpose algorithms and the largest known prime</h3></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/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div> In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer primality test</a> can determine whether a <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne number</a> (one less than a <a href="/wiki/Power_of_two" title="Power of two">power of two</a>) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.<a href="#cite_note-144">[140]</a> This is why since 1992 (as of December 2018<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Prime_number&action=edit">[update]</a>) the <a href="/wiki/Largest_known_prime" class="mw-redirect" title="Largest known prime">largest known prime</a> has always been a Mersenne prime.<a href="#cite_note-145">[141]</a> It is conjectured that there are infinitely many Mersenne primes.<a href="#cite_note-146">[142]</a> The following table gives the largest known primes of various types. Some of these primes have been found using <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a>. In 2009, the <a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a> project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits.<a href="#cite_note-147">[143]</a> The <a href="/wiki/Electronic_Frontier_Foundation" title="Electronic Frontier Foundation">Electronic Frontier Foundation</a> also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.<a href="#cite_note-148">[144]</a> <table class="wikitable"> <tbody><tr> <th>Type </th> <th>Prime </th> <th>Number of decimal digits </th> <th>Date </th> <th>Found by </th></tr> <tr> <td><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne prime</a> </td> <td>2136,279,841 − 1 </td> <td style="text-align:right;">41,024,320 </td> <td>October 21, 2024<a href="#cite_note-GIMPS-2024-1">[1]</a> </td> <td>Luke Durant, <a href="/wiki/Great_Internet_Mersenne_Prime_Search" title="Great Internet Mersenne Prime Search">Great Internet Mersenne Prime Search</a> </td></tr> <tr> <td><a href="/wiki/Proth_prime" title="Proth prime">Proth prime</a> </td> <td>10,223 × 231,172,165 + 1 </td> <td style="text-align:right;">9,383,761 </td> <td>October 31, 2016<a href="#cite_note-149">[145]</a> </td> <td>Péter Szabolcs, <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a><a href="#cite_note-150">[146]</a> </td></tr> <tr> <td><a href="/wiki/Factorial_prime" title="Factorial prime">factorial prime</a> </td> <td>208,003! − 1 </td> <td style="text-align:right;">1,015,843 </td> <td>July 2016 </td> <td>Sou Fukui<a href="#cite_note-151">[147]</a> </td></tr> <tr> <td><a href="/wiki/Primorial_prime" title="Primorial prime">primorial prime</a><a href="#cite_note-153">[e]</a> </td> <td>1,098,133# − 1 </td> <td style="text-align:right;">476,311 </td> <td>March 2012 </td> <td>James P. Burt, <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a><a href="#cite_note-154">[149]</a> </td></tr> <tr> <td><a href="/wiki/Twin_prime" title="Twin prime">twin primes</a> </td> <td>2,996,863,034,895 × 21,290,000 ± 1 </td> <td style="text-align:right;">388,342 </td> <td>September 2016 </td> <td>Tom Greer, <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a><a href="#cite_note-155">[150]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Integer_factorization">Integer factorization</h3></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/Integer_factorization" title="Integer factorization">Integer factorization</a></div> Given a composite integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the task of providing one (or all) prime factors is referred to as factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. It is significantly more difficult than primality testing,<a href="#cite_note-156">[151]</a> and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and <a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho algorithm</a> can be used to find very small factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">,<a href="#cite_note-p._220-126">[124]</a> and <a href="/wiki/Elliptic_curve_factorization" class="mw-redirect" title="Elliptic curve factorization">elliptic curve factorization</a> can be effective when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> has factors of moderate size.<a href="#cite_note-157">[152]</a> Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the <a href="/wiki/Quadratic_sieve" title="Quadratic sieve">quadratic sieve</a> and <a href="/wiki/General_number_field_sieve" title="General number field sieve">general number field sieve</a>. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the <a href="/wiki/Special_number_field_sieve" title="Special number field sieve">special number field sieve</a>.<a href="#cite_note-158">[153]</a> As of December 2019<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Prime_number&action=edit">[update]</a> the <a href="/wiki/Integer_factorization_records" title="Integer factorization records">largest number known to have been factored</a> by a general-purpose algorithm is <a href="/wiki/RSA-240" class="mw-redirect" title="RSA-240">RSA-240</a>, which has 240 decimal digits (795 bits) and is the product of two large primes.<a href="#cite_note-159">[154]</a> <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a> can factor any integer in a polynomial number of steps on a <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computer</a>.<a href="#cite_note-160">[155]</a> However, current technology can only run this algorithm for very small numbers. As of October 2012<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Prime_number&action=edit">[update]</a>, the largest number that has been factored by a quantum computer running Shor's algorithm is 21.<a href="#cite_note-161">[156]</a> <div class="mw-heading mw-heading3"><h3 id="Other_computational_applications">Other computational applications</h3></div> Several <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a> algorithms, such as <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a> and the <a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman key exchange</a>, are based on large prime numbers (2048-<a href="/wiki/Bit" title="Bit">bit</a> primes are common).<a href="#cite_note-162">[157]</a> RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> than to calculate <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><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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> (assumed <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>) if only the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> is known.<a href="#cite_note-ent-7-35">[33]</a> The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for <a href="/wiki/Modular_exponentiation" title="Modular exponentiation">modular exponentiation</a> (computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{b}{\bmod {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{b}{\bmod {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdcfa653d33b1ef6fa21a5a5fc365f6a7e0dea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.855ex; height:2.676ex;" alt="{\displaystyle a^{b}{\bmod {c}}}">), while the reverse operation (the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a>) is thought to be a hard problem.<a href="#cite_note-163">[158]</a> Prime numbers are frequently used for <a href="/wiki/Hash_table" title="Hash table">hash tables</a>. For instance the original method of Carter and Wegman for <a href="/wiki/Universal_hashing" title="Universal hashing">universal hashing</a> was based on computing <a href="/wiki/Hash_function" title="Hash function">hash functions</a> by choosing random <a href="/wiki/Linear_function" title="Linear function">linear functions</a> modulo large prime numbers. Carter and Wegman generalized this method to <a href="/wiki/K-independent_hashing" title="K-independent hashing"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-independent hashing</a> by using higher-degree polynomials, again modulo large primes.<a href="#cite_note-164">[159]</a> As well as in the hash function, prime numbers are used for the hash table size in <a href="/wiki/Quadratic_probing" title="Quadratic probing">quadratic probing</a> based hash tables to ensure that the probe sequence covers the whole table.<a href="#cite_note-165">[160]</a> Some <a href="/wiki/Checksum" title="Checksum">checksum</a> methods are based on the mathematics of prime numbers. For instance the checksums used in <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">International Standard Book Numbers</a> are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits.<a href="#cite_note-166">[161]</a> Another checksum method, <a href="/wiki/Adler-32" title="Adler-32">Adler-32</a>, uses arithmetic modulo 65521, the largest prime number less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{16}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{16}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e0dd3c0e42794174d2dbcb9a3ee2c6d69299d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.039ex; height:2.676ex;" alt="{\displaystyle 2^{16}}">.<a href="#cite_note-167">[162]</a> Prime numbers are also used in <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a> including <a href="/wiki/Linear_congruential_generator" title="Linear congruential generator">linear congruential generators</a><a href="#cite_note-168">[163]</a> and the <a href="/wiki/Mersenne_Twister" title="Mersenne Twister">Mersenne Twister</a>.<a href="#cite_note-169">[164]</a> <div class="mw-heading mw-heading2"><h2 id="Other_applications">Other applications</h2></div> Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that <a href="/wiki/No-three-in-line_problem" title="No-three-in-line problem">no three are in a line</a>, or so that every triangle formed by three of the points <a href="/wiki/Heilbronn_triangle_problem" title="Heilbronn triangle problem">has large area</a>.<a href="#cite_note-170">[165]</a> Another example is <a href="/wiki/Eisenstein%27s_criterion" title="Eisenstein's criterion">Eisenstein's criterion</a>, a test for whether a <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">polynomial is irreducible</a> based on divisibility of its coefficients by a prime number and its square.<a href="#cite_note-171">[166]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sum_of_knots3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/220px-Sum_of_knots3.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/330px-Sum_of_knots3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/440px-Sum_of_knots3.svg.png 2x" data-file-width="300" data-file-height="120" /></a><figcaption>The connected sum of two prime knots</figcaption></figure> The concept of a prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the <a href="/wiki/Prime_field" class="mw-redirect" title="Prime field">prime field</a> of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a <a href="/wiki/Finite_field" title="Finite field">finite field</a> with a prime number of elements, whence the name.<a href="#cite_note-172">[167]</a> Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, a <a href="/wiki/Prime_knot" title="Prime knot">prime knot</a> is a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> that is indecomposable in the sense that it cannot be written as the <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.<a href="#cite_note-173">[168]</a> The <a href="/wiki/Prime_decomposition_(3-manifold)" class="mw-redirect" title="Prime decomposition (3-manifold)">prime decomposition of 3-manifolds</a> is another example of this type.<a href="#cite_note-174">[169]</a> Beyond mathematics and computing, prime numbers have potential connections to <a href="/wiki/Quantum_mechanic" class="mw-redirect" title="Quantum mechanic">quantum mechanics</a>, and have been used metaphorically in the arts and literature. They have also been used in <a href="/wiki/Evolutionary_biology" title="Evolutionary biology">evolutionary biology</a> to explain the life cycles of <a href="/wiki/Cicada" title="Cicada">cicadas</a>. <div class="mw-heading mw-heading3"><h3 id="Constructible_polygons_and_polygon_partitions">Constructible polygons and polygon partitions</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pentagon_construct.gif" class="mw-file-description"><img alt="Construction of a regular pentagon using straightedge and compass" src="//upload.wikimedia.org/wikipedia/commons/7/76/Pentagon_construct.gif" decoding="async" width="180" height="180" class="mw-file-element" data-file-width="180" data-file-height="180" /></a><figcaption>Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a <a href="/wiki/Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat prime</a>.</figcaption></figure> <a href="/wiki/Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat primes</a> are primes of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}=2^{2^{k}}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}=2^{2^{k}}+1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc6edcaa62bbd72af19a441f8cdba0629f37eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.408ex; height:3.343ex;" alt="{\displaystyle F_{k}=2^{2^{k}}+1,}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> a <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integer</a>.<a href="#cite_note-175">[170]</a> They are named after <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime,<a href="#cite_note-kls-176">[171]</a> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64647603e8358bf2b07099963d5ac2d8b75ee9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{5}}"> is composite and so are all other Fermat numbers that have been verified as of 2017.<a href="#cite_note-177">[172]</a> A <a href="/wiki/Regular_polygon" title="Regular polygon">regular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-gon</a> is <a href="/wiki/Constructible_polygon" title="Constructible polygon">constructible using straightedge and compass</a> if and only if the odd prime factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> (if any) are distinct Fermat primes.<a href="#cite_note-kls-176">[171]</a> Likewise, a regular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-gon may be constructed using straightedge, compass, and an <a href="/wiki/Angle_trisection" title="Angle trisection">angle trisector</a> if and only if the prime factors of <a href="/wiki/Regular_polygon" title="Regular polygon"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></a> are any number of copies of 2 or 3 together with a (possibly empty) set of distinct <a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont primes</a>, primes of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{a}3^{b}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{a}3^{b}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06f1cc50f358fed2f88c9789d2a15fc5c003dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.367ex; height:2.843ex;" alt="{\displaystyle 2^{a}3^{b}+1}">.<a href="#cite_note-178">[173]</a> It is possible to partition any convex polygon into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> smaller convex polygons of equal area and equal perimeter, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is a <a href="/wiki/Prime_power" title="Prime power">power of a prime number</a>, but this is not known for other values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-179">[174]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3></div> Beginning with the work of <a href="/wiki/Hugh_Lowell_Montgomery" title="Hugh Lowell Montgomery">Hugh Montgomery</a> and <a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Freeman Dyson</a> in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of <a href="/wiki/Quantum_system" class="mw-redirect" title="Quantum system">quantum systems</a>.<a href="#cite_note-180">[175]</a><a href="#cite_note-181">[176]</a> Prime numbers are also significant in <a href="/wiki/Quantum_information_science" title="Quantum information science">quantum information science</a>, thanks to mathematical structures such as <a href="/wiki/Mutually_unbiased_bases" title="Mutually unbiased bases">mutually unbiased bases</a> and <a href="/wiki/SIC-POVM" title="SIC-POVM">symmetric informationally complete positive-operator-valued measures</a>.<a href="#cite_note-182">[177]</a><a href="#cite_note-183">[178]</a> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3></div> The evolutionary strategy used by <a href="/wiki/Cicada" title="Cicada">cicadas</a> of the genus <a href="/wiki/Magicicada" class="mw-redirect" title="Magicicada">Magicicada</a> makes use of prime numbers.<a href="#cite_note-184">[179]</a> These insects spend most of their lives as <a href="/wiki/Larva" title="Larva">grubs</a> underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles.<a href="#cite_note-185">[180]</a><a href="#cite_note-186">[181]</a> In contrast, the multi-year periods between flowering in <a href="/wiki/Bamboo" title="Bamboo">bamboo</a> plants are hypothesized to be <a href="/wiki/Smooth_number" title="Smooth number">smooth numbers</a>, having only small prime numbers in their factorizations.<a href="#cite_note-187">[182]</a> <div class="mw-heading mw-heading3"><h3 id="Arts_and_literature">Arts and literature</h3></div> Prime numbers have influenced many artists and writers. The French <a href="/wiki/Composer" title="Composer">composer</a> <a href="/wiki/Olivier_Messiaen" title="Olivier Messiaen">Olivier Messiaen</a> used prime numbers to create ametrical music through "natural phenomena". In works such as <a href="/wiki/La_Nativit%C3%A9_du_Seigneur" title="La Nativité du Seigneur">La Nativité du Seigneur</a> (1935) and <a href="/wiki/Quatre_%C3%A9tudes_de_rythme" class="mw-redirect" title="Quatre études de rythme">Quatre études de rythme</a> (1949–1950), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".<a href="#cite_note-188">[183]</a> In his science fiction novel <a href="/wiki/Contact_(novel)" title="Contact (novel)">Contact</a>, scientist <a href="/wiki/Carl_Sagan" title="Carl Sagan">Carl Sagan</a> suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer <a href="/wiki/Frank_Drake" title="Frank Drake">Frank Drake</a> in 1975.<a href="#cite_note-189">[184]</a> In the novel <a href="/wiki/The_Curious_Incident_of_the_Dog_in_the_Night-Time" title="The Curious Incident of the Dog in the Night-Time">The Curious Incident of the Dog in the Night-Time</a> by <a href="/wiki/Mark_Haddon" title="Mark Haddon">Mark Haddon</a>, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with <a href="/wiki/Asperger_syndrome" title="Asperger syndrome">Asperger syndrome</a>.<a href="#cite_note-190">[185]</a> Prime numbers are used as a metaphor for loneliness and isolation in the <a href="/wiki/Paolo_Giordano" title="Paolo Giordano">Paolo Giordano</a> novel <a href="/wiki/The_Solitude_of_Prime_Numbers_(novel)" title="The Solitude of Prime Numbers (novel)">The Solitude of Prime Numbers</a>, in which they are portrayed as "outsiders" among integers.<a href="#cite_note-191">[186]</a> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-30"><a href="#cite_ref-30">^</a> A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.<a href="#cite_note-29">[29]</a> </li> <li id="cite_note-pure-34">^ <a href="#cite_ref-pure_34-0">a</a> <a href="#cite_ref-pure_34-1">b</a> For instance, Beiler writes that number theorist <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a> loved his <a href="/wiki/Ideal_number" title="Ideal number">ideal numbers</a>, closely related to the primes, "because they had not soiled themselves with any practical applications",<a href="#cite_note-32">[31]</a> and Katz writes that <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a>, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as <a href="/wiki/Geometry" title="Geometry">geometry</a> that had already shown themselves to be useful.<a href="#cite_note-33">[32]</a> </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> In this test, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"> term is negative if <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><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}"> is a square modulo the given (supposed) prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, and positive otherwise. More generally, for non-prime values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"> term is the (negated) <a href="/wiki/Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a>, which can be calculated using <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>. </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> Indeed, much of the analysis of elliptic curve primality proving is based on the assumption that the input to the algorithm has already passed a probabilistic test.<a href="#cite_note-atkin-morain-137">[134]</a> </li> <li id="cite_note-153"><a href="#cite_ref-153">^</a> The <a href="/wiki/Primorial" title="Primorial">primorial</a> function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\#}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167c3481839df4ace6689a25e170c5e0c0d5551e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle n\#}">, yields the product of the prime numbers up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, and a <a href="/wiki/Primorial_prime" title="Primorial prime">primorial prime</a> is a prime of one of the forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\#\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi mathvariant="normal">#</mi> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\#\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0d1d93846f485187cb017391c93d5cc2147d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.333ex; height:2.509ex;" alt="{\displaystyle n\#\pm 1}">.<a href="#cite_note-152">[148]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></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-GIMPS-2024-1">^ <a href="#cite_ref-GIMPS-2024_1-0">a</a> <a href="#cite_ref-GIMPS-2024_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mersenne.org/primes/?press=M136279841">"GIMPS Discovers Largest Known Prime Number: 2136,279,841 − 1"</a>. Mersenne Research, Inc. 21 October 2024. Retrieved 21 October 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mersenne+Research%2C+Inc.&rft.atitle=GIMPS+Discovers+Largest+Known+Prime+Number%3A+2%3Csup%3E136%2C279%2C841%3C%2Fsup%3E+%E2%88%92+1&rft.date=2024-10-21&rft_id=https%3A%2F%2Fwww.mersenne.org%2Fprimes%2F%3Fpress%3DM136279841&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSparkes2024" class="citation journal cs1">Sparkes, Matthew (November 2, 2024). "Amateur sleuth finds largest-known prime number". New Scientist: 19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Scientist&rft.atitle=Amateur+sleuth+finds+largest-known+prime+number&rft.pages=19&rft.date=2024-11-02&rft.aulast=Sparkes&rft.aufirst=Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardiner1997" class="citation book cs1"><a href="/wiki/Tony_Gardiner" title="Tony Gardiner">Gardiner, Anthony</a> (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalolym1997gard">The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965–1996</a>. Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalolym1997gard/page/26">26</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850105-3" title="Special:BookSources/978-0-19-850105-3"><bdi>978-0-19-850105-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Olympiad+Handbook%3A+An+Introduction+to+Problem+Solving+Based+on+the+First+32+British+Mathematical+Olympiads+1965%E2%80%931996&rft.pages=26&rft.pub=Oxford+University+Press&rft.date=1997&rft.isbn=978-0-19-850105-3&rft.aulast=Gardiner&rft.aufirst=Anthony&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalolym1997gard&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenderson2014" class="citation book cs1">Henderson, Anne (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uy-yGVRUilMC&pg=PA62">Dyslexia, Dyscalculia and Mathematics: A practical guide</a> (2nd ed.). Routledge. p. 62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-63662-2" title="Special:BookSources/978-1-136-63662-2"><bdi>978-1-136-63662-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dyslexia%2C+Dyscalculia+and+Mathematics%3A+A+practical+guide&rft.pages=62&rft.edition=2nd&rft.pub=Routledge&rft.date=2014&rft.isbn=978-1-136-63662-2&rft.aulast=Henderson&rft.aufirst=Anne&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Duy-yGVRUilMC%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdler1960" class="citation book cs1"><a href="/wiki/Irving_Adler" title="Irving Adler">Adler, Irving</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/giantgoldenbooko00adle">The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space</a>. Golden Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/giantgoldenbooko00adle/page/16">16</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/6975809">6975809</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Giant+Golden+Book+of+Mathematics%3A+Exploring+the+World+of+Numbers+and+Space&rft.pages=16&rft.pub=Golden+Press&rft.date=1960&rft_id=info%3Aoclcnum%2F6975809&rft.aulast=Adler&rft.aufirst=Irving&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgiantgoldenbooko00adle&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeff2000" class="citation book cs1">Leff, Lawrence S. 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Mathematical Association of America. p. 42. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-584-3" title="Special:BookSources/978-0-88385-584-3"><bdi>978-0-88385-584-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Euler+Did+Even+More&rft.pages=42&rft.pub=Mathematical+Association+of+America&rft.date=2014&rft.isbn=978-0-88385-584-3&rft.aulast=Sandifer&rft.aufirst=C.+Edward&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3c6iBQAAQBAJ%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoshy2002" class="citation book cs1">Koshy, Thomas (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA369">Elementary Number Theory with Applications</a>. Academic Press. p. 369. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-421171-1" title="Special:BookSources/978-0-12-421171-1"><bdi>978-0-12-421171-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory+with+Applications&rft.pages=369&rft.pub=Academic+Press&rft.date=2002&rft.isbn=978-0-12-421171-1&rft.aulast=Koshy&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-9pg-4Pa19IC%26pg%3DPA369&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYuan2002" class="citation book cs1">Yuan, Wang (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g4jVCgAAQBAJ&pg=PA21">Goldbach Conjecture</a>. Series In Pure Mathematics. Vol. 4 (2nd ed.). World Scientific. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4487-52-8" title="Special:BookSources/978-981-4487-52-8"><bdi>978-981-4487-52-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Goldbach+Conjecture&rft.series=Series+In+Pure+Mathematics&rft.pages=21&rft.edition=2nd&rft.pub=World+Scientific&rft.date=2002&rft.isbn=978-981-4487-52-8&rft.aulast=Yuan&rft.aufirst=Wang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg4jVCgAAQBAJ%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNarkiewicz2000" class="citation book cs1">Narkiewicz, Wladyslaw (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VVr3EuiHU0YC&pg=PA11">"1.2 Sum of Reciprocals of Primes"</a>. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics. Springer. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66289-1" title="Special:BookSources/978-3-540-66289-1"><bdi>978-3-540-66289-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.2+Sum+of+Reciprocals+of+Primes&rft.btitle=The+Development+of+Prime+Number+Theory%3A+From+Euclid+to+Hardy+and+Littlewood&rft.series=Springer+Monographs+in+Mathematics&rft.pages=11&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-540-66289-1&rft.aulast=Narkiewicz&rft.aufirst=Wladyslaw&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVVr3EuiHU0YC%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTchebychev1852" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Tchebychev, P.</a> (1852). <a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf">"Mémoire sur les nombres premiers"</a> (PDF). Journal de mathématiques pures et appliquées. Série 1 (in French): 366–390.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+math%C3%A9matiques+pures+et+appliqu%C3%A9es&rft.atitle=M%C3%A9moire+sur+les+nombres+premiers.&rft.pages=366-390&rft.date=1852&rft.aulast=Tchebychev&rft.aufirst=P.&rft_id=http%3A%2F%2Fsites.mathdoc.fr%2FJMPA%2FPDF%2FJMPA_1852_1_17_A19_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988">. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol2000" class="citation book cs1"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aiDyBwAAQBAJ&pg=PA1">"A centennial history of the prime number theorem"</a>. In Bambah, R.P.; Dumir, V.C.; Hans-Gill, R.J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 1–14. <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=1764793">1764793</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+centennial+history+of+the+prime+number+theorem&rft.btitle=Number+Theory&rft.place=Basel&rft.series=Trends+in+Mathematics&rft.pages=1-14&rft.pub=Birkh%C3%A4user&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1764793%23id-name%3DMR&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaiDyBwAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <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=3yoBCAAAQBAJ&pg=PA146">"7. Dirichlet's Theorem on Primes in Arithmetical Progressions"</a>. Introduction to Analytic Number Theory. New York; Heidelberg: Springer-Verlag. pp. 146–156. <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=0434929">0434929</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7.+Dirichlet%27s+Theorem+on+Primes+in+Arithmetical+Progressions&rft.btitle=Introduction+to+Analytic+Number+Theory&rft.place=New+York%3B+Heidelberg&rft.pages=146-156&rft.pub=Springer-Verlag&rft.date=1976&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0434929%23id-name%3DMR&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3yoBCAAAQBAJ%26pg%3DPA146&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChabert2012" class="citation book cs1">Chabert, Jean-Luc (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XcDqCAAAQBAJ&pg=PA261">A History of Algorithms: From the Pebble to the Microchip</a>. Springer. p. 261. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-18192-4" title="Special:BookSources/978-3-642-18192-4"><bdi>978-3-642-18192-4</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+Algorithms%3A+From+the+Pebble+to+the+Microchip&rft.pages=261&rft.pub=Springer&rft.date=2012&rft.isbn=978-3-642-18192-4&rft.aulast=Chabert&rft.aufirst=Jean-Luc&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXcDqCAAAQBAJ%26pg%3DPA261&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosen2000" class="citation book cs1">Rosen, Kenneth H. (2000). "Theorem 9.20. Proth's Primality Test". Elementary Number Theory and Its Applications (4th ed.). Addison-Wesley. p. 342. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-87073-2" title="Special:BookSources/978-0-201-87073-2"><bdi>978-0-201-87073-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theorem+9.20.+Proth%27s+Primality+Test&rft.btitle=Elementary+Number+Theory+and+Its+Applications&rft.pages=342&rft.edition=4th&rft.pub=Addison-Wesley&rft.date=2000&rft.isbn=978-0-201-87073-2&rft.aulast=Rosen&rft.aufirst=Kenneth+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooperHodges2016" class="citation book cs1">Cooper, S. Barry; Hodges, Andrew (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h12cCwAAQBAJ&pg=PA37">The Once and Future Turing</a>. Cambridge University Press. pp. 37–38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-01083-3" title="Special:BookSources/978-1-107-01083-3"><bdi>978-1-107-01083-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Once+and+Future+Turing&rft.pages=37-38&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=978-1-107-01083-3&rft.aulast=Cooper&rft.aufirst=S.+Barry&rft.au=Hodges%2C+Andrew&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh12cCwAAQBAJ%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <a href="#CITEREFRosen2000">Rosen 2000</a>, p. 245. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeiler1999" class="citation book cs1">Beiler, Albert H. (1999) [1966]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA2">Recreations in the Theory of Numbers: The Queen of Mathematics Entertains</a>. Dover. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-21096-4" title="Special:BookSources/978-0-486-21096-4"><bdi>978-0-486-21096-4</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/444171535">444171535</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recreations+in+the+Theory+of+Numbers%3A+The+Queen+of+Mathematics+Entertains&rft.pages=2&rft.pub=Dover&rft.date=1999&rft_id=info%3Aoclcnum%2F444171535&rft.isbn=978-0-486-21096-4&rft.aulast=Beiler&rft.aufirst=Albert+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNbbbL9gMJ88C%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2004" class="citation journal cs1">Katz, Shaul (2004). "Berlin roots – Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem". Science in Context. 17 (1–2): 199–234. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0269889704000092">10.1017/S0269889704000092</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=2089305">2089305</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:145575536">145575536</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+in+Context&rft.atitle=Berlin+roots+%E2%80%93+Zionist+incarnation%3A+the+ethos+of+pure+mathematics+and+the+beginnings+of+the+Einstein+Institute+of+Mathematics+at+the+Hebrew+University+of+Jerusalem&rft.volume=17&rft.issue=1%E2%80%932&rft.pages=199-234&rft.date=2004&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2089305%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A145575536%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0269889704000092&rft.aulast=Katz&rft.aufirst=Shaul&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-ent-7-35">^ <a href="#cite_ref-ent-7_35-0">a</a> <a href="#cite_ref-ent-7_35-1">b</a> <a href="#cite_ref-ent-7_35-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKraftWashington2014" class="citation book cs1">Kraft, James S.; Washington, Lawrence C. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA7">Elementary Number Theory</a>. Textbooks in mathematics. CRC Press. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-0269-0" title="Special:BookSources/978-1-4987-0269-0"><bdi>978-1-4987-0269-0</bdi></a>.</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.series=Textbooks+in+mathematics&rft.pages=7&rft.pub=CRC+Press&rft.date=2014&rft.isbn=978-1-4987-0269-0&rft.aulast=Kraft&rft.aufirst=James+S.&rft.au=Washington%2C+Lawrence+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4NAqBgAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBauer2013" class="citation book cs1">Bauer, Craig P. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA468">Secret History: The Story of Cryptology</a>. Discrete Mathematics and Its Applications. CRC Press. p. 468. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-6186-1" title="Special:BookSources/978-1-4665-6186-1"><bdi>978-1-4665-6186-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Secret+History%3A+The+Story+of+Cryptology&rft.series=Discrete+Mathematics+and+Its+Applications&rft.pages=468&rft.pub=CRC+Press&rft.date=2013&rft.isbn=978-1-4665-6186-1&rft.aulast=Bauer&rft.aufirst=Craig+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEBkEGAOlCDsC%26pg%3DPA468&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleeWagon1991" class="citation book cs1"><a href="/wiki/Victor_Klee" title="Victor Klee">Klee, Victor</a>; <a href="/wiki/Stan_Wagon" title="Stan Wagon">Wagon, Stan</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tRdoIhHh3moC&pg=PA224">Old and New Unsolved Problems in Plane Geometry and Number Theory</a>. Dolciani mathematical expositions. Vol. 11. Cambridge University Press. p. 224. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-315-3" title="Special:BookSources/978-0-88385-315-3"><bdi>978-0-88385-315-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Old+and+New+Unsolved+Problems+in+Plane+Geometry+and+Number+Theory&rft.series=Dolciani+mathematical+expositions&rft.pages=224&rft.pub=Cambridge+University+Press&rft.date=1991&rft.isbn=978-0-88385-315-3&rft.aulast=Klee&rft.aufirst=Victor&rft.au=Wagon%2C+Stan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtRdoIhHh3moC%26pg%3DPA224&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-neale-18-47-38">^ <a href="#cite_ref-neale-18-47_38-0">a</a> <a href="#cite_ref-neale-18-47_38-1">b</a> <a href="#CITEREFNeale2017">Neale 2017</a>, pp. 18, 47. </li> <li id="cite_note-crxk-34-39">^ <a href="#cite_ref-crxk-34_39-0">a</a> <a href="#cite_ref-crxk-34_39-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwellReddickXiongKeller2012" class="citation journal cs1">Caldwell, Chris K.; Reddick, Angela; Xiong, Yeng; Keller, Wilfrid (2012). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html">"The history of the primality of one: a selection of sources"</a>. <a href="/wiki/Journal_of_Integer_Sequences" title="Journal of Integer Sequences">Journal of Integer Sequences</a>. 15 (9): Article 12.9.8. <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=3005523">3005523</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+Integer+Sequences&rft.atitle=The+history+of+the+primality+of+one%3A+a+selection+of+sources&rft.volume=15&rft.issue=9&rft.pages=Article+12.9.8&rft.date=2012&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3005523%23id-name%3DMR&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft.au=Reddick%2C+Angela&rft.au=Xiong%2C+Yeng&rft.au=Keller%2C+Wilfrid&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL15%2FCaldwell2%2Fcald6.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> For a selection of quotes from and about the ancient Greek positions on the status of 1 and 2, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6. </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarán1981" class="citation book cs1">Tarán, Leonardo (1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cUPXqSb7V1wC&pg=PA35">Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary</a>. Philosophia Antiqua : A Series of Monographs on Ancient Philosophy. Vol. 39. Brill. pp. 35–38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-04-06505-5" title="Special:BookSources/978-90-04-06505-5"><bdi>978-90-04-06505-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Speusippus+of+Athens%3A+A+Critical+Study+With+a+Collection+of+the+Related+Texts+and+Commentary&rft.series=Philosophia+Antiqua+%3A+A+Series+of+Monographs+on+Ancient+Philosophy&rft.pages=35-38&rft.pub=Brill&rft.date=1981&rft.isbn=978-90-04-06505-5&rft.aulast=Tar%C3%A1n&rft.aufirst=Leonardo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcUPXqSb7V1wC%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <a href="#CITEREFCaldwellReddickXiongKeller2012">Caldwell et al. 2012</a>, pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet. </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="#CITEREFCaldwellReddickXiongKeller2012">Caldwell et al. 2012</a>, pp. 6–7. </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <a href="#CITEREFCaldwellReddickXiongKeller2012">Caldwell et al. 2012</a>, p. 15. </li> <li id="cite_note-cx-44">^ <a href="#cite_ref-cx_44-0">a</a> <a href="#cite_ref-cx_44-1">b</a> <a href="#cite_ref-cx_44-2">c</a> <a href="#cite_ref-cx_44-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwellXiong2012" class="citation journal cs1">Caldwell, Chris K.; Xiong, Yeng (2012). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf">"What is the smallest prime?"</a> (PDF). <a href="/wiki/Journal_of_Integer_Sequences" title="Journal of Integer Sequences">Journal of Integer Sequences</a>. 15 (9): Article 12.9.7. <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=3005530">3005530</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+Integer+Sequences&rft.atitle=What+is+the+smallest+prime%3F&rft.volume=15&rft.issue=9&rft.pages=Article+12.9.7&rft.date=2012&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3005530%23id-name%3DMR&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft.au=Xiong%2C+Yeng&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL15%2FCaldwell1%2Fcald5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEConwayGuy1996130-45"><a href="#cite_ref-FOOTNOTEConwayGuy1996130_45-0">^</a> <a href="#CITEREFConwayGuy1996">Conway & Guy 1996</a>, pp. 130. </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiesel1994" class="citation book cs1"><a href="/wiki/Hans_Riesel" title="Hans Riesel">Riesel, Hans</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA36">Prime Numbers and Computer Methods for Factorization</a> (2nd ed.). Basel, Switzerland: Birkhäuser. p. 36. <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%2F978-1-4612-0251-6">10.1007/978-1-4612-0251-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3743-9" title="Special:BookSources/978-0-8176-3743-9"><bdi>978-0-8176-3743-9</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=1292250">1292250</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers+and+Computer+Methods+for+Factorization&rft.place=Basel%2C+Switzerland&rft.pages=36&rft.edition=2nd&rft.pub=Birkh%C3%A4user&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1292250%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0251-6&rft.isbn=978-0-8176-3743-9&rft.aulast=Riesel&rft.aufirst=Hans&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DITvaBwAAQBAJ%26pg%3DPA36&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-cg-bon-129-130-47">^ <a href="#cite_ref-cg-bon-129-130_47-0">a</a> <a href="#cite_ref-cg-bon-129-130_47-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayGuy1996" class="citation book cs1"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John Horton</a>; <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw">The Book of Numbers</a>. New York: Copernicus. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw/page/129">129–130</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-4072-3">10.1007/978-1-4612-4072-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97993-9" title="Special:BookSources/978-0-387-97993-9"><bdi>978-0-387-97993-9</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=1411676">1411676</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Book+of+Numbers&rft.place=New+York&rft.pages=129-130&rft.pub=Copernicus&rft.date=1996&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1411676%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-4072-3&rft.isbn=978-0-387-97993-9&rft.aulast=Conway&rft.aufirst=John+Horton&rft.au=Guy%2C+Richard+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbookofnumbers0000conw&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> For the totient, see <a href="#CITEREFSierpiński1988">Sierpiński 1988</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA245">p. 245</a>. For the sum of divisors, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSandifer2007" class="citation book cs1">Sandifer, C. Edward (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sohHs7ExOsYC&pg=PA59">How Euler Did It</a>. MAA Spectrum. Mathematical Association of America. p. 59. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-563-8" title="Special:BookSources/978-0-88385-563-8"><bdi>978-0-88385-563-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Euler+Did+It&rft.series=MAA+Spectrum&rft.pages=59&rft.pub=Mathematical+Association+of+America&rft.date=2007&rft.isbn=978-0-88385-563-8&rft.aulast=Sandifer&rft.aufirst=C.+Edward&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsohHs7ExOsYC%26pg%3DPA59&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2011" class="citation book cs1">Smith, Karl J. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Di0HyCgDYq8C&pg=PA188">The Nature of Mathematics</a> (12th ed.). Cengage Learning. p. 188. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-538-73758-6" title="Special:BookSources/978-0-538-73758-6"><bdi>978-0-538-73758-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+of+Mathematics&rft.pages=188&rft.edition=12th&rft.pub=Cengage+Learning&rft.date=2011&rft.isbn=978-0-538-73758-6&rft.aulast=Smith&rft.aufirst=Karl+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDi0HyCgDYq8C%26pg%3DPA188&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a href="#CITEREFDudley1978">Dudley 1978</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA16">Section 2, Theorem 2, p. 16</a>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeale2017" class="citation book cs1"><a href="/wiki/Vicky_Neale" title="Vicky Neale">Neale, Vicky</a> (2017). <a href="/wiki/Closing_the_Gap:_The_Quest_to_Understand_Prime_Numbers" title="Closing the Gap: The Quest to Understand Prime Numbers">Closing the Gap: The Quest to Understand Prime Numbers</a>. Oxford University Press. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T7Q1DwAAQBAJ&pg=PA107">p. 107</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-109243-5" title="Special:BookSources/978-0-19-109243-5"><bdi>978-0-19-109243-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Closing+the+Gap%3A+The+Quest+to+Understand+Prime+Numbers&rft.pages=p.+107&rft.pub=Oxford+University+Press&rft.date=2017&rft.isbn=978-0-19-109243-5&rft.aulast=Neale&rft.aufirst=Vicky&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFdu_Sautoy2003" class="citation book cs1"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a> (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/musicofprimessea00dusa">The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics</a>. Harper Collins. p. <a rel="nofollow" class="external text" href="https://archive.org/details/musicofprimessea00dusa/page/23">23</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-06-093558-0" title="Special:BookSources/978-0-06-093558-0"><bdi>978-0-06-093558-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Music+of+the+Primes%3A+Searching+to+Solve+the+Greatest+Mystery+in+Mathematics&rft.pages=23&rft.pub=Harper+Collins&rft.date=2003&rft.isbn=978-0-06-093558-0&rft.aulast=du+Sautoy&rft.aufirst=Marcus&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmusicofprimessea00dusa&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <a href="#CITEREFDudley1978">Dudley 1978</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA15">Section 2, Lemma 5, p. 15</a>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHiggins1998" class="citation book cs1">Higgins, Peter M. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LeYH8P8S9oQC&pg=PA77">Mathematics for the Curious</a>. Oxford University Press. pp. 77–78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-150050-3" title="Special:BookSources/978-0-19-150050-3"><bdi>978-0-19-150050-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+the+Curious&rft.pages=77-78&rft.pub=Oxford+University+Press&rft.date=1998&rft.isbn=978-0-19-150050-3&rft.aulast=Higgins&rft.aufirst=Peter+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLeYH8P8S9oQC%26pg%3DPA77&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman2000" class="citation book cs1">Rotman, Joseph J. (2000). A First Course in Abstract Algebra (2nd ed.). Prentice Hall. Problem 1.40, p. 56. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-011584-3" title="Special:BookSources/978-0-13-011584-3"><bdi>978-0-13-011584-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Abstract+Algebra&rft.pages=Problem+1.40%2C+p.+56&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=2000&rft.isbn=978-0-13-011584-3&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <a rel="nofollow" class="external text" href="http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0722.pdf">Letter</a> in <a href="/wiki/Latin" title="Latin">Latin</a> from Goldbach to Euler, July 1730. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFurstenberg1955" class="citation journal cs1"><a href="/wiki/Hillel_Furstenberg" title="Hillel Furstenberg">Furstenberg, Harry</a> (1955). "On the infinitude of primes". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 62 (5): 353. <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%2F2307043">10.2307/2307043</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/2307043">2307043</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=0068566">0068566</a>.</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=On+the+infinitude+of+primes&rft.volume=62&rft.issue=5&rft.pages=353&rft.date=1955&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0068566%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2307043%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2307043&rft.aulast=Furstenberg&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim2004" class="citation book cs1"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SvnTBwAAQBAJ&pg=PA5">The little book of bigger primes</a>. Berlin; New York: Springer-Verlag. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20169-6" title="Special:BookSources/978-0-387-20169-6"><bdi>978-0-387-20169-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+little+book+of+bigger+primes&rft.place=Berlin%3B+New+York&rft.pages=4&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-0-387-20169-6&rft.aulast=Ribenboim&rft.aufirst=Paulo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSvnTBwAAQBAJ%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, Book IX, Proposition 20. See <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html">David Joyce's English translation of Euclid's proof</a> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliamson1782" class="citation book cs1">Williamson, James (1782). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=umn.31951000084215o;view=1up;seq=95">The Elements of Euclid, With Dissertations</a>. Oxford: <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. p. 63. <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/642232959">642232959</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elements+of+Euclid%2C+With+Dissertations&rft.place=Oxford&rft.pages=63&rft.pub=Clarendon+Press&rft.date=1782&rft_id=info%3Aoclcnum%2F642232959&rft.aulast=Williamson&rft.aufirst=James&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dumn.31951000084215o%3Bview%3D1up%3Bseq%3D95&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVardi1991" class="citation book cs1">Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-52989-0" title="Special:BookSources/978-0-201-52989-0"><bdi>978-0-201-52989-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Recreations+in+Mathematica&rft.pages=82-89&rft.pub=Addison-Wesley&rft.date=1991&rft.isbn=978-0-201-52989-0&rft.aulast=Vardi&rft.aufirst=Ilan&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-matiyasevich-59">^ <a href="#cite_ref-matiyasevich_59-0">a</a> <a href="#cite_ref-matiyasevich_59-1">b</a> <a href="#cite_ref-matiyasevich_59-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatiyasevich1999" class="citation book cs1"><a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Matiyasevich, Yuri V.</a> (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oLKlk5o6WroC&pg=PA13">"Formulas for prime numbers"</a>. In <a href="/wiki/Sergei_Tabachnikov" title="Sergei Tabachnikov">Tabachnikov, Serge</a> (ed.). Kvant Selecta: Algebra and Analysis. Vol. II. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. pp. 13–24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1915-9" title="Special:BookSources/978-0-8218-1915-9"><bdi>978-0-8218-1915-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Formulas+for+prime+numbers&rft.btitle=Kvant+Selecta%3A+Algebra+and+Analysis&rft.pages=13-24&rft.pub=American+Mathematical+Society&rft.date=1999&rft.isbn=978-0-8218-1915-9&rft.aulast=Matiyasevich&rft.aufirst=Yuri+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoLKlk5o6WroC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMackinnon1987" class="citation journal cs1">Mackinnon, Nick (June 1987). "Prime number formulae". <a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 71 (456): 113–114. <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%2F3616496">10.2307/3616496</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/3616496">3616496</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:171537609">171537609</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Prime+number+formulae&rft.volume=71&rft.issue=456&rft.pages=113-114&rft.date=1987-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A171537609%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3616496%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3616496&rft.aulast=Mackinnon&rft.aufirst=Nick&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWright1951" class="citation journal cs1"><a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E.M.</a> (1951). "A prime-representing function". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 58 (9): 616–618. <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%2F2306356">10.2307/2306356</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/2306356">2306356</a>.</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=A+prime-representing+function&rft.volume=58&rft.issue=9&rft.pages=616-618&rft.date=1951&rft_id=info%3Adoi%2F10.2307%2F2306356&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2306356%23id-name%3DJSTOR&rft.aulast=Wright&rft.aufirst=E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <a href="#CITEREFGuy2013">Guy 2013</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7">p. vii</a>. </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFGuy2013">Guy 2013</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PA105">C1 Goldbach's conjecture, pp. 105–107</a>. </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOliveira_e_SilvaHerzogPardi2014" class="citation journal cs1">Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2013-02787-1">"Empirical verification of the even Goldbach conjecture and computation of prime gaps up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot 10^{18}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 10^{18}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddafb9809111d88d71982711a1ec9cde3e62e901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.043ex; height:2.676ex;" alt="{\displaystyle 4\cdot 10^{18}}">"</a>. <a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a>. 83 (288): 2033–2060. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2013-02787-1">10.1090/S0025-5718-2013-02787-1</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=3194140">3194140</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Empirical+verification+of+the+even+Goldbach+conjecture+and+computation+of+prime+gaps+up+to+MATH+RENDER+ERROR&rft.volume=83&rft.issue=288&rft.pages=2033-2060&rft.date=2014&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-2013-02787-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3194140%23id-name%3DMR&rft.aulast=Oliveira+e+Silva&rft.aufirst=Tom%C3%A1s&rft.au=Herzog%2C+Siegfried&rft.au=Pardi%2C+Silvio&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0025-5718-2013-02787-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <a href="#CITEREFTao2009">Tao 2009</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NxnVAwAAQBAJ&pg=PA239">3.1 Structure and randomness in the prime numbers, pp. 239–247</a>. See especially p. 239. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFGuy2013">Guy 2013</a>, p. 159. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamaré1995" class="citation journal cs1">Ramaré, Olivier (1995). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220209175544/http://www.numdam.org/item/?id=ASNSP_1995_4_22_4_645_0">"On Šnirel'man's constant"</a>. Annali della Scuola Normale Superiore di Pisa. 22 (4): 645–706. <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=1375315">1375315</a>. Archived from <a rel="nofollow" class="external text" href="https://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0">the original</a> on 2022-02-09. Retrieved 2018-01-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annali+della+Scuola+Normale+Superiore+di+Pisa&rft.atitle=On+%C5%A0nirel%27man%27s+constant&rft.volume=22&rft.issue=4&rft.pages=645-706&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1375315%23id-name%3DMR&rft.aulast=Ramar%C3%A9&rft.aufirst=Olivier&rft_id=https%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASNSP_1995_4_22_4_645_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRassias2017" class="citation book cs1">Rassias, Michael Th. (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ibwpDwAAQBAJ&pg=PP6">Goldbach's Problem: Selected Topics</a>. Cham: Springer. p. vii. <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%2F978-3-319-57914-6">10.1007/978-3-319-57914-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-57912-2" title="Special:BookSources/978-3-319-57912-2"><bdi>978-3-319-57912-2</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=3674356">3674356</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Goldbach%27s+Problem%3A+Selected+Topics&rft.place=Cham&rft.pages=vii&rft.pub=Springer&rft.date=2017&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3674356%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-319-57914-6&rft.isbn=978-3-319-57912-2&rft.aulast=Rassias&rft.aufirst=Michael+Th.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DibwpDwAAQBAJ%26pg%3DPP6&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a href="#CITEREFKoshy2002">Koshy 2002</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA109">Theorem 2.14, p. 109</a>. <a href="#CITEREFRiesel1994">Riesel 1994</a> gives a similar argument using the <a href="/wiki/Primorial" title="Primorial">primorial</a> in place of the factorial. </li> <li id="cite_note-riesel-gaps-70">^ <a href="#cite_ref-riesel-gaps_70-0">a</a> <a href="#cite_ref-riesel-gaps_70-1">b</a> <a href="#CITEREFRiesel1994">Riesel 1994</a>, "<a rel="nofollow" class="external text" href="https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA78">Large gaps between consecutive primes</a>", pp. 78–79. </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A100964"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A100964">"Sequence A100964 (Smallest prime number that begins a prime gap of at least 2n)"</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA100964%26%23x20%3B%28Smallest+prime+number+that+begins+a+prime+gap+of+at+least+2n%29&rft_id=https%3A%2F%2Foeis.org%2FA100964&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-rib-gaps-72">^ <a href="#cite_ref-rib-gaps_72-0">a</a> <a href="#cite_ref-rib-gaps_72-1">b</a> <a href="#cite_ref-rib-gaps_72-2">c</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, Gaps between primes, pp. 186–192. </li> <li id="cite_note-rib-183-73">^ <a href="#cite_ref-rib-183_73-0">a</a> <a href="#cite_ref-rib-183_73-1">b</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, p. 183. </li> <li id="cite_note-chan-74"><a href="#cite_ref-chan_74-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChan1996" class="citation journal cs1">Chan, Joel (February 1996). "Prime time!". Math Horizons. 3 (3): 23–25. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10724117.1996.11974965">10.1080/10724117.1996.11974965</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/25678057">25678057</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math+Horizons&rft.atitle=Prime+time%21&rft.volume=3&rft.issue=3&rft.pages=23-25&rft.date=1996-02&rft_id=info%3Adoi%2F10.1080%2F10724117.1996.11974965&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25678057%23id-name%3DJSTOR&rft.aulast=Chan&rft.aufirst=Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate". </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, Prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-tuples conjecture, pp. 201–202. </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <a href="#CITEREFSandifer2007">Sandifer 2007</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sohHs7ExOsYC&pg=PA205">Chapter 35, Estimating the Basel problem, pp. 205–208</a>. </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOgilvyAnderson1988" class="citation book cs1"><a href="/wiki/C._Stanley_Ogilvy" title="C. Stanley Ogilvy">Ogilvy, C.S.</a>; Anderson, J.T. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=efbaDLlTXvMC&pg=PA29">Excursions in Number Theory</a>. Dover Publications Inc. pp. 29–35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-25778-5" title="Special:BookSources/978-0-486-25778-5"><bdi>978-0-486-25778-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Excursions+in+Number+Theory&rft.pages=29-35&rft.pub=Dover+Publications+Inc.&rft.date=1988&rft.isbn=978-0-486-25778-5&rft.aulast=Ogilvy&rft.aufirst=C.S.&rft.au=Anderson%2C+J.T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DefbaDLlTXvMC%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <a href="#CITEREFApostol1976">Apostol 1976</a>, Section 1.6, Theorem 1.13 </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <a href="#CITEREFApostol1976">Apostol 1976</a>, Section 4.8, Theorem 4.12 </li> <li id="cite_note-mtb-invitation-80">^ <a href="#cite_ref-mtb-invitation_80-0">a</a> <a href="#cite_ref-mtb-invitation_80-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillerTakloo-Bighash2006" class="citation book cs1">Miller, Steven J.; Takloo-Bighash, Ramin (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kLz4z8iwKiwC&pg=PA43">An Invitation to Modern Number Theory</a>. Princeton University Press. pp. 43–44. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12060-7" title="Special:BookSources/978-0-691-12060-7"><bdi>978-0-691-12060-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Invitation+to+Modern+Number+Theory&rft.pages=43-44&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0-691-12060-7&rft.aulast=Miller&rft.aufirst=Steven+J.&rft.au=Takloo-Bighash%2C+Ramin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkLz4z8iwKiwC%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <a href="#CITEREFCrandallPomerance2005">Crandall & Pomerance 2005</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA6">p. 6</a>. </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <a href="#CITEREFCrandallPomerance2005">Crandall & Pomerance 2005</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA152">Section 3.7, Counting primes, pp. 152–162</a>. </li> <li id="cite_note-cranpom10-83">^ <a href="#cite_ref-cranpom10_83-0">a</a> <a href="#cite_ref-cranpom10_83-1">b</a> <a href="#CITEREFCrandallPomerance2005">Crandall & Pomerance 2005</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA10">p. 10</a>. </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFdu_Sautoy2011" class="citation book cs1"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a> (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=snaUbkIb8SEC&pg=PA50">"What are the odds that your telephone number is prime?"</a>. The Number Mysteries: A Mathematical Odyssey through Everyday Life. St. Martin's Press. pp. 50–52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-230-12028-0" title="Special:BookSources/978-0-230-12028-0"><bdi>978-0-230-12028-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+are+the+odds+that+your+telephone+number+is+prime%3F&rft.btitle=The+Number+Mysteries%3A+A+Mathematical+Odyssey+through+Everyday+Life&rft.pages=50-52&rft.pub=St.+Martin%27s+Press&rft.date=2011&rft.isbn=978-0-230-12028-0&rft.aulast=du+Sautoy&rft.aufirst=Marcus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsnaUbkIb8SEC%26pg%3DPA50&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <a href="#CITEREFApostol1976">Apostol 1976</a>, Section 4.6, Theorem 4.7 </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandShen2003" class="citation book cs1"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, Israel M.</a>; Shen, Alexander (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA37">Algebra</a>. Springer. p. 37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3677-7" title="Special:BookSources/978-0-8176-3677-7"><bdi>978-0-8176-3677-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=37&rft.pub=Springer&rft.date=2003&rft.isbn=978-0-8176-3677-7&rft.aulast=Gelfand&rft.aufirst=Israel+M.&rft.au=Shen%2C+Alexander&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZ9z7iliyFD0C%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMollin1997" class="citation book cs1">Mollin, Richard A. (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fsaa3MUUQYkC&pg=PA76">Fundamental Number Theory with Applications</a>. Discrete Mathematics and Its Applications. CRC Press. p. 76. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-3987-5" title="Special:BookSources/978-0-8493-3987-5"><bdi>978-0-8493-3987-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Number+Theory+with+Applications&rft.series=Discrete+Mathematics+and+Its+Applications&rft.pages=76&rft.pub=CRC+Press&rft.date=1997&rft.isbn=978-0-8493-3987-5&rft.aulast=Mollin&rft.aufirst=Richard+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFsaa3MUUQYkC%26pg%3DPA76&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> <a href="#CITEREFCrandallPomerance2005">Crandall & Pomerance 2005</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA">Theorem 1.1.5, p. 12</a>. </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenTao2008" class="citation journal cs1"><a href="/wiki/Ben_J._Green" class="mw-redirect" title="Ben J. Green">Green, Ben</a>; <a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> (2008). "The primes contain arbitrarily long arithmetic progressions". <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. 167 (2): 481–547. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.NT/0404188">math.NT/0404188</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2008.167.481">10.4007/annals.2008.167.481</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:1883951">1883951</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=The+primes+contain+arbitrarily+long+arithmetic+progressions&rft.volume=167&rft.issue=2&rft.pages=481-547&rft.date=2008&rft_id=info%3Aarxiv%2Fmath.NT%2F0404188&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1883951%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4007%2Fannals.2008.167.481&rft.aulast=Green&rft.aufirst=Ben&rft.au=Tao%2C+Terence&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHua2009" class="citation book cs1">Hua, L. K. (2009) [1965]. Additive Theory of Prime Numbers. Translations of Mathematical Monographs. Vol. 13. Providence, RI: American Mathematical Society. pp. 176–177. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4942-2" title="Special:BookSources/978-0-8218-4942-2"><bdi>978-0-8218-4942-2</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=0194404">0194404</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/824812353">824812353</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Additive+Theory+of+Prime+Numbers&rft.place=Providence%2C+RI&rft.series=Translations+of+Mathematical+Monographs&rft.pages=176-177&rft.pub=American+Mathematical+Society&rft.date=2009&rft_id=info%3Aoclcnum%2F824812353&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0194404%23id-name%3DMR&rft.isbn=978-0-8218-4942-2&rft.aulast=Hua&rft.aufirst=L.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> The sequence of these primes, starting at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" 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=1}"> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" 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=0}">, is listed by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLavaBalzarotti2010" class="citation book cs1 cs1-prop-foreign-lang-source">Lava, Paolo Pietro; Balzarotti, Giorgio (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YfsSAAAAQBAJ&pg=PA133">"Chapter 33. Formule fortunate"</a>. 103 curiosità matematiche: Teoria dei numeri, delle cifre e delle relazioni nella matematica contemporanea (in Italian). Ulrico Hoepli Editore S.p.A. p. 133. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-88-203-5804-4" title="Special:BookSources/978-88-203-5804-4"><bdi>978-88-203-5804-4</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+33.+Formule+fortunate&rft.btitle=103+curiosit%C3%A0+matematiche%3A+Teoria+dei+numeri%2C+delle+cifre+e+delle+relazioni+nella+matematica+contemporanea&rft.pages=133&rft.pub=Ulrico+Hoepli+Editore+S.p.A.&rft.date=2010&rft.isbn=978-88-203-5804-4&rft.aulast=Lava&rft.aufirst=Paolo+Pietro&rft.au=Balzarotti%2C+Giorgio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYfsSAAAAQBAJ%26pg%3DPA133&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChamberland2015" class="citation book cs1">Chamberland, Marc (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=n9iqBwAAQBAJ&pg=PA213">"The Heegner numbers"</a>. Single Digits: In Praise of Small Numbers. Princeton University Press. pp. 213–215. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-6569-7" title="Special:BookSources/978-1-4008-6569-7"><bdi>978-1-4008-6569-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Heegner+numbers&rft.btitle=Single+Digits%3A+In+Praise+of+Small+Numbers&rft.pages=213-215&rft.pub=Princeton+University+Press&rft.date=2015&rft.isbn=978-1-4008-6569-7&rft.aulast=Chamberland&rft.aufirst=Marc&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dn9iqBwAAQBAJ%26pg%3DPA213&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-guy-a1-93">^ <a href="#cite_ref-guy-a1_93-0">a</a> <a href="#cite_ref-guy-a1_93-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2013" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA7">"A1 Prime values of quadratic functions"</a>. Unsolved Problems in Number Theory. Problem Books in Mathematics (3rd ed.). Springer. pp. 7–10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-26677-0" title="Special:BookSources/978-0-387-26677-0"><bdi>978-0-387-26677-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A1+Prime+values+of+quadratic+functions&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.series=Problem+Books+in+Mathematics&rft.pages=7-10&rft.edition=3rd&rft.pub=Springer&rft.date=2013&rft.isbn=978-0-387-26677-0&rft.aulast=Guy&rft.aufirst=Richard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1BnoBwAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatterson1988" class="citation book cs1">Patterson, S. J. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IdHLCgAAQBAJ&pg=PA1">An introduction to the theory of the Riemann zeta-function</a>. Cambridge Studies in Advanced Mathematics. Vol. 14. Cambridge University Press, Cambridge. p. 1. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511623707">10.1017/CBO9780511623707</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-33535-5" title="Special:BookSources/978-0-521-33535-5"><bdi>978-0-521-33535-5</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=0933558">0933558</a>.</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+the+Riemann+zeta-function&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pages=1&rft.pub=Cambridge+University+Press%2C+Cambridge&rft.date=1988&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D933558%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511623707&rft.isbn=978-0-521-33535-5&rft.aulast=Patterson&rft.aufirst=S.+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIdHLCgAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinChoiRooneyWeirathmueller2008" class="citation book cs1"><a href="/wiki/Peter_Borwein" title="Peter Borwein">Borwein, Peter</a>; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA10">The Riemann hypothesis: A resource for the afficionado and virtuoso alike</a>. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York: Springer. pp. 10–11. <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%2F978-0-387-72126-2">10.1007/978-0-387-72126-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72125-5" title="Special:BookSources/978-0-387-72125-5"><bdi>978-0-387-72125-5</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=2463715">2463715</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Riemann+hypothesis%3A+A+resource+for+the+afficionado+and+virtuoso+alike&rft.place=New+York&rft.series=CMS+Books+in+Mathematics%2FOuvrages+de+Math%C3%A9matiques+de+la+SMC&rft.pages=10-11&rft.pub=Springer&rft.date=2008&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2463715%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-72126-2&rft.isbn=978-0-387-72125-5&rft.aulast=Borwein&rft.aufirst=Peter&rft.au=Choi%2C+Stephen&rft.au=Rooney%2C+Brendan&rft.au=Weirathmueller%2C+Andrea&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQm1aZA-UwX4C%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> <a href="#CITEREFSandifer2007">Sandifer 2007</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sohHs7ExOsYC&pg=PA191">pp. 191–193</a>. </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> <a href="#CITEREFBorweinChoiRooneyWeirathmueller2008">Borwein et al. 2008</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA15">Conjecture 2.7 (the Riemann hypothesis), p. 15</a>. </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <a href="#CITEREFPatterson1988">Patterson 1988</a>, p. 7. </li> <li id="cite_note-bcrw18-99">^ <a href="#cite_ref-bcrw18_99-0">a</a> <a href="#cite_ref-bcrw18_99-1">b</a> <a href="#CITEREFBorweinChoiRooneyWeirathmueller2008">Borwein et al. 2008</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA18">p. 18.</a> </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <a href="#CITEREFNathanson2000">Nathanson 2000</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PA289">Chapter 9, The prime number theorem, pp. 289–324</a>. </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZagier1977" class="citation journal cs1"><a href="/wiki/Don_Zagier" title="Don Zagier">Zagier, Don</a> (1977). "The first 50 million prime numbers". <a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a>. 1 (S2): 7–19. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf03351556">10.1007/bf03351556</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:37866599">37866599</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=The+first+50+million+prime+numbers&rft.volume=1&rft.issue=S2&rft.pages=7-19&rft.date=1977&rft_id=info%3Adoi%2F10.1007%2Fbf03351556&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A37866599%23id-name%3DS2CID&rft.aulast=Zagier&rft.aufirst=Don&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> See especially pp. 14–16. </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <a href="#CITEREFKraftWashington2014">Kraft & Washington (2014)</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VG9YBQAAQBAJ&pg=PA96">Proposition 5.3</a>, p. 96. </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShahriari2017" class="citation book cs1"><a href="/wiki/Shahriar_Shahriari" title="Shahriar Shahriari">Shahriari, Shahriar</a> (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20">Algebra in Action: A Course in Groups, Rings, and Fields</a>. Pure and Applied Undergraduate Texts. Vol. 27. American Mathematical Society. pp. 20–21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-2849-5" title="Special:BookSources/978-1-4704-2849-5"><bdi>978-1-4704-2849-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+in+Action%3A+A+Course+in+Groups%2C+Rings%2C+and+Fields&rft.series=Pure+and+Applied+Undergraduate+Texts&rft.pages=20-21&rft.pub=American+Mathematical+Society&rft.date=2017&rft.isbn=978-1-4704-2849-5&rft.aulast=Shahriari&rft.aufirst=Shahriar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGJwxDwAAQBAJ%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <a href="#CITEREFDudley1978">Dudley 1978</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28">Theorem 3, p. 28</a>. </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <a href="#CITEREFShahriari2017">Shahriari 2017</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27">pp. 27–28</a>. </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21. </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, The property of Giuga, pp. 21–22. </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, The theorem of Wilson, p. 21. </li> <li id="cite_note-childress-109">^ <a href="#cite_ref-childress_109-0">a</a> <a href="#cite_ref-childress_109-1">b</a> <a href="#cite_ref-childress_109-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChildress2009" class="citation book cs1">Childress, Nancy (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RYdy4PCJYosC&pg=PA8">Class Field Theory</a>. Universitext. Springer, New York. pp. 8–11. <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%2F978-0-387-72490-4">10.1007/978-0-387-72490-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72489-8" title="Special:BookSources/978-0-387-72489-8"><bdi>978-0-387-72489-8</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=2462595">2462595</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Class+Field+Theory&rft.series=Universitext&rft.pages=8-11&rft.pub=Springer%2C+New+York&rft.date=2009&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2462595%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-72490-4&rft.isbn=978-0-387-72489-8&rft.aulast=Childress&rft.aufirst=Nancy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRYdy4PCJYosC%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> See also p. 64. </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEricksonVazzanaGarth2016" class="citation book cs1">Erickson, Marty; Vazzana, Anthony; Garth, David (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QpLwCgAAQBAJ&pg=PA200">Introduction to Number Theory</a>. Textbooks in Mathematics (2nd ed.). Boca Raton, FL: CRC Press. p. 200. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-1749-6" title="Special:BookSources/978-1-4987-1749-6"><bdi>978-1-4987-1749-6</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=3468748">3468748</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Number+Theory&rft.place=Boca+Raton%2C+FL&rft.series=Textbooks+in+Mathematics&rft.pages=200&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2016&rft.isbn=978-1-4987-1749-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3468748%23id-name%3DMR&rft.aulast=Erickson&rft.aufirst=Marty&rft.au=Vazzana%2C+Anthony&rft.au=Garth%2C+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQpLwCgAAQBAJ%26pg%3DPA200&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeil1995" class="citation book cs1"><a href="/wiki/Andr%C3%A9_Weil" title="André Weil">Weil, André</a> (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/basicnumbertheor00weil_866">Basic Number Theory</a>. Classics in Mathematics. Berlin: Springer-Verlag. p. <a rel="nofollow" class="external text" href="https://archive.org/details/basicnumbertheor00weil_866/page/n56">43</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58655-5" title="Special:BookSources/978-3-540-58655-5"><bdi>978-3-540-58655-5</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=1344916">1344916</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Number+Theory&rft.place=Berlin&rft.series=Classics+in+Mathematics&rft.pages=43&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-3-540-58655-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1344916%23id-name%3DMR&rft.aulast=Weil&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbasicnumbertheor00weil_866&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> Note however that some authors such as <a href="#CITEREFChildress2009">Childress (2009)</a> instead use "place" to mean an equivalence class of norms. </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoch1997" class="citation book cs1">Koch, H. (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wt1sCQAAQBAJ&pg=PA136">Algebraic Number Theory</a>. Berlin: Springer-Verlag. p. 136. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.309.8812">10.1.1.309.8812</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-58095-6">10.1007/978-3-642-58095-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63003-6" title="Special:BookSources/978-3-540-63003-6"><bdi>978-3-540-63003-6</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=1474965">1474965</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Theory&rft.place=Berlin&rft.pages=136&rft.pub=Springer-Verlag&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.309.8812%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1474965%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-58095-6&rft.isbn=978-3-540-63003-6&rft.aulast=Koch&rft.aufirst=H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dwt1sCQAAQBAJ%26pg%3DPA136&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLauritzen2003" class="citation book cs1">Lauritzen, Niels (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BdAbcje-TZUC&pg=PA127">Concrete Abstract Algebra: From numbers to Gröbner bases</a>. Cambridge: Cambridge University Press. p. 127. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511804229">10.1017/CBO9780511804229</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53410-9" title="Special:BookSources/978-0-521-53410-9"><bdi>978-0-521-53410-9</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=2014325">2014325</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Abstract+Algebra%3A+From+numbers+to+Gr%C3%B6bner+bases&rft.place=Cambridge&rft.pages=127&rft.pub=Cambridge+University+Press&rft.date=2003&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2014325%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511804229&rft.isbn=978-0-521-53410-9&rft.aulast=Lauritzen&rft.aufirst=Niels&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBdAbcje-TZUC%26pg%3DPA127&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <a href="#CITEREFLauritzen2003">Lauritzen 2003</a>, Corollary 3.5.14, p. 133; Lemma 3.5.18, p. 136. </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> <a href="#CITEREFKraftWashington2014">Kraft & Washington 2014</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA297">Section 12.1, Sums of two squares, pp. 297–301</a>. </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbud1995" class="citation book cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995). Commutative Algebra. Graduate Texts in Mathematics. Vol. 150. Berlin; New York: Springer-Verlag. Section 3.3. <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%2F978-1-4612-5350-1">10.1007/978-1-4612-5350-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94268-1" title="Special:BookSources/978-0-387-94268-1"><bdi>978-0-387-94268-1</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=1322960">1322960</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Algebra&rft.place=Berlin%3B+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pages=Section+3.3&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1322960%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-5350-1&rft.isbn=978-0-387-94268-1&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShafarevich2013" class="citation book cs1"><a href="/wiki/Igor_Shafarevich" title="Igor Shafarevich">Shafarevich, Igor R.</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zDW8BAAAQBAJ&pg=PA5">"Definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad53059b6a9d6c309ab164aa4eaed021844b5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.78ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} A}">"</a>. Basic Algebraic Geometry 2: Schemes and Complex Manifolds (3rd ed.). Springer, Heidelberg. p. 5. <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%2F978-3-642-38010-5">10.1007/978-3-642-38010-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-38009-9" title="Special:BookSources/978-3-642-38009-9"><bdi>978-3-642-38009-9</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=3100288">3100288</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Definition+of+MATH+RENDER+ERROR&rft.btitle=Basic+Algebraic+Geometry+2%3A+Schemes+and+Complex+Manifolds&rft.pages=5&rft.edition=3rd&rft.pub=Springer%2C+Heidelberg&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3100288%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-38010-5&rft.isbn=978-3-642-38009-9&rft.aulast=Shafarevich&rft.aufirst=Igor+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzDW8BAAAQBAJ%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirch1999" class="citation book cs1"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 322. Berlin: Springer-Verlag. Section I.8, p. 50. <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%2F978-3-662-03983-0">10.1007/978-3-662-03983-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65399-8" title="Special:BookSources/978-3-540-65399-8"><bdi>978-3-540-65399-8</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=1697859">1697859</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Theory&rft.place=Berlin&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+%5BFundamental+Principles+of+Mathematical+Sciences%5D&rft.pages=Section+I.8%2C+p.+50&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1697859%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-662-03983-0&rft.isbn=978-3-540-65399-8&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-119"><a href="#cite_ref-119">^</a> <a href="#CITEREFNeukirch1999">Neukirch 1999</a>, Section I.7, p. 38 </li> <li id="cite_note-120"><a href="#cite_ref-120">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStevenhagenLenstra1996" class="citation journal cs1">Stevenhagen, P.; <a href="/wiki/Hendrik_Lenstra" title="Hendrik Lenstra">Lenstra, H.W. Jr.</a> (1996). "Chebotarëv and his density theorem". <a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a>. 18 (2): 26–37. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.116.9409">10.1.1.116.9409</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03027290">10.1007/BF03027290</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=1395088">1395088</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:14089091">14089091</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=Chebotar%C3%ABv+and+his+density+theorem&rft.volume=18&rft.issue=2&rft.pages=26-37&rft.date=1996&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.116.9409%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1395088%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14089091%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF03027290&rft.aulast=Stevenhagen&rft.aufirst=P.&rft.au=Lenstra%2C+H.W.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2018" class="citation book cs1"><a href="/wiki/Marshall_Hall_(mathematician)" title="Marshall Hall (mathematician)">Hall, Marshall</a> (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K8hEDwAAQBAJ">The Theory of Groups</a>. Dover Books on Mathematics. Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-81690-6" title="Special:BookSources/978-0-486-81690-6"><bdi>978-0-486-81690-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups&rft.series=Dover+Books+on+Mathematics&rft.pub=Courier+Dover+Publications&rft.date=2018&rft.isbn=978-0-486-81690-6&rft.aulast=Hall&rft.aufirst=Marshall&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK8hEDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for Burnside's theorem see p. 143. </li> <li id="cite_note-122"><a href="#cite_ref-122">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryantSangwin2008" class="citation book cs1">Bryant, John; Sangwin, Christopher J. (2008). <a href="/wiki/How_Round_Is_Your_Circle" class="mw-redirect" title="How Round Is Your Circle">How Round is Your Circle?: Where Engineering and Mathematics Meet</a>. Princeton University Press. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iIN_2WjBH1cC&pg=PA178">p. 178</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-13118-4" title="Special:BookSources/978-0-691-13118-4"><bdi>978-0-691-13118-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Round+is+Your+Circle%3F%3A+Where+Engineering+and+Mathematics+Meet&rft.pages=p.+178&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-13118-4&rft.aulast=Bryant&rft.aufirst=John&rft.au=Sangwin%2C+Christopher+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-123"><a href="#cite_ref-123">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy2012" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, Godfrey Harold</a> (2012) [1940]. <a href="/wiki/A_Mathematician%27s_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EkY2im6xkVkC&pg=PA140">140</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42706-7" title="Special:BookSources/978-0-521-42706-7"><bdi>978-0-521-42706-7</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/922010634">922010634</a>. <q>No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematician%27s+Apology&rft.pages=140&rft.pub=Cambridge+University+Press&rft.date=2012&rft_id=info%3Aoclcnum%2F922010634&rft.isbn=978-0-521-42706-7&rft.aulast=Hardy&rft.aufirst=Godfrey+Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiblin1993" class="citation book cs1"><a href="/wiki/Peter_Giblin" title="Peter Giblin">Giblin, Peter</a> (1993). <a rel="nofollow" class="external text" href="https://archive.org/details/primesprogrammin0000gibl">Primes and Programming</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/primesprogrammin0000gibl/page/39">39</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-40988-9" title="Special:BookSources/978-0-521-40988-9"><bdi>978-0-521-40988-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primes+and+Programming&rft.pages=39&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=978-0-521-40988-9&rft.aulast=Giblin&rft.aufirst=Peter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprimesprogrammin0000gibl&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-125"><a href="#cite_ref-125">^</a> <a href="#CITEREFGiblin1993">Giblin 1993</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/primesprogrammin0000gibl/page/54">p. 54</a> </li> <li id="cite_note-p._220-126">^ <a href="#cite_ref-p._220_126-0">a</a> <a href="#cite_ref-p._220_126-1">b</a> <a href="#CITEREFRiesel1994">Riesel 1994</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA220">p. 220</a>. </li> <li id="cite_note-127"><a href="#cite_ref-127">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBullynck2010" class="citation journal cs1">Bullynck, Maarten (2010). <a rel="nofollow" class="external text" href="https://hal-univ-paris8.archives-ouvertes.fr/hal-01103903/">"A history of factor tables with notes on the birth of number theory 1657–1817"</a>. Revue d'Histoire des Mathématiques. 16 (2): 133–216.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revue+d%27Histoire+des+Math%C3%A9matiques&rft.atitle=A+history+of+factor+tables+with+notes+on+the+birth+of+number+theory+1657%E2%80%931817&rft.volume=16&rft.issue=2&rft.pages=133-216&rft.date=2010&rft.aulast=Bullynck&rft.aufirst=Maarten&rft_id=https%3A%2F%2Fhal-univ-paris8.archives-ouvertes.fr%2Fhal-01103903%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-128"><a href="#cite_ref-128">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWagstaff2013" class="citation book cs1"><a href="/wiki/Samuel_S._Wagstaff_Jr." title="Samuel S. Wagstaff Jr.">Wagstaff, Samuel S. Jr.</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rowCAQAAQBAJ&pg=PA191">The Joy of Factoring</a>. Student mathematical library. Vol. 68. American Mathematical Society. p. 191. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1048-3" title="Special:BookSources/978-1-4704-1048-3"><bdi>978-1-4704-1048-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Joy+of+Factoring&rft.series=Student+mathematical+library&rft.pages=191&rft.pub=American+Mathematical+Society&rft.date=2013&rft.isbn=978-1-4704-1048-3&rft.aulast=Wagstaff&rft.aufirst=Samuel+S.+Jr.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrowCAQAAQBAJ%26pg%3DPA191&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-129"><a href="#cite_ref-129">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrandallPomerance2005" class="citation book cs1"><a href="/wiki/Richard_Crandall" title="Richard Crandall">Crandall, Richard</a>; <a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Pomerance, Carl</a> (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA121">Prime Numbers: A Computational Perspective</a> (2nd ed.). Springer. p. 121. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-25282-7" title="Special:BookSources/978-0-387-25282-7"><bdi>978-0-387-25282-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers%3A+A+Computational+Perspective&rft.pages=121&rft.edition=2nd&rft.pub=Springer&rft.date=2005&rft.isbn=978-0-387-25282-7&rft.aulast=Crandall&rft.aufirst=Richard&rft.au=Pomerance%2C+Carl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRbEz-_D7sAUC%26pg%3DPA121&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-130"><a href="#cite_ref-130">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarach-ColtonTsai2015" class="citation conference cs1"><a href="/wiki/Martin_Farach-Colton" title="Martin Farach-Colton">Farach-Colton, Martín</a>; Tsai, Meng-Tsung (2015). "On the complexity of computing prime tables". In Elbassioni, Khaled; Makino, Kazuhisa (eds.). Algorithms and Computation: 26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9-11, 2015, Proceedings. Lecture Notes in Computer Science. Vol. 9472. Springer. pp. 677–688. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1504.05240">1504.05240</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-48971-0_57">10.1007/978-3-662-48971-0_57</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-48970-3" title="Special:BookSources/978-3-662-48970-3"><bdi>978-3-662-48970-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=On+the+complexity+of+computing+prime+tables&rft.btitle=Algorithms+and+Computation%3A+26th+International+Symposium%2C+ISAAC+2015%2C+Nagoya%2C+Japan%2C+December+9-11%2C+2015%2C+Proceedings&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=677-688&rft.pub=Springer&rft.date=2015&rft_id=info%3Aarxiv%2F1504.05240&rft_id=info%3Adoi%2F10.1007%2F978-3-662-48971-0_57&rft.isbn=978-3-662-48970-3&rft.aulast=Farach-Colton&rft.aufirst=Mart%C3%ADn&rft.au=Tsai%2C+Meng-Tsung&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-131"><a href="#cite_ref-131">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreaves2013" class="citation book cs1">Greaves, George (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G0TtCAAAQBAJ&pg=PA1">Sieves in Number Theory</a>. Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge). Vol. 43. Springer. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-04658-6" title="Special:BookSources/978-3-662-04658-6"><bdi>978-3-662-04658-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sieves+in+Number+Theory&rft.series=Ergebnisse+der+Mathematik+und+ihrer+Grenzgebiete+%283.+Folge%29&rft.pages=1&rft.pub=Springer&rft.date=2013&rft.isbn=978-3-662-04658-6&rft.aulast=Greaves&rft.aufirst=George&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG0TtCAAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-hromkovic-132">^ <a href="#cite_ref-hromkovic_132-0">a</a> <a href="#cite_ref-hromkovic_132-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHromkovič2001" class="citation book cs1"><a href="/wiki/Juraj_Hromkovi%C4%8D" title="Juraj Hromkovič">Hromkovič, Juraj</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nkeqCAAAQBAJ&pg=PA383">"5.5 Bibliographic Remarks"</a>. Algorithmics for Hard Problems. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin. pp. 383–385. <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%2F978-3-662-04616-6">10.1007/978-3-662-04616-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66860-2" title="Special:BookSources/978-3-540-66860-2"><bdi>978-3-540-66860-2</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=1843669">1843669</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:31159492">31159492</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.5+Bibliographic+Remarks&rft.btitle=Algorithmics+for+Hard+Problems&rft.series=Texts+in+Theoretical+Computer+Science.+An+EATCS+Series&rft.pages=383-385&rft.pub=Springer-Verlag%2C+Berlin&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F978-3-662-04616-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1843669%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A31159492%23id-name%3DS2CID&rft.isbn=978-3-540-66860-2&rft.aulast=Hromkovi%C4%8D&rft.aufirst=Juraj&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnkeqCAAAQBAJ%26pg%3DPA383&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-koblitz-134">^ <a href="#cite_ref-koblitz_134-0">a</a> <a href="#cite_ref-koblitz_134-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoblitz1987" class="citation book cs1"><a href="/wiki/Neal_Koblitz" title="Neal Koblitz">Koblitz, Neal</a> (1987). "Chapter V. Primality and Factoring". A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. Vol. 114. Springer-Verlag, New York. pp. 112–149. <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%2F978-1-4684-0310-7_5">10.1007/978-1-4684-0310-7_5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96576-5" title="Special:BookSources/978-0-387-96576-5"><bdi>978-0-387-96576-5</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=0910297">0910297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+V.+Primality+and+Factoring&rft.btitle=A+Course+in+Number+Theory+and+Cryptography&rft.series=Graduate+Texts+in+Mathematics&rft.pages=112-149&rft.pub=Springer-Verlag%2C+New+York&rft.date=1987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D910297%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4684-0310-7_5&rft.isbn=978-0-387-96576-5&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPieprzykHardjonoSeberry2013" class="citation book cs1">Pieprzyk, Josef; Hardjono, Thomas; <a href="/wiki/Jennifer_Seberry" title="Jennifer Seberry">Seberry, Jennifer</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BG2rCAAAQBAJ&pg=PA51">"2.3.9 Probabilistic Computations"</a>. Fundamentals of Computer Security. Springer. pp. 51–52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-07324-7" title="Special:BookSources/978-3-662-07324-7"><bdi>978-3-662-07324-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.3.9+Probabilistic+Computations&rft.btitle=Fundamentals+of+Computer+Security&rft.pages=51-52&rft.pub=Springer&rft.date=2013&rft.isbn=978-3-662-07324-7&rft.aulast=Pieprzyk&rft.aufirst=Josef&rft.au=Hardjono%2C+Thomas&rft.au=Seberry%2C+Jennifer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBG2rCAAAQBAJ%26pg%3DPA51&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-tao-aks-136">^ <a href="#cite_ref-tao-aks_136-0">a</a> <a href="#cite_ref-tao-aks_136-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTao2010" class="citation book cs1"><a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> (2010). <a rel="nofollow" class="external text" href="https://terrytao.wordpress.com/2009/08/11/the-aks-primality-test/">"1.11 The AKS primality test"</a>. 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Wagstaff, Jr.">Wagstaff, Jr., Samuel S.</a> (October 1980). <a rel="nofollow" class="external text" href="http://mpqs.free.fr/LucasPseudoprimes.pdf">"Lucas Pseudoprimes"</a> (PDF). <a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a>. 35 (152): 1391–1417. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1980-0583518-6">10.1090/S0025-5718-1980-0583518-6</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/2006406">2006406</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=0583518">0583518</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Lucas+Pseudoprimes&rft.volume=35&rft.issue=152&rft.pages=1391-1417&rft.date=1980-10&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D583518%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2006406%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-1980-0583518-6&rft.aulast=Baillie&rft.aufirst=Robert&rft.au=Wagstaff%2C+Jr.%2C+Samuel+S.&rft_id=http%3A%2F%2Fmpqs.free.fr%2FLucasPseudoprimes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-monier-143">^ <a href="#cite_ref-monier_143-0">a</a> <a href="#cite_ref-monier_143-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonier1980" class="citation journal cs1">Monier, Louis (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2880%2990007-9">"Evaluation and comparison of two efficient probabilistic primality testing algorithms"</a>. <a href="/wiki/Theoretical_Computer_Science_(journal)" title="Theoretical Computer Science (journal)">Theoretical Computer Science</a>. 12 (1): 97–108. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2880%2990007-9">10.1016/0304-3975(80)90007-9</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=0582244">0582244</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Evaluation+and+comparison+of+two+efficient+probabilistic+primality+testing+algorithms&rft.volume=12&rft.issue=1&rft.pages=97-108&rft.date=1980&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2880%2990007-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D582244%23id-name%3DMR&rft.aulast=Monier&rft.aufirst=Louis&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0304-3975%252880%252990007-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTao2009" class="citation book cs1"><a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> (2009). <a rel="nofollow" class="external text" href="https://terrytao.wordpress.com/2008/10/02/the-lucas-lehmer-test-for-mersenne-primes/">"1.7 The Lucas–Lehmer test for Mersenne primes"</a>. Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: American Mathematical Society. pp. 36–41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4883-8" title="Special:BookSources/978-0-8218-4883-8"><bdi>978-0-8218-4883-8</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=2523047">2523047</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.7+The+Lucas%E2%80%93Lehmer+test+for+Mersenne+primes&rft.btitle=Poincar%C3%A9%27s+legacies%2C+pages+from+year+two+of+a+mathematical+blog.+Part+I&rft.place=Providence%2C+RI&rft.pages=36-41&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4883-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2523047%23id-name%3DMR&rft.aulast=Tao&rft.aufirst=Terence&rft_id=https%3A%2F%2Fterrytao.wordpress.com%2F2008%2F10%2F02%2Fthe-lucas-lehmer-test-for-mersenne-primes%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> <a href="#CITEREFKraftWashington2014">Kraft & Washington 2014</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA41">p. 41</a>. </li> <li id="cite_note-146"><a href="#cite_ref-146">^</a> For instance see <a href="#CITEREFGuy2013">Guy 2013</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA13">A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{n}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4e701ab91db138d631d50abf2452c3991e3d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.274ex; height:2.509ex;" alt="{\displaystyle k\cdot 2^{n}+1}">. pp. 13–21</a>. </li> <li id="cite_note-147"><a href="#cite_ref-147">^</a> <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.eff.org/press/archives/2009/10/14-0">"Record 12-Million-Digit Prime Number Nets $100,000 Prize"</a>. 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Retrieved 2010-01-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=EFF+Cooperative+Computing+Awards&rft.pub=Electronic+Frontier+Foundation&rft.date=2008-02-29&rft_id=https%3A%2F%2Fwww.eff.org%2Fawards%2Fcoop&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-149"><a href="#cite_ref-149">^</a> <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.primegrid.com/download/SOB-31172165.pdf">"PrimeGrid's Seventeen or Bust Subproject"</a> (PDF). 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Retrieved 2017-01-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Prime+Pages&rft.atitle=The+Top+Twenty%3A+Factorial&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=http%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D30&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-152"><a href="#cite_ref-152">^</a> <a href="#CITEREFRibenboim2004">Ribenboim 2004</a>, p. 4. </li> <li id="cite_note-154"><a href="#cite_ref-154">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwell" class="citation web cs1">Caldwell, Chris K. <a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=5">"The Top Twenty: Primorial"</a>. <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">The Prime Pages</a>. Retrieved 2017-01-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Prime+Pages&rft.atitle=The+Top+Twenty%3A+Primorial&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=http%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-155"><a href="#cite_ref-155">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaldwell" class="citation web cs1">Caldwell, Chris K. <a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=1">"The Top Twenty: Twin Primes"</a>. <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">The Prime Pages</a>. Retrieved 2017-01-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Prime+Pages&rft.atitle=The+Top+Twenty%3A+Twin+Primes&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=http%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> <a href="#CITEREFKraftWashington2014">Kraft & Washington 2014</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA275">p. 275</a>. </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoffsteinPipherSilverman2014" class="citation book cs1">Hoffstein, Jeffrey; <a href="/wiki/Jill_Pipher" title="Jill Pipher">Pipher, Jill</a>; <a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">Silverman, Joseph H.</a> (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cbl_BAAAQBAJ&pg=PA329">An Introduction to Mathematical Cryptography</a>. Undergraduate Texts in Mathematics (2nd ed.). Springer. p. 329. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-1711-2" title="Special:BookSources/978-1-4939-1711-2"><bdi>978-1-4939-1711-2</bdi></a>.</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+Mathematical+Cryptography&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=329&rft.edition=2nd&rft.pub=Springer&rft.date=2014&rft.isbn=978-1-4939-1711-2&rft.aulast=Hoffstein&rft.aufirst=Jeffrey&rft.au=Pipher%2C+Jill&rft.au=Silverman%2C+Joseph+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dcbl_BAAAQBAJ%26pg%3DPA329&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-158"><a href="#cite_ref-158">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPomerance1996" class="citation journal cs1"><a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Pomerance, Carl</a> (1996). "A tale of two sieves". <a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>. 43 (12): 1473–1485. <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=1416721">1416721</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=A+tale+of+two+sieves&rft.volume=43&rft.issue=12&rft.pages=1473-1485&rft.date=1996&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1416721%23id-name%3DMR&rft.aulast=Pomerance&rft.aufirst=Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-159"><a href="#cite_ref-159">^</a> Emmanuel Thomé, <a rel="nofollow" class="external text" href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;fd743373.1912">“795-bit factoring and discrete logarithms,”</a> December 2, 2019. </li> <li id="cite_note-160"><a href="#cite_ref-160">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRieffelPolak2011" class="citation book cs1"><a href="/wiki/Eleanor_Rieffel" title="Eleanor Rieffel">Rieffel, Eleanor G.</a>; Polak, Wolfgang H. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iYX6AQAAQBAJ&pg=PA163">"Chapter 8. Shor's Algorithm"</a>. <a href="/wiki/Quantum_Computing:_A_Gentle_Introduction" title="Quantum Computing: A Gentle Introduction">Quantum Computing: A Gentle Introduction</a>. 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(12 October 2012). "Experimental realization of Shor's quantum factoring algorithm using qubit recycling". Nature Photonics. 6 (11): 773–776. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1111.4147">1111.4147</a>. <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/2012NaPho...6..773M">2012NaPho...6..773M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphoton.2012.259">10.1038/nphoton.2012.259</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:46546101">46546101</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Photonics&rft.atitle=Experimental+realization+of+Shor%27s+quantum+factoring+algorithm+using+qubit+recycling&rft.volume=6&rft.issue=11&rft.pages=773-776&rft.date=2012-10-12&rft_id=info%3Aarxiv%2F1111.4147&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A46546101%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fnphoton.2012.259&rft_id=info%3Abibcode%2F2012NaPho...6..773M&rft.aulast=Mart%C3%ADn-L%C3%B3pez&rft.aufirst=Enrique&rft.au=Laing%2C+Anthony&rft.au=Lawson%2C+Thomas&rft.au=Alvarez%2C+Roberto&rft.au=Zhou%2C+Xiao-Qi&rft.au=O%27Brien%2C+Jeremy+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-162"><a href="#cite_ref-162">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChirgwin2016" class="citation news cs1">Chirgwin, Richard (October 9, 2016). <a rel="nofollow" class="external text" href="https://www.theregister.co.uk/2016/10/09/crypto_needs_more_transparency_researchers_warn/">"Crypto needs more transparency, researchers warn"</a>. <a href="/wiki/The_Register" title="The Register">The Register</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Register&rft.atitle=Crypto+needs+more+transparency%2C+researchers+warn&rft.date=2016-10-09&rft.aulast=Chirgwin&rft.aufirst=Richard&rft_id=https%3A%2F%2Fwww.theregister.co.uk%2F2016%2F10%2F09%2Fcrypto_needs_more_transparency_researchers_warn%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-163"><a href="#cite_ref-163">^</a> <a href="#CITEREFHoffsteinPipherSilverman2014">Hoffstein, Pipher & Silverman 2014</a>, Section 2.3, Diffie–Hellman key exchange, pp. 65–67. </li> <li id="cite_note-164"><a href="#cite_ref-164">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="/wiki/Ron_Rivest" title="Ron Rivest">Rivest, Ronald L.</a>; <a href="/wiki/Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2001) [1990]. "11.3 Universal hashing". <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (2nd ed.). MIT Press and McGraw-Hill. pp. 232–236. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-03293-7" title="Special:BookSources/0-262-03293-7"><bdi>0-262-03293-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=11.3+Universal+hashing&rft.btitle=Introduction+to+Algorithms&rft.pages=232-236&rft.edition=2nd&rft.pub=MIT+Press+and+McGraw-Hill&rft.date=2001&rft.isbn=0-262-03293-7&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-independent hashing see problem 11–4, p. 251. For the credit to Carter and Wegman, see the chapter notes, p. 252. </li> <li id="cite_note-165"><a href="#cite_ref-165">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodrichTamassia2006" class="citation book cs1"><a href="/wiki/Michael_T._Goodrich" title="Michael T. Goodrich">Goodrich, Michael T.</a>; <a href="/wiki/Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (2006). Data Structures & Algorithms in Java (4th ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-73884-8" title="Special:BookSources/978-0-471-73884-8"><bdi>978-0-471-73884-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+Structures+%26+Algorithms+in+Java&rft.edition=4th&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-471-73884-8&rft.aulast=Goodrich&rft.aufirst=Michael+T.&rft.au=Tamassia%2C+Roberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> See "Quadratic probing", p. 382, and exercise C–9.9, p. 415. </li> <li id="cite_note-166"><a href="#cite_ref-166">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirtland2001" class="citation book cs1"><a href="/wiki/Joseph_Kirtland" title="Joseph Kirtland">Kirtland, Joseph</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z8eka35WUb8C&pg=PA43">Identification Numbers and Check Digit Schemes</a>. Classroom Resource Materials. Vol. 18. 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(May 1996). <a rel="nofollow" class="external text" href="https://datatracker.ietf.org/doc/html/rfc1950">ZLIB Compressed Data Format Specification version 3.3</a>. Network Working Group. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17487%2FRFC1950">10.17487/RFC1950</a>. <a href="/wiki/Request_for_Comments" title="Request for Comments">RFC</a> <a rel="nofollow" class="external text" href="https://datatracker.ietf.org/doc/html/rfc1950">1950</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=ZLIB+Compressed+Data+Format+Specification+version+3.3&rft.pub=Network+Working+Group&rft.date=1996-05&rft_id=info%3Adoi%2F10.17487%2FRFC1950&rft.aulast=Deutsch&rft.aufirst=P.&rft_id=https%3A%2F%2Fdatatracker.ietf.org%2Fdoc%2Fhtml%2Frfc1950&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-168"><a href="#cite_ref-168">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1998" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1998). 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"Mersenne Twister: A 623-dimensionally equidistributed uniform pseudo-random number generator". 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Retrieved 2018-01-25.</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=Why+Eisenstein+proved+the+Eisenstein+criterion+and+why+Sch%C3%B6nemann+discovered+it+first&rft.volume=118&rft.issue=1&rft.pages=3-31&rft.date=2011&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.398.3440%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15978494%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4169%2Famer.math.monthly.118.01.003&rft.aulast=Cox&rft.aufirst=David+A.&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fupload_library%2F22%2FFord%2FCox-2012.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-172"><a href="#cite_ref-172">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2002" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002). 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Retrieved February 22, 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=National+Geographic&rft.atitle=Bamboo+Mathematicians&rft.date=2015-05-15&rft.aulast=Zimmer&rft.aufirst=Carl&rft_id=https%3A%2F%2Fwww.nationalgeographic.com%2Fscience%2Farticle%2Fbamboo-mathematicians&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-188"><a href="#cite_ref-188">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHill1995" class="citation book cs1">Hill, Peter Jensen, ed. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7ag3ymWqvfgC&pg=PT225">The Messiaen companion</a>. Portland, OR: Amadeus Press. 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Washington, DC: Mathematical Association of America. pp. 3–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-548-5" title="Special:BookSources/978-0-88385-548-5"><bdi>978-0-88385-548-5</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=2085842">2085842</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Prime+Numbers+and+the+Search+for+Extraterrestrial+Intelligence&rft.btitle=Mathematical+Adventures+for+Students+and+Amateurs&rft.place=Washington%2C+DC&rft.series=MAA+Spectrum&rft.pages=3-6&rft.pub=Mathematical+Association+of+America&rft.date=2004&rft.isbn=978-0-88385-548-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2085842%23id-name%3DMR&rft.aulast=Pomerance&rft.aufirst=Carl&rft_id=https%3A%2F%2Fgauss.dartmouth.edu%2F~carlp%2FPDF%2Fextraterrestrial.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> <li id="cite_note-190"><a href="#cite_ref-190">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrrlScientist2010" class="citation news cs1">GrrlScientist (September 16, 2010). <a rel="nofollow" class="external text" href="https://www.theguardian.com/science/punctuated-equilibrium/2010/sep/16/curious-incident-dog-night-time">"The Curious Incident of the Dog in the Night-Time"</a>. 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Sunday Book Review. <a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=Counting+on+Each+Other&rft.date=2010-04-09&rft.aulast=Schillinger&rft.aufirst=Liesl&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2010%2F04%2F11%2Fbooks%2Freview%2FSchillinger-t.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output 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.sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-abovebelow"> Prime number at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects">sister projects</a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/41px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/54px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /><a href="https://en.wiktionary.org/wiki/Special:Search/Prime_number" class="extiw" title="wikt:Special:Search/Prime number">Definitions</a> from Wiktionary</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /><a href="https://commons.wikimedia.org/wiki/Category:Prime_numbers" class="extiw" title="c:Category:Prime numbers">Media</a> from Commons</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/27px-Wikinews-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/41px-Wikinews-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/54px-Wikinews-logo.svg.png 2x" data-file-width="759" data-file-height="415" /><a href="https://en.wikinews.org/wiki/Category:Prime_numbers" class="extiw" title="n:Category:Prime numbers">News</a> from Wikinews</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /><a href="https://en.wikiquote.org/wiki/Prime_number" class="extiw" title="q:Prime number">Quotations</a> from Wikiquote</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" decoding="async" width="26" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/39px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/51px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /><a href="https://en.wikisource.org/wiki/Special:Search/Prime_number" class="extiw" title="s:Special:Search/Prime number">Texts</a> from Wikisource</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /><a href="https://en.wikibooks.org/wiki/A_Guide_to_the_GRE/Prime_Numbers" class="extiw" title="b:A Guide to the GRE/Prime Numbers">Textbooks</a> from Wikibooks</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /><a href="https://en.wikiversity.org/wiki/Special:Search/Prime_number" class="extiw" title="v:Special:Search/Prime number">Resources</a> from Wikiversity</li></ul></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Prime_number">"Prime number"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Prime+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPrime_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrime+number" class="Z3988"></li> <li>Caldwell, Chris, The <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a> at <a rel="nofollow" class="external text" href="http://primes.utm.edu/">primes.utm.edu</a>.</li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p003hyf5">Prime Numbers</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)">In Our Time</a> at the <a href="/wiki/BBC" title="BBC">BBC</a>.</li> <li><a rel="nofollow" class="external text" href="http://plus.maths.org/issue49/package/index.html">Plus teacher and student package: prime numbers</a> from Plus, the free online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Generators_and_calculators">Generators and calculators</h3></div> <ul><li><a rel="nofollow" class="external text" href="http://www.javascripter.net/math/calculators/primefactorscalculator.htm">Prime factors calculator</a> can factorize any positive integer up to 20 digits.</li> <li><a rel="nofollow" class="external text" href="https://www.alpertron.com.ar/ECM.HTM">Fast Online primality test with factorization</a> makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).</li> <li><a rel="nofollow" class="external text" href="http://www.bigprimes.net/">Huge database of prime numbers</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.primos.mat.br/indexen.html">Prime Numbers up to 1 trillion</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210227001026/http://www.primos.mat.br/indexen.html">Archived</a> 2021-02-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist 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class="external text" href="http://id.worldcat.org/fast/1041241/">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</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 rel="nofollow" class="external text" href="https://d-nb.info/gnd/4047263-2">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85093218">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11932592t">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11932592t">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00571462">Japan</a></li><li><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph139050&CON_LNG=ENG">Czech Republic</a></li><li><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000232578&P_CON_LNG=ENG">Latvia</a></li><li><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538747905171">Israel</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" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#ffb;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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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 class="mw-selflink selflink">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><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, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Number_theory" title="Category:Number theory">Category</a></li> <li><img alt="" 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" /> <a href="/wiki/List_of_number_theory_topics" title="List of number theory topics">List of topics</a></li> <li><img alt="" 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" /> <a href="/wiki/List_of_recreational_number_theory_topics" title="List of recreational number theory topics">List of recreational topics</a></li> <li><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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a> <a href="https://en.wikibooks.org/wiki/Number_Theory" class="extiw" title="wikibooks:Number Theory">Wikibook</a></li> <li><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> <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="Divisibility-based_sets_of_integers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Divisor_classes" title="Template:Divisor classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Divisor_classes" title="Template talk:Divisor classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Divisor_classes" title="Special:EditPage/Template:Divisor classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Divisibility-based_sets_of_integers" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</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/Integer_factorization" title="Integer factorization">Integer factorization</a></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function</a></li> <li><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</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 class="mw-selflink selflink">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</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/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</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/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</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/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</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/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</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/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</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="Prime_number_classes" style="padding:3px"><table class="nowraplinks 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:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</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/Fermat_number" title="Fermat number">Fermat (22n + 1)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (2p − 1)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (22p−1 − 1)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff (2p + 1)/3</a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (k·2n + 1)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (n! ± 1)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (pn# ± 1)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (pn# + 1)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (4n + 1)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (2m·3n + 1)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (x4 + y4)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (2m ± 2n ± 1)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (n·2n + 1)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (n·2n − 1)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (x3 − y3)/(x − y)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (xy + yx)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (3·2n − 1)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams ((b−1)·bn − 1)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (⌊A3n⌋)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</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/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</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/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</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/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit (10n − 1)/9</a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple">k-tuples</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/Twin_prime" title="Twin prime">Twin (p, p + 2)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (p, p + 2 or p + 4, p + 6)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (p, p + 2, p + 6, p + 8)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (p, p + 4)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (p, p + 6)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (consecutive p − n, p, p + n)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)</a></li> <li><a href="/wiki/Chen_prime" title="Chen prime">Chen</a></li> <li><a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">Sophie Germain/Safe (p, 2p + 1)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By size</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Megaprime" title="Megaprime">Mega (1,000,000+ digits)</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known</a> <ul><li><a href="/wiki/List_of_largest_known_primes_and_probable_primes" title="List of largest known primes and probable primes">list</a></li></ul></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></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/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Gaussian_integer#Gaussian_primes" title="Gaussian integer">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></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/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</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/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</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/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></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="Classes_of_natural_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form a × 2b ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</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/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</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/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</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/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</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/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</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/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</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/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</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/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</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/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</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/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</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/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</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/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</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/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</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/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</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/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</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/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</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/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</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/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</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/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</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/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td 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1,\n [\"CITEREFChamberland2015\"] = 1,\n [\"CITEREFChan1996\"] = 1,\n [\"CITEREFChildress2009\"] = 1,\n [\"CITEREFChirgwin2016\"] = 1,\n [\"CITEREFConwayGuy1996\"] = 1,\n [\"CITEREFCooperHodges2016\"] = 1,\n [\"CITEREFCox2011\"] = 1,\n [\"CITEREFCrandallPomerance2005\"] = 1,\n [\"CITEREFDeutsch1996\"] = 1,\n [\"CITEREFDudley1978\"] = 1,\n [\"CITEREFEisenbud1995\"] = 1,\n [\"CITEREFEricksonVazzanaGarth2016\"] = 1,\n [\"CITEREFFarach-ColtonTsai2015\"] = 1,\n [\"CITEREFFaticoni2012\"] = 1,\n [\"CITEREFFurstenberg1955\"] = 1,\n [\"CITEREFGardiner1997\"] = 1,\n [\"CITEREFGelfandShen2003\"] = 1,\n [\"CITEREFGiblin1993\"] = 1,\n [\"CITEREFGillings1974\"] = 1,\n [\"CITEREFGleason1988\"] = 1,\n [\"CITEREFGolesSchulzMarkus2001\"] = 1,\n [\"CITEREFGoodrichTamassia2006\"] = 1,\n [\"CITEREFGreaves2013\"] = 1,\n [\"CITEREFGreenTao2008\"] = 1,\n [\"CITEREFGrrlScientist2010\"] = 1,\n [\"CITEREFGuy2013\"] = 1,\n [\"CITEREFHall2018\"] = 1,\n [\"CITEREFHardy2012\"] = 1,\n [\"CITEREFHayes2003\"] = 1,\n [\"CITEREFHenderson2014\"] = 1,\n [\"CITEREFHiggins1998\"] = 1,\n [\"CITEREFHill1995\"] = 1,\n [\"CITEREFHoffsteinPipherSilverman2014\"] = 1,\n [\"CITEREFHromkovič2001\"] = 1,\n [\"CITEREFHua2009\"] = 1,\n [\"CITEREFKatz2004\"] = 1,\n [\"CITEREFKirtland2001\"] = 1,\n [\"CITEREFKleeWagon1991\"] = 1,\n [\"CITEREFKnuth1998\"] = 1,\n [\"CITEREFKoblitz1987\"] = 1,\n [\"CITEREFKoch1997\"] = 1,\n [\"CITEREFKoshy2002\"] = 1,\n [\"CITEREFKraftWashington2014\"] = 1,\n [\"CITEREFKřížekLucaSomer2001\"] = 1,\n [\"CITEREFLang2002\"] = 1,\n [\"CITEREFLauritzen2003\"] = 1,\n [\"CITEREFLavaBalzarotti2010\"] = 1,\n [\"CITEREFLeff2000\"] = 1,\n [\"CITEREFLenstraPomerance2019\"] = 1,\n [\"CITEREFMackinnon1987\"] = 1,\n [\"CITEREFMartín-LópezLaingLawsonAlvarez2012\"] = 1,\n [\"CITEREFMatiyasevich1999\"] = 1,\n [\"CITEREFMatsumotoNishimura1998\"] = 1,\n [\"CITEREFMillerTakloo-Bighash2006\"] = 1,\n [\"CITEREFMilnor1962\"] = 1,\n [\"CITEREFMollin1997\"] = 1,\n [\"CITEREFMollin2002\"] = 1,\n [\"CITEREFMonier1980\"] = 1,\n [\"CITEREFMorain2007\"] = 1,\n [\"CITEREFNarkiewicz2000\"] = 1,\n [\"CITEREFNathanson2000\"] = 1,\n [\"CITEREFNeale2017\"] = 1,\n [\"CITEREFNeukirch1999\"] = 1,\n [\"CITEREFOgilvyAnderson1988\"] = 1,\n [\"CITEREFOliveira_e_SilvaHerzogPardi2014\"] = 1,\n [\"CITEREFPatterson1988\"] = 1,\n [\"CITEREFPeterson1999\"] = 1,\n [\"CITEREFPieprzykHardjonoSeberry2013\"] = 1,\n [\"CITEREFPomerance1982\"] = 1,\n [\"CITEREFPomerance1996\"] = 1,\n [\"CITEREFPomerance2004\"] = 1,\n [\"CITEREFPomeranceSelfridgeWagstaff,_Jr.1980\"] = 1,\n [\"CITEREFRamaré1995\"] = 1,\n [\"CITEREFRassias2017\"] = 1,\n [\"CITEREFRibenboim2004\"] = 1,\n [\"CITEREFRieffelPolak2011\"] = 1,\n [\"CITEREFRiesel1994\"] = 1,\n [\"CITEREFRosen2000\"] = 1,\n [\"CITEREFRoth1951\"] = 1,\n [\"CITEREFRotman2000\"] = 1,\n [\"CITEREFSandifer2007\"] = 1,\n [\"CITEREFSandifer2014\"] = 1,\n [\"CITEREFSchillinger2010\"] = 1,\n [\"CITEREFSchubert1949\"] = 1,\n [\"CITEREFShafarevich2013\"] = 1,\n [\"CITEREFShahriari2017\"] = 1,\n [\"CITEREFSierpiński1964\"] = 1,\n [\"CITEREFSierpiński1988\"] = 1,\n [\"CITEREFSmith2011\"] = 1,\n [\"CITEREFSparkes2024\"] = 1,\n [\"CITEREFStevenhagenLenstra1996\"] = 1,\n [\"CITEREFStillwell1997\"] = 1,\n [\"CITEREFStillwell2010\"] = 1,\n [\"CITEREFTao2009\"] = 1,\n [\"CITEREFTao2010\"] = 1,\n [\"CITEREFTarán1981\"] = 1,\n [\"CITEREFTchebychev1852\"] = 1,\n [\"CITEREFVardi1991\"] = 1,\n [\"CITEREFWagstaff2013\"] = 1,\n [\"CITEREFWeil1995\"] = 1,\n [\"CITEREFWilliamson1782\"] = 1,\n [\"CITEREFWright1951\"] = 1,\n [\"CITEREFYuan2002\"] = 1,\n [\"CITEREFZagier1977\"] = 1,\n [\"CITEREFZhu2010\"] = 1,\n [\"CITEREFZiegler2004\"] = 1,\n [\"CITEREFZiegler2015\"] = 1,\n [\"CITEREFZimmer2015\"] = 1,\n [\"CITEREFdu_Sautoy2003\"] = 1,\n [\"CITEREFdu_Sautoy2011\"] = 1,\n}\ntemplate_list = table#1 {\n [\"11\"] = 1,\n [\"13\"] = 1,\n [\"As of\"] = 5,\n [\"Authority control\"] = 1,\n [\"Cite IETF\"] = 1,\n [\"Cite OEIS\"] = 1,\n [\"Cite book\"] = 80,\n [\"Cite conference\"] = 1,\n [\"Cite journal\"] = 39,\n [\"Cite magazine\"] = 1,\n [\"Cite news\"] = 4,\n [\"Cite web\"] = 9,\n [\"Classes of natural numbers\"] = 1,\n [\"Divisor classes\"] = 1,\n [\"Efn\"] = 6,\n [\"Further\"] = 2,\n [\"Good article\"] = 1,\n [\"Harvnb\"] = 47,\n [\"Harvtxt\"] = 3,\n [\"In Our Time\"] = 1,\n [\"Introduction to Algorithms\"] = 1,\n [\"Lang\"] = 1,\n [\"MacTutor Biography\"] = 1,\n [\"Main\"] = 14,\n [\"Mvar\"] = 1,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 9,\n [\"Number theory\"] = 1,\n [\"OEIS\"] = 1,\n [\"Portal bar\"] = 1,\n [\"Pp-vandalism\"] = 1,\n [\"Prime number classes\"] = 1,\n [\"Redirect\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn\"] = 1,\n [\"Short description\"] = 1,\n [\"Sister project links\"] = 1,\n [\"SpringerEOM\"] = 1,\n [\"Transl\"] = 1,\n [\"Webarchive\"] = 1,\n [\"Wide image\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["?","320","19.3"],["dataWrapper 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