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class="vector-toc-list"> <li id="toc-Brun's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Brun's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Brun's theorem</span> </div> </a> <ul id="toc-Brun's_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Twin_prime_conjecture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Twin_prime_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Twin prime conjecture</span> </div> </a> <ul id="toc-Twin_prime_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_theorems_weaker_than_the_twin_prime_conjecture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_theorems_weaker_than_the_twin_prime_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Other theorems weaker than the twin prime conjecture</span> </div> </a> <ul id="toc-Other_theorems_weaker_than_the_twin_prime_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjectures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conjectures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conjectures</span> </div> </a> <button aria-controls="toc-Conjectures-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Conjectures subsection</span> </button> <ul id="toc-Conjectures-sublist" class="vector-toc-list"> <li id="toc-First_Hardy–Littlewood_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_Hardy–Littlewood_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>First Hardy–Littlewood conjecture</span> </div> </a> <ul id="toc-First_Hardy–Littlewood_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polignac's_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polignac's_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Polignac's conjecture</span> </div> </a> <ul id="toc-Polignac's_conjecture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Large_twin_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Large_twin_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Large twin primes</span> </div> </a> <ul id="toc-Large_twin_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_elementary_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_elementary_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other elementary properties</span> </div> </a> <ul id="toc-Other_elementary_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isolated_prime" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isolated_prime"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Isolated prime</span> </div> </a> <ul id="toc-Isolated_prime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Twin prime</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 44 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-44" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">44 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF%D8%A7%D9%86_%D8%A3%D9%88%D9%84%D9%8A%D8%A7%D9%86_%D8%AA%D9%88%D8%A3%D9%85" title="عددان أوليان توأم – Arabic" lang="ar" hreflang="ar" data-title="عددان أوليان توأم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%81%E0%A6%97%E0%A7%8D%E0%A6%AE_%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="যুগ্ম মৌলিক সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="যুগ্ম মৌলিক সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Siang-si%E2%81%BF_s%C3%B2%CD%98-s%C3%B2%CD%98" title="Siang-siⁿ sò͘-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Siang-siⁿ sò͘-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B8_%D1%87%D0%B8%D1%81%D0%BB%D0%B0_%D0%B1%D0%BB%D0%B8%D0%B7%D0%BD%D0%B0%D1%86%D0%B8" title="Прости числа близнаци – Bulgarian" lang="bg" hreflang="bg" data-title="Прости числа близнаци" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombres_primers_bessons" title="Nombres primers bessons – Catalan" lang="ca" hreflang="ca" data-title="Nombres primers bessons" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%99%C4%95%D0%BA%C4%95%D1%80_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%D1%81%D0%B5%D0%BC" title="Йĕкĕр хисепсем – Chuvash" lang="cv" hreflang="cv" data-title="Йĕкĕр хисепсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADseln%C3%A1_dvojice" title="Prvočíselná dvojice – Czech" lang="cs" hreflang="cs" data-title="Prvočíselná dvojice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Primtalstvillinger" title="Primtalstvillinger – Danish" lang="da" hreflang="da" data-title="Primtalstvillinger" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Primzahlzwilling" title="Primzahlzwilling – German" lang="de" hreflang="de" data-title="Primzahlzwilling" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%AF%CE%B4%CF%85%CE%BC%CE%BF%CE%B9_%CF%80%CF%81%CF%8E%CF%84%CE%BF%CE%B9_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CE%AF" title="Δίδυμοι πρώτοι αριθμοί – Greek" lang="el" hreflang="el" data-title="Δίδυμοι πρώτοι αριθμοί" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_primo_gemelo" title="Número primo gemelo – Spanish" lang="es" hreflang="es" data-title="Número primo gemelo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/%C4%9Cemela_primo" title="Ĝemela primo – Esperanto" lang="eo" hreflang="eo" data-title="Ĝemela primo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_lehen_biki" title="Zenbaki lehen biki – Basque" lang="eu" hreflang="eu" data-title="Zenbaki lehen biki" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%A7%D9%88%D9%84_%D8%AF%D9%88%D9%82%D9%84%D9%88" title="اعداد اول دوقلو – Persian" lang="fa" hreflang="fa" data-title="اعداد اول دوقلو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombres_premiers_jumeaux" title="Nombres premiers jumeaux – French" lang="fr" hreflang="fr" data-title="Nombres premiers jumeaux" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Primo_xemelgo" title="Primo xemelgo – Galician" lang="gl" hreflang="gl" data-title="Primo xemelgo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EB%91%A5%EC%9D%B4_%EC%86%8C%EC%88%98" title="쌍둥이 소수 – Korean" lang="ko" hreflang="ko" data-title="쌍둥이 소수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D5%BE%D5%B8%D6%80%D5%B5%D5%A1%D5%AF%D5%B6%D5%A5%D6%80_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Երկվորյակներ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Երկվորյակներ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_prima_kembar" title="Bilangan prima kembar – Indonesian" lang="id" hreflang="id" data-title="Bilangan prima kembar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_primi_gemelli" title="Numeri primi gemelli – Italian" lang="it" hreflang="it" data-title="Numeri primi gemelli" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99%D7%99%D7%9D_%D7%AA%D7%90%D7%95%D7%9E%D7%99%D7%9D" title="ראשוניים תאומים – Hebrew" lang="he" hreflang="he" data-title="ראשוניים תאומים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_primm_giumej" title="Numer primm giumej – Lombard" lang="lmo" hreflang="lmo" data-title="Numer primm giumej" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ikerpr%C3%ADm" title="Ikerprím – Hungarian" lang="hu" hreflang="hu" data-title="Ikerprím" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D1%85%D1%8D%D1%80_%D0%B0%D0%BD%D1%85%D0%BD%D1%8B_%D1%82%D0%BE%D0%BE%D0%BD%D1%83%D1%83%D0%B4" title="Ихэр анхны тоонууд – Mongolian" lang="mn" hreflang="mn" data-title="Ихэр анхны тоонууд" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Priemtweeling" title="Priemtweeling – Dutch" lang="nl" hreflang="nl" data-title="Priemtweeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E5%AD%90%E7%B4%A0%E6%95%B0" title="双子素数 – Japanese" lang="ja" hreflang="ja" data-title="双子素数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tvillingprimtall" title="Tvillingprimtall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tvillingprimtall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Primtalltweeschen" title="Primtalltweeschen – Low German" lang="nds" hreflang="nds" data-title="Primtalltweeschen" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_bli%C5%BAniacze" title="Liczby bliźniacze – Polish" lang="pl" hreflang="pl" data-title="Liczby bliźniacze" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmeros_primos_g%C3%A9meos" title="Números primos gémeos – Portuguese" lang="pt" hreflang="pt" data-title="Números primos gémeos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ksh mw-list-item"><a href="https://ksh.wikipedia.org/wiki/Primzalzwilling" title="Primzalzwilling – Colognian" lang="ksh" hreflang="ksh" data-title="Primzalzwilling" data-language-autonym="Ripoarisch" data-language-local-name="Colognian" class="interlanguage-link-target"><span>Ripoarisch</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Numere_prime_gemene" title="Numere prime gemene – Romanian" lang="ro" hreflang="ro" data-title="Numere prime gemene" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0-%D0%B1%D0%BB%D0%B8%D0%B7%D0%BD%D0%B5%D1%86%D1%8B" title="Числа-близнецы – Russian" lang="ru" hreflang="ru" data-title="Числа-близнецы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Twin_prime" title="Twin prime – Simple English" lang="en-simple" hreflang="en-simple" data-title="Twin prime" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pra%C5%A1tevilski_dvoj%C4%8Dek" title="Praštevilski dvojček – Slovenian" lang="sl" hreflang="sl" data-title="Praštevilski dvojček" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%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_%D8%AF%D9%88%D8%A7%D9%86%DB%95" title="ژمارەی سەرەتایی دوانە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی سەرەتایی دوانە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Alkulukupari" title="Alkulukupari – Finnish" lang="fi" hreflang="fi" data-title="Alkulukupari" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Primtalstvilling" title="Primtalstvilling – Swedish" lang="sv" hreflang="sv" data-title="Primtalstvilling" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a 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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">Prime 2 more or 2 less than another prime</div> <p>A <b>twin prime</b> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair <span class="nowrap"><span class="texhtml"> (17, 19)</span> </span> or <span class="nowrap"><span class="texhtml">(41, 43)</span>.</span> In other words, a twin prime is a prime that has a <a href="/wiki/Prime_gap" title="Prime gap">prime gap</a> of two. Sometimes the term <i>twin prime</i> is used for a pair of twin primes; an alternative name for this is <b>prime twin</b> or <b>prime pair</b>. </p><p>Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called <b>twin prime conjecture</b>) or if there is a largest pair. The breakthrough<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> work of <a href="/wiki/Yitang_Zhang" title="Yitang Zhang">Yitang Zhang</a> in 2013, as well as work by <a href="/wiki/James_Maynard_(mathematician)" class="mw-redirect" title="James Maynard (mathematician)">James Maynard</a>, <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> and others, has made substantial progress towards <a href="/wiki/Mathematical_proof" title="Mathematical proof">proving</a> that there are infinitely many twin primes, but at present this remains unsolved.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> </p> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Are there infinitely many twin primes?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usually the pair <span class="texhtml">(2, 3)</span> is not considered to be a pair of twin primes.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Since 2 is the only <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes. </p><p>The first several twin prime pairs are </p> <dl><dd><span class="texhtml"> (3, 5), (5, 7), (11, 13),</span> <span class="texhtml"> (17, 19), (29, 31), (41, 43),</span> <span class="texhtml"> (59, 61), (71, 73), (101, 103),</span> <span class="texhtml"> (107, 109), (137, 139), ...</span> <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A077800" class="extiw" title="oeis:A077800">A077800</a></span>.</dd></dl> <p>Five is the only prime that belongs to two pairs, as every twin prime pair greater than <span class="texhtml"> (3, 5) </span> is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (6n-1,6n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>6</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (6n-1,6n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ef6ecf66cae3fd21dca4ec214c92c1aacf13c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.963ex; height:2.843ex;" alt="{\displaystyle (6n-1,6n+1)}"></span> for some <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>; that is, the number between the two primes is a multiple of 6.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12. </p> <div class="mw-heading mw-heading3"><h3 id="Brun's_theorem"><span id="Brun.27s_theorem"></span>Brun's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=2" title="Edit section: Brun's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Brun%27s_theorem" title="Brun's theorem">Brun's theorem</a></div> <p>In 1915, <a href="/wiki/Viggo_Brun" title="Viggo Brun">Viggo Brun</a> showed that the sum of <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of the twin primes was <a href="/wiki/Convergent_series" title="Convergent series">convergent</a>.<sup id="cite_ref-Brun-1915_5-0" class="reference"><a href="#cite_note-Brun-1915-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This famous result, called <a href="/wiki/Brun%27s_theorem" title="Brun's theorem">Brun's theorem</a>, was the first use of the <a href="/wiki/Brun_sieve" title="Brun sieve">Brun sieve</a> and helped initiate the development of modern <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a>. The modern version of Brun's argument can be used to show that the number of twin primes less than <span class="texhtml mvar" style="font-style:italic;">N</span> does not exceed </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {CN}{(\log N)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mi>N</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {CN}{(\log N)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc127b60a14577b2ec56fe78c13190a01b7f8104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.122ex; height:6.176ex;" alt="{\displaystyle {\frac {CN}{(\log N)^{2}}}}"></span></dd></dl> <p>for some absolute constant <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">C</span> > 0.</span><sup id="cite_ref-Bateman-Diamond-2004_6-0" class="reference"><a href="#cite_note-Bateman-Diamond-2004-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In fact, it is bounded above by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8C_{2}N}{(\log N)^{2}}}\left[1+\operatorname {\mathcal {O}} \left({\frac {\log \log N}{\log N}}\right)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>N</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8C_{2}N}{(\log N)^{2}}}\left[1+\operatorname {\mathcal {O}} \left({\frac {\log \log N}{\log N}}\right)\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d0e9f973c16b61d8e6bc1d9039bd47fdefc622" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.89ex; height:6.343ex;" alt="{\displaystyle {\frac {8C_{2}N}{(\log N)^{2}}}\left[1+\operatorname {\mathcal {O}} \left({\frac {\log \log N}{\log N}}\right)\right],}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec545f7870665e1028b7492746848d149878808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}"></span> is the <i>twin prime constant</i> (slightly less than 2/3), <a href="#First_Hardy–Littlewood_conjecture">given below</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Twin_prime_conjecture">Twin prime conjecture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=3" title="Edit section: Twin prime conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The question of whether there exist infinitely many twin primes has been one of the great <a href="/wiki/Open_problem" title="Open problem">open questions</a> in <a href="/wiki/Number_theory" title="Number theory">number theory</a> for many years. This is the content of the <i>twin prime conjecture</i>, which states that there are infinitely many primes <span class="texhtml mvar" style="font-style:italic;">p</span> such that <span class="nowrap"> <span class="texhtml"><i>p</i> + 2</span> </span> is also prime. In 1849, <a href="/wiki/Alphonse_de_Polignac" title="Alphonse de Polignac">de Polignac</a> made the more general conjecture that for every natural number <span class="texhtml mvar" style="font-style:italic;">k</span>, there are infinitely many primes <span class="texhtml mvar" style="font-style:italic;">p</span> such that <span class="nowrap"> <span class="texhtml"><i>p</i> + 2<i>k</i></span> </span> is also prime.<sup id="cite_ref-dePolignac-1849_8-0" class="reference"><a href="#cite_note-dePolignac-1849-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The <span class="nowrap">case <span class="texhtml mvar" style="font-style:italic;">k</span> = 1</span> of <a href="/wiki/De_Polignac%27s_conjecture" class="mw-redirect" title="De Polignac's conjecture">de Polignac's conjecture</a> is the twin prime conjecture. </p><p>A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. </p><p>On 17 April 2013, <a href="/wiki/Yitang_Zhang" title="Yitang Zhang">Yitang Zhang</a> announced a proof that there exists an <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml mvar" style="font-style:italic;">N</span> that is less than 70 million, where there are infinitely many pairs of primes that differ by <span class="texhtml mvar" style="font-style:italic;">N</span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Zhang's paper was accepted in early May 2013.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> subsequently proposed a <a href="/wiki/Polymath_Project" title="Polymath Project">Polymath Project</a> collaborative effort to optimize Zhang's bound.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>One year after Zhang's announcement, the bound had been reduced to 246, where it remains.<sup id="cite_ref-nielsen-bd-gaps_12-0" class="reference"><a href="#cite_note-nielsen-bd-gaps-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by <a href="/wiki/James_Maynard_(mathematician)" class="mw-redirect" title="James Maynard (mathematician)">James Maynard</a> and <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a>. This second approach also gave bounds for the smallest <span class="nowrap"> <span class="texhtml"><i>f</i> (<i>m</i>)</span> </span> needed to guarantee that infinitely many intervals of width <span class="texhtml"><i>f</i> (<i>m</i>)</span> contain at least <span class="texhtml mvar" style="font-style:italic;">m</span> primes. Moreover (see also the next section) assuming the <a href="/wiki/Elliott%E2%80%93Halberstam_conjecture" title="Elliott–Halberstam conjecture">Elliott–Halberstam conjecture</a> and its generalized form, the Polymath Project wiki states that the bound is 12 and 6, respectively.<sup id="cite_ref-nielsen-bd-gaps_12-1" class="reference"><a href="#cite_note-nielsen-bd-gaps-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>A strengthening of <a href="/wiki/Goldbach%E2%80%99s_conjecture" class="mw-redirect" title="Goldbach’s conjecture">Goldbach’s conjecture</a>, if proved, would also prove there is an infinite number of twin primes, as would the existence of <a href="/wiki/Siegel_zero" title="Siegel zero">Siegel zeroes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_theorems_weaker_than_the_twin_prime_conjecture">Other theorems weaker than the twin prime conjecture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=4" title="Edit section: Other theorems weaker than the twin prime conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1940, <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> showed that there is a <a href="/wiki/Mathematical_constant" title="Mathematical constant">constant</a> <span class="texhtml"><i>c</i> < 1</span> and infinitely many primes <span class="texhtml mvar" style="font-style:italic;">p</span> such that <span class="texhtml"><i>p</i>′ − <i>p</i> < <i>c</i> ln <i>p</i></span> where <span class="texhtml mvar" style="font-style:italic;">p′</span> denotes the next prime after <span class="texhtml mvar" style="font-style:italic;">p</span>. What this means is that we can find infinitely many intervals that contain two primes <span class="texhtml">(<i>p</i>, <i>p</i>′)</span> as long as we let these intervals grow slowly in size as we move to bigger and bigger primes. Here, "grow slowly" means that the length of these intervals can grow <a href="/wiki/Logarithm" title="Logarithm">logarithmically</a>. This result was successively improved; in 1986 <a href="/wiki/Helmut_Maier" title="Helmut Maier">Helmut Maier</a> showed that a constant <span class="texhtml"><i>c</i> < 0.25</span> can be used. In 2004 <a href="/wiki/Daniel_Goldston" title="Daniel Goldston">Daniel Goldston</a> and <a href="/wiki/Cem_Y%C4%B1ld%C4%B1r%C4%B1m" title="Cem Yıldırım">Cem Yıldırım</a> showed that the constant could be improved further to <span class="nowrap"><span class="texhtml"><i>c</i> = 0.085786... </span>.</span> In 2005, <a href="/wiki/Daniel_Goldston" title="Daniel Goldston">Goldston</a>, <a href="/wiki/J%C3%A1nos_Pintz" title="János Pintz">Pintz</a>, and <a href="/wiki/Cem_Y%C4%B1ld%C4%B1r%C4%B1m" title="Cem Yıldırım">Yıldırım</a> established that <span class="texhtml mvar" style="font-style:italic;">c</span> can be chosen to be arbitrarily small,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }\left({\frac {p_{n+1}-p_{n}}{\log p_{n}}}\right)=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }\left({\frac {p_{n+1}-p_{n}}{\log p_{n}}}\right)=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4f7113bb5de531620cfd48665317f2e2478ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.118ex; height:6.176ex;" alt="{\displaystyle \liminf _{n\to \infty }\left({\frac {p_{n+1}-p_{n}}{\log p_{n}}}\right)=0~.}"></span></dd></dl> <p>On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example, <span class="nowrap"><span class="texhtml"><i>c</i> ln ln <i>p</i> </span>.</span> </p><p>By assuming the <a href="/wiki/Elliott%E2%80%93Halberstam_conjecture" title="Elliott–Halberstam conjecture">Elliott–Halberstam conjecture</a> or a slightly weaker version, they were able to show that there are infinitely many <span class="texhtml mvar" style="font-style:italic;">n</span> such that at least two of <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml"><i>n</i> + 2</span>, <span class="texhtml"><i>n</i> + 6</span>, <span class="texhtml"><i>n</i> + 8</span>, <span class="texhtml"><i>n</i> + 12</span>, <span class="texhtml"><i>n</i> + 18</span>, or <span class="texhtml"><i>n</i> + 20</span> are prime. Under a stronger hypothesis they showed that for infinitely many <span class="texhtml mvar" style="font-style:italic;">n</span>, at least two of <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml"><i>n</i> + 2</span>, <span class="texhtml"><i>n</i> + 4</span>, and <span class="texhtml"><i>n</i> + 6</span> are prime. </p><p>The result of <a href="/wiki/Yitang_Zhang" title="Yitang Zhang">Yitang Zhang</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n})<N~\mathrm {with} ~N=7\times 10^{7},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mtext> </mtext> <mi>N</mi> <mo>=</mo> <mn>7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n})<N~\mathrm {with} ~N=7\times 10^{7},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27ce7e56dff3f253533a104ca33b0932170c1b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.831ex; height:4.176ex;" alt="{\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n})<N~\mathrm {with} ~N=7\times 10^{7},}"></span></dd></dl> <p>is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard have reduced the bound: the <a href="/wiki/Limit_inferior" class="mw-redirect" title="Limit inferior">limit inferior</a> is at most 246.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Conjectures">Conjectures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=5" title="Edit section: Conjectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="First_Hardy–Littlewood_conjecture"><span id="First_Hardy.E2.80.93Littlewood_conjecture"></span>First Hardy–Littlewood conjecture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=6" title="Edit section: First Hardy–Littlewood conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/First_Hardy%E2%80%93Littlewood_conjecture" title="First Hardy–Littlewood conjecture">first Hardy–Littlewood conjecture</a> (named after <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> and <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">John Littlewood</a>) is a generalization of the twin prime conjecture. It is concerned with the distribution of <a href="/wiki/Prime_constellation" class="mw-redirect" title="Prime constellation">prime constellations</a>, including twin primes, in analogy to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. Let <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ce2e4bf80be87c950bf4bc31f98a9798805bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle \pi _{2}(x)}"></span>⁠</span> denote the number of primes <span class="texhtml"><i>p</i> ≤ <i>x</i></span> such that <span class="texhtml"><i>p</i> + 2</span> is also prime. Define the <i>twin prime constant</i> <span class="texhtml"><i>C</i><sub>2</sub></span> as<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}=\prod _{\textstyle {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>p</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> <mo>,</mo> </mrow> </mrow> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>0.660161815846869573927812110014</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}=\prod _{\textstyle {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e90d98357fa8da31df09c666a1f04ea73965ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:73.025ex; height:8.843ex;" alt="{\displaystyle C_{2}=\prod _{\textstyle {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .}"></span> (Here the product extends over all prime numbers <span class="texhtml"><i>p</i> ≥ 3</span>.) Then a special case of the first Hardy-Littlewood conjecture is that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∼</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd45d571221e688ff4fd06986ca3044c0de00cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.105ex; height:6.176ex;" alt="{\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}}"></span> in the sense that the quotient of the two expressions <a href="/wiki/Limit_of_a_function" title="Limit of a function">tends to</a> 1 as <span class="texhtml mvar" style="font-style:italic;">x</span> approaches infinity.<sup id="cite_ref-Bateman-Diamond-2004_6-1" class="reference"><a href="#cite_note-Bateman-Diamond-2004-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (The second ~ is not part of the conjecture and is proven by <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>.) </p><p>The conjecture can be justified (but not proven) by assuming that <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\ln t}}}"> <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> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\ln t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df22aaee28667b7b958e3b271ad1a90142967ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.188ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{\ln t}}}"></span>⁠</span> describes the <a href="/wiki/Density_function" class="mw-redirect" title="Density function">density function</a> of the prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ce2e4bf80be87c950bf4bc31f98a9798805bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle \pi _{2}(x)}"></span>⁠</span> above. </p><p>The fully general first Hardy–Littlewood conjecture on <a href="/wiki/Prime_k-tuple" title="Prime k-tuple">prime <span class="texhtml mvar" style="font-style:italic;">k</span>-tuples</a> (not given here) implies that the <a href="/wiki/Second_Hardy%E2%80%93Littlewood_conjecture" title="Second Hardy–Littlewood conjecture"><i>second</i> Hardy–Littlewood conjecture</a> is false. </p><p>This conjecture has been extended by <a href="/wiki/Dickson%27s_conjecture" title="Dickson's conjecture">Dickson's conjecture</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Polignac's_conjecture"><span id="Polignac.27s_conjecture"></span>Polignac's conjecture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=7" title="Edit section: Polignac's conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Twin_prime" title="Special:EditPage/Twin prime">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Twin+prime%22">"Twin prime"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Twin+prime%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Twin+prime%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Twin+prime%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Twin+prime%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Twin+prime%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><a href="/wiki/Polignac%27s_conjecture" title="Polignac's conjecture">Polignac's conjecture</a> from 1849 states that for every positive even integer <span class="texhtml mvar" style="font-style:italic;">k</span>, there are infinitely many consecutive prime pairs <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">p′</span> such that <span class="texhtml"><i>p</i>′ − <i>p</i> = <i>k</i></span> (i.e. there are infinitely many <a href="/wiki/Prime_gap" title="Prime gap">prime gaps</a> of size <span class="texhtml mvar" style="font-style:italic;">k</span>). The case <span class="texhtml"><i>k</i> = 2</span> is the <b>twin prime conjecture</b>. The conjecture has not yet been proven or disproven for any specific value of <span class="texhtml mvar" style="font-style:italic;">k</span>, but Zhang's result proves that it is true for at least one (currently unknown) value of <span class="texhtml mvar" style="font-style:italic;">k</span>. Indeed, if such a <span class="texhtml mvar" style="font-style:italic;">k</span> did not exist, then for any positive even natural number <span class="texhtml mvar" style="font-style:italic;">N</span> there are at most finitely many <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n+1}-p_{n}=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n+1}-p_{n}=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da4534758a23ffd9d7bfd26e1c42630f1cc491c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.945ex; height:2.343ex;" alt="{\displaystyle p_{n+1}-p_{n}=m}"></span> for all <span class="texhtml"><i>m</i> < <i>N</i></span> and so for <span class="texhtml mvar" style="font-style:italic;">n</span> large enough we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n+1}-p_{n}>N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n+1}-p_{n}>N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8301df6150e47fa42db3d629577a59e590d2e36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.615ex; height:2.509ex;" alt="{\displaystyle p_{n+1}-p_{n}>N,}"></span> which would contradict Zhang's result.<sup id="cite_ref-dePolignac-1849_8-1" class="reference"><a href="#cite_note-dePolignac-1849-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Large_twin_primes">Large twin primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=8" title="Edit section: Large twin primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beginning in 2007, two <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> projects, <a href="/wiki/Twin_Prime_Search" title="Twin Prime Search">Twin Prime Search</a> and <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a>, have produced several record-largest twin primes. As of August 2022<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Twin_prime&action=edit">[update]</a></sup>, the current largest twin prime pair known is <span class="nowrap"> 2996863034895 × 2<sup>1290000</sup> ± 1 ,</span><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> with 388,342 decimal digits. It was discovered in September 2016.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>There are 808,675,888,577,436 twin prime pairs below 10<sup><span class="nowrap"><span data-sort-value="7001180000000000000♠"></span>18</span></sup>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>An empirical analysis of all prime pairs up to 4.35 × 10<sup><span class="nowrap"><span data-sort-value="7001150000000000000♠"></span>15</span></sup> shows that if the number of such pairs less than <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml"><i>f</i> (<i>x</i>) ·<i>x</i> /(log <i>x</i>)<sup>2</sup> </span> then <span class="texhtml"><i>f</i> (<i>x</i>)</span> is about 1.7 for small <span class="texhtml mvar" style="font-style:italic;">x</span> and decreases towards about 1.3 as <span class="texhtml mvar" style="font-style:italic;">x</span> tends to infinity. The limiting value of <span class="texhtml"><i>f</i> (<i>x</i>)</span> is conjectured to equal twice the twin prime constant (<span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A114907" class="extiw" title="oeis:A114907">A114907</a></span>) (not to be confused with <a href="/wiki/Brun%27s_constant" class="mw-redirect" title="Brun's constant">Brun's constant</a>), according to the Hardy–Littlewood conjecture. </p> <div class="mw-heading mw-heading2"><h2 id="Other_elementary_properties">Other elementary properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=9" title="Edit section: Other elementary properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every third <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a <a href="/wiki/Chen_prime" title="Chen prime">Chen prime</a>. </p><p>If <i>m</i> − 4 or <i>m</i> + 6 is also prime then the three primes are called a <a href="/wiki/Prime_triplet" title="Prime triplet">prime triplet</a>. </p><p>It has been proven<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> that the pair (<i>m</i>, <i>m</i> + 2) is a twin prime if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4((m-1)!+1)\equiv -m{\pmod {m(m+2)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>m</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4((m-1)!+1)\equiv -m{\pmod {m(m+2)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de640e4dc7dbc45825f8c01a82e7e67662f4d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.645ex; height:2.843ex;" alt="{\displaystyle 4((m-1)!+1)\equiv -m{\pmod {m(m+2)}}.}"></span></dd></dl> <p>For a twin prime pair of the form (6<i>n</i> − 1, 6<i>n</i> + 1) for some natural number <i>n</i> > 1, <i>n</i> must end in the digit 0, 2, 3, 5, 7, or 8 (<span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A002822" class="extiw" title="oeis:A002822">A002822</a></span>). If <i>n</i> were to end in 1 or 6, 6<i>n</i> would end in 6, and 6<i>n</i> −1 would be a multiple of 5. This is not prime unless <i>n</i> = 1. Likewise, if <i>n</i> were to end in 4 or 9, 6<i>n</i> would end in 4, and 6<i>n</i> +1 would be a multiple of 5. The same rule applies modulo any prime <i>p</i> ≥ 5: If <i>n</i> ≡ ±6<sup>−1</sup> (mod <i>p</i>), then one of the pair will be divisible by <i>p</i> and will not be a twin prime pair unless 6<i>n</i> = <i>p</i> ±1. <i>p</i> = 5 just happens to produce particularly simple patterns in base 10. </p> <div class="mw-heading mw-heading2"><h2 id="Isolated_prime">Isolated prime</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=10" title="Edit section: Isolated prime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>isolated prime</b> (also known as <b>single prime</b> or <b>non-twin prime</b>) is a prime number <i>p</i> such that neither <i>p</i> − 2 nor <i>p</i> + 2 is prime. In other words, <i>p</i> is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both <a href="/wiki/Composite_number" title="Composite number">composite</a>. </p><p>The first few isolated primes are </p> <dl><dd><a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, <a href="/wiki/23_(number)" title="23 (number)">23</a>, <a href="/wiki/37_(number)" title="37 (number)">37</a>, <a href="/wiki/47_(number)" title="47 (number)">47</a>, <a href="/wiki/53_(number)" title="53 (number)">53</a>, <a href="/wiki/67_(number)" title="67 (number)">67</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>, ... <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A007510" class="extiw" title="oeis:A007510">A007510</a></span>.</dd></dl> <p>It follows from <a href="/wiki/Brun%27s_theorem" title="Brun's theorem">Brun's theorem</a> that <a href="/wiki/Almost_all#Meaning_in_number_theory" title="Almost all">almost all</a> primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold <i>n</i> and the number of all primes less than <i>n</i> tends to 1 as <i>n</i> tends to infinity. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin prime</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li> <li><a href="/wiki/Prime_k-tuple" title="Prime k-tuple">Prime <i>k</i>-tuple</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Prime quadruplet</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Prime triplet</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy prime</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFThomas,_Kelly_Devine2014" class="citation magazine cs1">Thomas, Kelly Devine (Summer 2014). <a rel="nofollow" class="external text" href="https://www.ias.edu/ideas/2014/zhang-breakthrough">"Yitang Zhang's spectacular mathematical journey"</a>. <i>The Institute Letter</i>. Princeton, NJ: <a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">Institute for Advanced Study</a> – via ias.edu.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Institute+Letter&rft.atitle=Yitang+Zhang%27s+spectacular+mathematical+journey&rft.ssn=summer&rft.date=2014&rft.au=Thomas%2C+Kelly+Devine&rft_id=https%3A%2F%2Fwww.ias.edu%2Fideas%2F2014%2Fzhang-breakthrough&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1">Tao, Terry, Ph.D. (presenter) (7 October 2014). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=pp06oGD4m00&t=425"><i>Small and large gaps between the primes</i></a> (video lecture). <a href="/wiki/University_of_California,_Los_Angeles" title="University of California, Los Angeles">UCLA</a> Department of Mathematics – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Small+and+large+gaps+between+the+primes&rft.pub=UCLA+Department+of+Mathematics&rft.date=2014-10-07&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dpp06oGD4m00%26t%3D425&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://primes.utm.edu/lists/small/100ktwins.txt">"The first 100,000 twin primes (only first member of pair)"</a> <span class="cs1-format">(plain text)</span>. Lists. <i>The Prime Pages (primes.utm.edu)</i>. Martin, TN: <a href="/wiki/University_of_Tennessee,_Martin" class="mw-redirect" title="University of Tennessee, Martin">U.T. 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Martin, TN: <a href="/wiki/University_of_Tennessee,_Martin" class="mw-redirect" title="University of Tennessee, Martin">U.T. Martin</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-09-27</span></span>.</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+%28primes.utm.edu%29&rft.atitle=Are+all+primes+%28past+2+and+3%29+of+the+forms+%3Cspan+class%3D%22texhtml+%22+%3E6n%2B1%3C%2Fspan%3E+and+%3Cspan+class%3D%22texhtml+%22+%3E6n%E2%88%921%3C%2Fspan%3E%3F&rft.aulast=Caldwell&rft.aufirst=Chris+K.&rft_id=https%3A%2F%2Fprimes.utm.edu%2Fnotes%2Ffaq%2Fsix.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-Brun-1915-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brun-1915_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrun1915" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Viggo_Brun" title="Viggo Brun">Brun, V.</a> (1915). "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" [On Goldbach's rule and the number of prime number pairs]. <i>Archiv for Mathematik og Naturvidenskab</i> (in German). <b>34</b> (8): 3–19. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0365-4524">0365-4524</a>. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:45.0330.16">45.0330.16</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+for+Mathematik+og+Naturvidenskab&rft.atitle=%C3%9Cber+das+Goldbachsche+Gesetz+und+die+Anzahl+der+Primzahlpaare&rft.volume=34&rft.issue=8&rft.pages=3-19&rft.date=1915&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A45.0330.16%23id-name%3DJFM&rft.issn=0365-4524&rft.aulast=Brun&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-Bateman-Diamond-2004-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bateman-Diamond-2004_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bateman-Diamond-2004_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatemanDiamond2004" class="citation book cs1"><a href="/wiki/Paul_T._Bateman" title="Paul T. 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OEIS Foundation<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-01</span></span>.</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%3BA007508%26%23x20%3B%28Number+of+twin+prime+pairs+below+10%3Csup%3E%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E%3C%2Fsup%3E%29&rft_id=https%3A%2F%2Foeis.org%2FA007508&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOliveira_e_Silva,_Tomás2008" class="citation web cs1">Oliveira e Silva, Tomás (7 April 2008). <a rel="nofollow" class="external text" href="http://www.ieeta.pt/~tos/primes.html">"Tables of values of <span class="texhtml"><i>π</i>(<i>x</i>)</span> and of <span class="texhtml"><i>π</i><sub>2</sub>(<i>x</i>)</span>"</a>. <a href="/wiki/Aveiro_University" class="mw-redirect" title="Aveiro University">Aveiro University</a><span class="reference-accessdate">. Retrieved <span class="nowrap">7 January</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tables+of+values+of+%3Cspan+class%3D%22texhtml+%22+%3E%26pi%3B%28x%29%3C%2Fspan%3E+and+of+%3Cspan+class%3D%22texhtml+%22+%3E%26pi%3B%3Csub%3E2%3C%2Fsub%3E%28x%29%3C%2Fspan%3E&rft.pub=Aveiro+University&rft.date=2008-04-07&rft.au=Oliveira+e+Silva%2C+Tom%C3%A1s&rft_id=http%3A%2F%2Fwww.ieeta.pt%2F~tos%2Fprimes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._A._Clement1949" class="citation journal cs1">P. A. Clement (January 1949). <a rel="nofollow" class="external text" href="http://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1949%20ClementCongruences%20for%20Sets%20of%20Primes.pdf">"Congruences for sets of primes"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>56</b> (1): 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/10.2307%2F2305816">10.2307/2305816</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/2305816">2305816</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=Congruences+for+sets+of+primes&rft.volume=56&rft.issue=1&rft.pages=23-25&rft.date=1949-01&rft_id=info%3Adoi%2F10.2307%2F2305816&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2305816%23id-name%3DJSTOR&rft.au=P.+A.+Clement&rft_id=http%3A%2F%2Fwww.math.stonybrook.edu%2F~moira%2Fmat331-spr10%2Fpapers%2F1949%2520ClementCongruences%2520for%2520Sets%2520of%2520Primes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloanePlouffe1995" class="citation book cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, Neil</a>; <a href="/wiki/Simon_Plouffe" title="Simon Plouffe">Plouffe, Simon</a> (1995). <i>The Encyclopedia of Integer Sequences</i>. San Diego, CA: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-558630-2" title="Special:BookSources/0-12-558630-2"><bdi>0-12-558630-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Encyclopedia+of+Integer+Sequences&rft.place=San+Diego%2C+CA&rft.pub=Academic+Press&rft.date=1995&rft.isbn=0-12-558630-2&rft.aulast=Sloane&rft.aufirst=Neil&rft.au=Plouffe%2C+Simon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twin_prime&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Twins">"Twins"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <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=Twins&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTwins&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://primes.utm.edu/top20/page.php?id=1">Top-20 Twin Primes</a> at Chris Caldwell's <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a></li> <li>Xavier Gourdon, Pascal Sebah: <a rel="nofollow" class="external text" href="http://numbers.computation.free.fr/Constants/Primes/twin.html"><i>Introduction to Twin Primes and Brun's Constant</i></a></li> <li><a rel="nofollow" class="external text" href="http://mersenneforum.org/showpost.php?p=96237&postcount=51">"Official press release"</a> of 58711-digit twin prime record</li> <li><span class="citation mathworld" id="Reference-Mathworld-Twin_Primes"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TwinPrimes.html">"Twin Primes"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Twin+Primes&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTwinPrimes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATwin+prime" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://arnflo.se/~site_files/Other/twinprimes">The 20 000 first twin primes</a></li> <li><a rel="nofollow" class="external text" href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records">Polymath: Bounded gaps between primes</a></li> <li><a rel="nofollow" class="external text" href="https://www.wired.com/wiredscience/2013/11/prime/">Sudden Progress on Prime Number Problem Has Mathematicians Buzzing</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 ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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.navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></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" 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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 (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>n</i></sup></sup> + 1</span>)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>p</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</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 <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></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"><i>k</i>-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 class="mw-selflink selflink">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i> + <i>a·n</i>, <i>n</i> = 0, 1, 2, 3, ...</span>)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (<span class="texhtml texhtml-big" style="font-size:110%;">consecutive <i>p</i> − <i>n</i>, <i>p</i>, <i>p</i> + <i>n</i></span>)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i> ± 1, 2<i>n</i> ± 1, 4<i>n</i> ± 1, …</span>)</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 (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> + 1</span>)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> ± 1, 4<i>p</i> ± 3, 8<i>p</i> ± 7, ...</span>)</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="Prime_number_conjectures" 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_conjectures" title="Template:Prime number conjectures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_conjectures" title="Template talk:Prime number conjectures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_conjectures" title="Special:EditPage/Template:Prime number conjectures"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_conjectures" style="font-size:114%;margin:0 4em">Prime number conjectures</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy%E2%80%93Littlewood_conjecture" class="mw-redirect" title="Hardy–Littlewood conjecture">Hardy–Littlewood</a> <ul><li><a href="/wiki/First_Hardy%E2%80%93Littlewood_conjecture" title="First Hardy–Littlewood conjecture">1st</a></li> <li><a href="/wiki/Second_Hardy%E2%80%93Littlewood_conjecture" title="Second Hardy–Littlewood conjecture">2nd</a></li></ul></li> <li><a href="/wiki/Agoh%E2%80%93Giuga_conjecture" title="Agoh–Giuga conjecture">Agoh–Giuga</a></li> <li><a href="/wiki/Andrica%27s_conjecture" title="Andrica's conjecture">Andrica's</a></li> <li><a href="/wiki/Artin%27s_conjecture_on_primitive_roots" title="Artin's conjecture on primitive roots">Artin's</a></li> <li><a href="/wiki/Bateman%E2%80%93Horn_conjecture" title="Bateman–Horn conjecture">Bateman–Horn</a></li> <li><a href="/wiki/Brocard%27s_conjecture" title="Brocard's conjecture">Brocard's</a></li> <li><a href="/wiki/Bunyakovsky_conjecture" title="Bunyakovsky conjecture">Bunyakovsky</a></li> <li><a href="/wiki/Chinese_hypothesis" title="Chinese hypothesis">Chinese hypothesis</a></li> <li><a href="/wiki/Cram%C3%A9r%27s_conjecture" title="Cramér's conjecture">Cramér's (Shanks')</a></li> <li><a href="/wiki/Dickson%27s_conjecture" title="Dickson's conjecture">Dickson's</a></li> <li><a href="/wiki/Elliott%E2%80%93Halberstam_conjecture" title="Elliott–Halberstam conjecture">Elliott–Halberstam</a></li> <li><a href="/wiki/Firoozbakht%27s_conjecture" title="Firoozbakht's conjecture">Firoozbakht's (Forgues', Nicholson's, Farhadian's)</a></li> <li><a href="/wiki/Gilbreath%27s_conjecture" title="Gilbreath's conjecture">Gilbreath's</a></li> <li><a href="/wiki/Grimm%27s_conjecture" title="Grimm's conjecture">Grimm's</a></li> <li><a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's problems</a> <ul><li><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's</a> <ul><li><a href="/wiki/Goldbach%27s_weak_conjecture" title="Goldbach's weak conjecture">weak</a></li></ul></li> <li><a href="/wiki/Legendre%27s_conjecture" title="Legendre's conjecture">Legendre's</a></li> <li><a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">Twin prime</a></li></ul></li> <li><a href="/wiki/Legendre%27s_constant" title="Legendre's constant">Legendre's constant</a></li> <li><a href="/wiki/Lemoine%27s_conjecture" title="Lemoine's conjecture">Lemoine's</a></li> <li><a href="/wiki/Mersenne_conjectures" title="Mersenne conjectures">Mersenne</a></li> <li><a href="/wiki/Oppermann%27s_conjecture" title="Oppermann's conjecture">Oppermann's</a></li> <li><a href="/wiki/Polignac%27s_conjecture" title="Polignac's conjecture">Polignac's</a></li> <li><a href="/wiki/P%C3%B3lya_conjecture" title="Pólya conjecture">Pólya</a></li> <li><a href="/wiki/Schinzel%27s_hypothesis_H" title="Schinzel's hypothesis H">Schinzel's hypothesis H</a></li> <li><a href="/wiki/Waring%27s_prime_number_conjecture" title="Waring's prime number conjecture">Waring's prime number</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Twin_prime&oldid=1259252991">https://en.wikipedia.org/w/index.php?title=Twin_prime&oldid=1259252991</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Classes_of_prime_numbers" title="Category:Classes of prime numbers">Classes of prime 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