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data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-Fermat-prímek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fermat-prímek"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Fermat-prímek</span> </div> </a> <ul id="toc-Fermat-prímek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tulajdonságok" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tulajdonságok"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tulajdonságok</span> </div> </a> <ul id="toc-Tulajdonságok-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prímteszt" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Prímteszt"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Prímteszt</span> </div> </a> <ul id="toc-Prímteszt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prímosztók" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Prímosztók"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Prímosztók</span> </div> </a> <ul id="toc-Prímosztók-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sejtések" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejtések"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Sejtések</span> </div> </a> <ul id="toc-Sejtések-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Források</span> </div> </a> <ul id="toc-Források-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Fermat-számok</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="Ugrás egy más nyelvű szócikkre. Elérhető 41 nyelven" > <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-41" 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">41 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fermat_number" title="Fermat number – angol" lang="en" hreflang="en" data-title="Fermat number" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/Fermat_t%C3%A6l" title="Fermat tæl – óangol" lang="ang" hreflang="ang" data-title="Fermat tæl" data-language-autonym="Ænglisc" data-language-local-name="óangol" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%81%D9%8A%D8%B1%D9%85%D8%A7" title="عدد فيرما – arab" lang="ar" hreflang="ar" data-title="عدد فيرما" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ferma_%C9%99d%C9%99dl%C9%99ri" title="Ferma ədədləri – azerbajdzsáni" lang="az" hreflang="az" data-title="Ferma ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BD%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число на Ферма – bolgár" lang="bg" hreflang="bg" data-title="Число на Ферма" data-language-autonym="Български" data-language-local-name="bolgár" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%AE%E0%A6%BE_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="ফার্মা সংখ্যা – bangla" lang="bn" hreflang="bn" data-title="ফার্মা সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Fermat" title="Nombre de Fermat – katalán" lang="ca" hreflang="ca" data-title="Nombre de Fermat" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</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_%D9%81%DB%8E%D8%B1%D9%85%D8%A7" title="ژمارەی فێرما – közép-ázsiai kurd" lang="ckb" hreflang="ckb" data-title="ژمارەی فێرما" data-language-autonym="کوردی" data-language-local-name="közép-ázsiai kurd" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fermatovo_%C4%8D%C3%ADslo" title="Fermatovo číslo – cseh" lang="cs" hreflang="cs" data-title="Fermatovo číslo" data-language-autonym="Čeština" data-language-local-name="cseh" 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/Fermatprimtal" title="Fermatprimtal – dán" lang="da" hreflang="da" data-title="Fermatprimtal" data-language-autonym="Dansk" data-language-local-name="dán" 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/Fermat-Zahl" title="Fermat-Zahl – német" lang="de" hreflang="de" data-title="Fermat-Zahl" data-language-autonym="Deutsch" data-language-local-name="német" 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%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%A6%CE%B5%CF%81%CE%BC%CE%AC" title="Αριθμός Φερμά – görög" lang="el" hreflang="el" data-title="Αριθμός Φερμά" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nombro_de_Fermat" title="Nombro de Fermat – eszperantó" lang="eo" hreflang="eo" data-title="Nombro de Fermat" data-language-autonym="Esperanto" data-language-local-name="eszperantó" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – spanyol" lang="es" hreflang="es" data-title="Número de Fermat" data-language-autonym="Español" data-language-local-name="spanyol" class="interlanguage-link-target"><span>Español</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_%D9%81%D8%B1%D9%85%D8%A7" title="اعداد فرما – perzsa" lang="fa" hreflang="fa" data-title="اعداد فرما" data-language-autonym="فارسی" data-language-local-name="perzsa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fermat%E2%80%99n_luku" title="Fermat’n luku – finn" lang="fi" hreflang="fi" data-title="Fermat’n luku" data-language-autonym="Suomi" data-language-local-name="finn" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Fermat" title="Nombre de Fermat – francia" lang="fr" hreflang="fr" data-title="Nombre de Fermat" data-language-autonym="Français" data-language-local-name="francia" 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/N%C3%BAmero_de_Fermat" title="Número de Fermat – gallego" lang="gl" hreflang="gl" data-title="Número de Fermat" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94" title="מספר פרמה – héber" lang="he" hreflang="he" data-title="מספר פרמה" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%A5%D6%80%D5%B4%D5%A1%D5%B5%D5%AB_%D5%A9%D5%AB%D5%BE" title="Ֆերմայի թիվ – örmény" lang="hy" hreflang="hy" data-title="Ֆերմայի թիվ" data-language-autonym="Հայերեն" data-language-local-name="örmény" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Fermat" title="Numero di Fermat – olasz" lang="it" hreflang="it" data-title="Numero di Fermat" data-language-autonym="Italiano" data-language-local-name="olasz" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9E%E3%83%BC%E6%95%B0" title="フェルマー数 – japán" lang="ja" hreflang="ja" data-title="フェルマー数" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%98%EB%A5%B4%EB%A7%88_%EC%88%98" title="페르마 수 – koreai" lang="ko" hreflang="ko" data-title="페르마 수" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ferma_skai%C4%8Dius" title="Ferma skaičius – litván" lang="lt" hreflang="lt" data-title="Ferma skaičius" data-language-autonym="Lietuvių" data-language-local-name="litván" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fermatgetal" title="Fermatgetal – holland" lang="nl" hreflang="nl" data-title="Fermatgetal" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fermattal" title="Fermattal – norvég (nynorsk)" lang="nn" hreflang="nn" data-title="Fermattal" data-language-autonym="Norsk nynorsk" data-language-local-name="norvég (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fermat-tallene" title="Fermat-tallene – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Fermat-tallene" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Fermata" title="Liczby Fermata – lengyel" lang="pl" hreflang="pl" data-title="Liczby Fermata" data-language-autonym="Polski" data-language-local-name="lengyel" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_%C3%ABd_Fermat" title="Nùmer ëd Fermat – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer ëd Fermat" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_de_Fermat" title="Número de Fermat – portugál" lang="pt" hreflang="pt" data-title="Número de Fermat" data-language-autonym="Português" data-language-local-name="portugál" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Число Ферма – orosz" lang="ru" hreflang="ru" data-title="Число Ферма" data-language-autonym="Русский" data-language-local-name="orosz" 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/Fermat_number" title="Fermat number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fermat number" 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/Fermatovo_pra%C5%A1tevilo" title="Fermatovo praštevilo – szlovén" lang="sl" hreflang="sl" data-title="Fermatovo praštevilo" data-language-autonym="Slovenščina" data-language-local-name="szlovén" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fermattal" title="Fermattal – svéd" lang="sv" hreflang="sv" data-title="Fermattal" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%83%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="ஃபெர்மா எண் – tamil" lang="ta" hreflang="ta" data-title="ஃபெர்மா எண்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%81%E0%B8%9F%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%B2" title="จำนวนแฟร์มา – thai" lang="th" hreflang="th" data-title="จำนวนแฟร์มา" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fermat_say%C4%B1lar%C4%B1" title="Fermat sayıları – török" lang="tr" hreflang="tr" data-title="Fermat sayıları" data-language-autonym="Türkçe" data-language-local-name="török" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B5%D1%80%D0%BC%D0%B0" title="Числа Ферма – ukrán" lang="uk" hreflang="uk" data-title="Числа Ферма" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_Fermat" title="Số Fermat – vietnámi" lang="vi" hreflang="vi" data-title="Số Fermat" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B2%BB%E9%A6%AC%E6%95%B8" title="費馬數 – kínai" lang="zh" hreflang="zh" data-title="費馬數" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B2%BB%E9%A6%AC%E6%95%B8" title="費馬數 – kantoni" lang="yue" hreflang="yue" data-title="費馬數" data-language-autonym="粵語" data-language-local-name="kantoni" 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							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Fermat-sz%C3%A1mok">ellenőrizve</a>: <i>2021. május 21.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><p>A <b>Fermat-számok</b> a <a href="/wiki/Matematika" title="Matematika">matematikában</a> elsőként <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> (ejtsd: pier dö fermá) által tanulmányozott (majd később róla elnevezett) pozitív egész számok, mégpedig a következő <a href="/wiki/Sorozat_(matematika)" title="Sorozat (matematika)">sorozat</a> elemei: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"></span>, ahol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> nemnegatív egész.</center> <p>Tehát ha egy kettőhatványt kettes hatványalapra emelünk és hozzáadunk egyet, Fermat-számot kapunk. </p><p>Az első nyolc Fermat-szám: </p> <dl><dd><i>F</i><sub>0</sub> = 2<sup>1</sup> + 1 = <a href="/wiki/3_(sz%C3%A1m)" title="3 (szám)">3</a></dd> <dd><i>F</i><sub>1</sub> = 2<sup>2</sup> + 1 = <a href="/wiki/5_(sz%C3%A1m)" title="5 (szám)">5</a></dd> <dd><i>F</i><sub>2</sub> = 2<sup>4</sup> + 1 = <a href="/wiki/17_(sz%C3%A1m)" title="17 (szám)">17</a></dd> <dd><i>F</i><sub>3</sub> = 2<sup>8</sup> + 1 = <a href="/wiki/257_(sz%C3%A1m)" title="257 (szám)">257</a></dd> <dd><i>F</i><sub>4</sub> = 2<sup>16</sup> + 1 = 65537</dd> <dd><i>F</i><sub>5</sub> = 2<sup>32</sup> + 1 = 4294967297 = 641 × 6700417</dd> <dd><i>F</i><sub>6</sub> = 2<sup>64</sup> + 1 = 18446744073709551617 = 274177 × 67280421310721</dd> <dd><i>F</i><sub>7</sub> = 2<sup>128</sup> + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721</dd></dl> <p>Jelenleg csak az első 12 Fermat-szám prímtényezőkre bontását ismerjük teljesen. Az ismert információk megtalálhatók a <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160210152415/http://www.prothsearch.net/fermat.html">(Prime Factors of Fermat Numbers)</a> lapon. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Fermat-prímek"><span id="Fermat-pr.C3.ADmek"></span>Fermat-prímek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=1" title="Szakasz szerkesztése: Fermat-prímek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Searchtool_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/14px-Searchtool_right.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/21px-Searchtool_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/28px-Searchtool_right.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></td><td><i>Bővebben: <a href="/wiki/Fermat-pr%C3%ADmek" title="Fermat-prímek">Fermat-prímek</a></i></td></tr></tbody></table> <p>Ha F<sub>n</sub> = 2<sup><i>n</i></sup> + 1 <a href="/wiki/Pr%C3%ADmsz%C3%A1mok" title="Prímszámok">prímszám</a>, akkor – amint ez megmutatható – <i>n</i> szükségképp 2-hatvány. Ugyanis ha nem lenne 2-nek egy hatványa, akkor létezne <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-nek 1-nél nagyobb páratlan osztója, mondjuk <i>n</i> = <i>ab,</i> ahol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60e5500fb07fa1c8f222e074904b6e9c91c63e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.258ex; height:2.343ex;" alt="{\displaystyle b\geq 3}"></span> páratlan és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06f20f72b40654833aff35ad637c3bb7c36fe5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.491ex; height:2.343ex;" alt="{\displaystyle a\geq 1}"></span> természetes számok, így pedig <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}+1=2^{(ab)}+1=(2^{a})^{b}+1\equiv (-1)^{b}+1=0{\pmod {(2^{a}+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>≡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}+1=2^{(ab)}+1=(2^{a})^{b}+1\equiv (-1)^{b}+1=0{\pmod {(2^{a}+1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8274c94f4e76c5fd4899b7d82f49dc520ea9da09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.3ex; height:3.343ex;" alt="{\displaystyle 2^{n}+1=2^{(ab)}+1=(2^{a})^{b}+1\equiv (-1)^{b}+1=0{\pmod {(2^{a}+1)}}}"></span>. Más szóval, minden 2<sup><i>n</i></sup> + 1 alakú prím Fermat-szám, így Fermat-prímnek nevezendő. Az ismert Fermat-prímek a következők: <i>F</i><sub>0</sub>, <i>F</i><sub>1</sub>, <i>F</i><sub>2</sub>, <i>F</i><sub>3</sub> és <i>F</i><sub>4</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Tulajdonságok"><span id="Tulajdons.C3.A1gok"></span>Tulajdonságok</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=2" title="Szakasz szerkesztése: Tulajdonságok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Fermat-számok sorozata eleget tesz a következő, rekurzív definícióra is alkalmas egyenlőségeknek (<i>n</i> ≥ 2-re), melyek mindegyike <a href="/wiki/Teljes_indukci%C3%B3" title="Teljes indukció">indukcióval</a> belátható: </p> <dl><dd><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 F_{n}=(F_{n-1}-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ff1c52a7a38d3afcce05117270671017d1b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.494ex; height:3.176ex;" alt="{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}"></span>,</dd> <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 F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d330d839625d9bec0ae45d2b7fc591e2a62065e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.308ex; height:3.509ex;" alt="{\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}"></span>,</dd> <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 F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99e131e1d0c14da739b86e0e145409caae43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.212ex; height:3.343ex;" alt="{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}"></span>,</dd> <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 F_{n}=F_{0}\cdots F_{n-1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42e25a368baa3cee4a5f1e1db0a00d38e197a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.674ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i=0}^{n-1}F_{i}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i=0}^{n-1}F_{i}+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7d01119b610d1ea408dd8ea542157f57629dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.771ex; height:7.343ex;" alt="{\displaystyle \prod _{i=0}^{n-1}F_{i}+2}"></span>.</dd></dl></dd></dl> <p>Az utolsó egyenlőség felhasználásával belátható <b>Goldbach tétele</b>; ti. bármely két Fermat-szám <a href="/wiki/Relat%C3%ADv_pr%C3%ADmek" title="Relatív prímek">relatív prím</a> (mellesleg, ebből meg bizonyítható, hogy végtelen sok prímszám van). </p><p>Azt viszonlyag könnyű belátni, hogy egyik Fermat-szám sem valódi prímhatvány. </p> <div class="mw-heading mw-heading2"><h2 id="Prímteszt"><span id="Pr.C3.ADmteszt"></span>Prímteszt</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=3" title="Szakasz szerkesztése: Prímteszt"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mint speciális alakú számokra oly gyakran, egyszerű <a href="/wiki/Pr%C3%ADmteszt" title="Prímteszt">prímteszt</a> van Fermat-számokra. Ha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>, akkor az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d9d4b1346ac464869a5cee356ae0468de70f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle F_{n}=2^{2^{n}}+1}"></span> Fermat-szám pontosan akkor (akkor és csak akkor) prím, ha </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e895adb2d79055dc3671fe60a2c376718e510c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.194ex; height:4.343ex;" alt="{\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}"></span></center> <p>teljesül. </p> <div class="mw-heading mw-heading2"><h2 id="Prímosztók"><span id="Pr.C3.ADmoszt.C3.B3k"></span>Prímosztók</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=4" title="Szakasz szerkesztése: Prímosztók"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ha <i>k</i> legalább 2, az <i>F<sub>k</sub></i> szám minden prímosztója <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2^{k+2}x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2^{k+2}x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe4e0242e100e2e38ea279fe6c7aacb26e58de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.042ex; height:3.009ex;" alt="{\displaystyle p=2^{k+2}x+1}"></span> alakú. Ennek bizonyításához legyen <i>d</i> 2 rendje mod <i>p</i>, azaz a legkisebb olyan kitevő, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{d}\equiv 1{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{d}\equiv 1{\pmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9653626a80eeb2135ae41e9a772e3934a313ff4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.369ex; height:3.176ex;" alt="{\displaystyle 2^{d}\equiv 1{\pmod {p}}}"></span>. </p><p>Mivel </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{k}}\equiv -1{\pmod {p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{k}}\equiv -1{\pmod {p}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173563dcc671b07806b51ddc0fe9042443ae44f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.646ex; height:3.509ex;" alt="{\displaystyle 2^{2^{k}}\equiv -1{\pmod {p}},}"></span></center> <p><i>d</i> nem osztja 2<sup><i>k</i></sup>-t de osztja 2<sup><i>k</i>+1</sup>-et. De ekkor <i>d</i> csak 2<sup><i>k</i>+1</sup> lehet. Másrészt a <a href="/wiki/Kis_Fermat-t%C3%A9tel" title="Kis Fermat-tétel">kis Fermat-tétel</a> miatt <i>d</i> osztója <i>p</i>-1-nek, azaz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2^{k+2}x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2^{k+2}x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe4e0242e100e2e38ea279fe6c7aacb26e58de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.042ex; height:3.009ex;" alt="{\displaystyle p=2^{k+2}x+1}"></span> alakú, tehát 8-cal osztva 1-et ad maradékul. Ezért a <a href="/wiki/Legendre-szimb%C3%B3lum" title="Legendre-szimbólum">Legendre-szimbólum</a> tulajdonságai miatt </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {p-1}{2}}\equiv \left({\frac {2}{p}}\right)\equiv 1{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {p-1}{2}}\equiv \left({\frac {2}{p}}\right)\equiv 1{\pmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6d16576e90c39701dcb71fde89a81f23f9bf8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.247ex; height:6.176ex;" alt="{\displaystyle 2^{\frac {p-1}{2}}\equiv \left({\frac {2}{p}}\right)\equiv 1{\pmod {p}}}"></span>,</center> <p>ahonnan adódik, hogy <i>d</i> osztja (<i>p</i>-1)/2-t, ami azonos azzal, amit állítottunk. </p><p>Ez megkönnyíti a Fermat-számok prímfelbontását, ami a <a href="/wiki/Mersenne-pr%C3%ADmek" title="Mersenne-prímek">Mersenne-prímek</a> kutatásához hasonló internetes-programozói versennyé kezd válni. Például ha <i>F</i><sub>5</sub>-öt próbáljuk felbontani, a prímosztókat 128<i>x</i>+1 alakban kell keresnünk. Az <i>x</i>=1,3,4 esetek kiesnek, mert ekkor 128<i>x</i>+1 összetett, <i>x</i>=2-re 257=<i>F</i><sub>3</sub>-t kapjuk, ami a fentiek szerint nem oszthatja <i>F</i><sub>5</sub>-öt tehát az első igazi eset <i>p</i>=641 és ez valóban osztó. </p> <div class="mw-heading mw-heading2"><h2 id="Sejtések"><span id="Sejt.C3.A9sek"></span>Sejtések</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=5" title="Szakasz szerkesztése: Sejtések"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sok sejtést lehet a Fermat-számokról felállítani és ezek mindegyike reménytelenül nehéz. Sejtjük, de nem tudjuk bizonyítani, hogy az ismerteken kívül nincs több prím. De még azt sem tudjuk, hogy végtelen sok összetett Fermat-szám van, hogy mind négyzetmentes, vagy akár, hogy végtelen sok négyzetmentes Fermat-szám van. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermat-sz%C3%A1mok&action=edit&section=6" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>17 Lectures on Fermat Numbers: From Number Theory to Geometry,</i> Michal Krizek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, 257 oldal <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/0387953329" title="Speciális:Könyvforrások/0387953329">ISBN 0387953329</a></li> <li>Courant-Robins: <i>Mi a matematika?</i></li></ul> <div class="noprint noviewer" style="overflow: hidden; clear: both;"><div style="margin-left:0; margin-right:2px;"><ul style="display:block; list-style-image:none; list-style-type:none; width:100%; vertical-align:middle; margin:0; padding:0; min-height: 27px;"><li style="float:left; min-height: 27px; line-height:25px; width:100%; margin:0; margin-top:.5em; margin-left:0; margin-right:0; padding:0; border:1px solid #CCF; background-color:#F0EEFF"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:P_cartesian_graph.svg" class="mw-file-description" title="matematika"><img alt="matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/25px-P_cartesian_graph.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/38px-P_cartesian_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/50px-P_cartesian_graph.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Matematikaportál</a></b> • összefoglaló, színes tartalomajánló lap</li></ul></div></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="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Fermat-számok&oldid=23869059">https://hu.wikipedia.org/w/index.php?title=Fermat-számok&oldid=23869059</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Nevezetes_sz%C3%A1msorozatok" title="Kategória:Nevezetes számsorozatok">Nevezetes számsorozatok</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A lap utolsó módosítása: 2021. május 15., 20:13</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. 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