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mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%9E%D7%A1%D7%A4%D7%A8+%D7%A4%D7%A8%D7%9E%D7%94" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%9E%D7%A1%D7%A4%D7%A8+%D7%A4%D7%A8%D7%9E%D7%94" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%9E%D7%A1%D7%A4%D7%A8+%D7%A4%D7%A8%D7%9E%D7%94" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%9E%D7%A1%D7%A4%D7%A8+%D7%A4%D7%A8%D7%9E%D7%94" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> 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</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="תוכן עניינים" 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="מצב תוכן העניינים" > <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">מצב תוכן העניינים</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">מספר פרמה</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="מעבר לערך בשפה אחרת. זמין ב־41 שפות" > <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 שפות</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 – אנגלית" lang="en" hreflang="en" data-title="Fermat number" data-language-autonym="English" data-language-local-name="אנגלית" 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 – אנגלית עתיקה" lang="ang" hreflang="ang" data-title="Fermat tæl" data-language-autonym="Ænglisc" data-language-local-name="אנגלית עתיקה" 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 – ערבית" lang="ar" hreflang="ar" data-title="عدد فيرما" data-language-autonym="العربية" data-language-local-name="ערבית" 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 – אזרית" lang="az" hreflang="az" data-title="Ferma ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="אזרית" 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 – בולגרית" lang="bg" hreflang="bg" data-title="Число на Ферма" data-language-autonym="Български" data-language-local-name="בולגרית" 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 – בנגלית" lang="bn" hreflang="bn" data-title="ফার্মা সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="בנגלית" 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 – קטלאנית" lang="ca" hreflang="ca" data-title="Nombre de Fermat" data-language-autonym="Català" data-language-local-name="קטלאנית" 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 – כורדית סוראנית" lang="ckb" hreflang="ckb" data-title="ژمارەی فێرما" data-language-autonym="کوردی" data-language-local-name="כורדית סוראנית" 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 – צ׳כית" lang="cs" hreflang="cs" data-title="Fermatovo číslo" data-language-autonym="Čeština" data-language-local-name="צ׳כית" 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 – דנית" lang="da" hreflang="da" data-title="Fermatprimtal" data-language-autonym="Dansk" data-language-local-name="דנית" 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 – גרמנית" lang="de" hreflang="de" data-title="Fermat-Zahl" data-language-autonym="Deutsch" data-language-local-name="גרמנית" 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 – יוונית" lang="el" hreflang="el" data-title="Αριθμός Φερμά" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" 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 – אספרנטו" lang="eo" hreflang="eo" data-title="Nombro de Fermat" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" 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 – ספרדית" lang="es" hreflang="es" data-title="Número de Fermat" data-language-autonym="Español" data-language-local-name="ספרדית" 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 – פרסית" lang="fa" hreflang="fa" data-title="اعداد فرما" data-language-autonym="فارسی" data-language-local-name="פרסית" 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 – פינית" lang="fi" hreflang="fi" data-title="Fermat’n luku" data-language-autonym="Suomi" data-language-local-name="פינית" 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 – צרפתית" lang="fr" hreflang="fr" data-title="Nombre de Fermat" data-language-autonym="Français" data-language-local-name="צרפתית" 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 – גליסית" lang="gl" hreflang="gl" data-title="Número de Fermat" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fermat-sz%C3%A1mok" title="Fermat-számok – הונגרית" lang="hu" hreflang="hu" data-title="Fermat-számok" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</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 – ארמנית" lang="hy" hreflang="hy" data-title="Ֆերմայի թիվ" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" 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 – איטלקית" lang="it" hreflang="it" data-title="Numero di Fermat" data-language-autonym="Italiano" data-language-local-name="איטלקית" 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 – יפנית" lang="ja" hreflang="ja" data-title="フェルマー数" data-language-autonym="日本語" data-language-local-name="יפנית" 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 – קוריאנית" lang="ko" hreflang="ko" data-title="페르마 수" data-language-autonym="한국어" data-language-local-name="קוריאנית" 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 – ליטאית" lang="lt" hreflang="lt" data-title="Ferma skaičius" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" 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 – הולנדית" lang="nl" hreflang="nl" data-title="Fermatgetal" data-language-autonym="Nederlands" data-language-local-name="הולנדית" 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 – נורווגית חדשה" lang="nn" hreflang="nn" data-title="Fermattal" data-language-autonym="Norsk nynorsk" data-language-local-name="נורווגית חדשה" 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 – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Fermat-tallene" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" 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 – פולנית" lang="pl" hreflang="pl" data-title="Liczby Fermata" data-language-autonym="Polski" data-language-local-name="פולנית" 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 – פיימונטית" lang="pms" hreflang="pms" data-title="Nùmer ëd Fermat" data-language-autonym="Piemontèis" data-language-local-name="פיימונטית" 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 – פורטוגזית" lang="pt" hreflang="pt" data-title="Número de Fermat" data-language-autonym="Português" data-language-local-name="פורטוגזית" 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 – רוסית" lang="ru" hreflang="ru" data-title="Число Ферма" data-language-autonym="Русский" data-language-local-name="רוסית" 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 – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Fermat number" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" 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 – סלובנית" lang="sl" hreflang="sl" data-title="Fermatovo praštevilo" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" 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 – שוודית" lang="sv" hreflang="sv" data-title="Fermattal" data-language-autonym="Svenska" data-language-local-name="שוודית" 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 – טמילית" lang="ta" hreflang="ta" data-title="ஃபெர்மா எண்" data-language-autonym="தமிழ்" data-language-local-name="טמילית" 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 – תאית" lang="th" hreflang="th" data-title="จำนวนแฟร์มา" data-language-autonym="ไทย" data-language-local-name="תאית" 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 – טורקית" lang="tr" hreflang="tr" data-title="Fermat sayıları" data-language-autonym="Türkçe" data-language-local-name="טורקית" 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 – אוקראינית" lang="uk" hreflang="uk" data-title="Числа Ферма" data-language-autonym="Українська" data-language-local-name="אוקראינית" 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 – וייטנאמית" lang="vi" hreflang="vi" data-title="Số Fermat" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" 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 – סינית" lang="zh" hreflang="zh" data-title="費馬數" data-language-autonym="中文" data-language-local-name="סינית" 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 – קנטונזית" lang="yue" hreflang="yue" data-title="費馬數" data-language-autonym="粵語" data-language-local-name="קנטונזית" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207264#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</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-rtl mw-parser-output" lang="he" dir="rtl"><p>ב<a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D" title="תורת המספרים">תורת המספרים</a>, <b>מספרי פרמה</b> הם <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">מספרים טבעיים</a> מהצורה <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^{\left(2^{n}\right)}+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"> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=2^{\left(2^{n}\right)}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6795d6f59d20ba59b07b6341156a8b3e0ab840c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.275ex; height:3.176ex;" alt="{\displaystyle F_{n}=2^{\left(2^{n}\right)}+1}"></span>, כאשר <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"> <mspace width="thinmathspace" /> <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/a798350c07bbc4b91b4190498a18535149176d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle \,n}"></span> הוא <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A9%D7%9C%D7%9D" title="מספר שלם">מספר שלם</a> לא שלילי. המספרים קרויים על שם ה<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%90%D7%99" title="מתמטיקאי">מתמטיקאי</a> ה<a href="/wiki/%D7%A6%D7%A8%D7%A4%D7%AA" title="צרפת">צרפתי</a> <a href="/wiki/%D7%A4%D7%99%D7%99%D7%A8_%D7%93%D7%94_%D7%A4%D7%A8%D7%9E%D7%94" title="פייר דה פרמה">פייר דה פרמה</a> שחקר אותם לראשונה. </p><p>ניתן להוכיח שכאשר <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5130547f888adb43fdcf0d7d935a8e9695655e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.771ex; height:2.509ex;" alt="{\displaystyle \,{2^{n}}+1}"></span> הוא <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99" title="מספר ראשוני">מספר ראשוני</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <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/a798350c07bbc4b91b4190498a18535149176d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle \,n}"></span> חייב להיות <a href="/wiki/%D7%97%D7%96%D7%A7%D7%94_%D7%A9%D7%9C_2" class="mw-redirect" title="חזקה של 2">חזקה של 2</a>. מכאן: כל מספר ראשוני מהצורה <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,{2^{n}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5130547f888adb43fdcf0d7d935a8e9695655e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.771ex; height:2.509ex;" alt="{\displaystyle \,{2^{n}}+1}"></span> הוא מספר פרמה, ומספר ראשוני כזה קרוי <b>ראשוני פרמה</b>. פרמה הבחין שחמשת המספרים הראשונים בסדרה, </p> <dl><dd><i>F</i><sub>0</sub> = 2<sup>1</sup> + 1 = 3</dd> <dd><i>F</i><sub>1</sub> = 2<sup>2</sup> + 1 = 5</dd> <dd><i>F</i><sub>2</sub> = 2<sup>4</sup> + 1 = 17</dd> <dd><i>F</i><sub>3</sub> = 2<sup>8</sup> + 1 = 257</dd> <dd><i>F</i><sub>4</sub> = 2<sup>16</sup> + 1 = 65,537,</dd></dl> <p>הם ראשוניים, וקבע (בסביבות <a href="/wiki/1640" title="1640">1640</a>) שכל המספרים בסדרה הם ראשוניים. כמאה שנים אחר-כך גילה <a href="/wiki/%D7%9C%D7%90%D7%95%D7%A0%D7%A8%D7%93_%D7%90%D7%95%D7%99%D7%9C%D7%A8" title="לאונרד אוילר">לאונרד אוילר</a> שהמספר הבא בסדרה אינו ראשוני: <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_{5}=2^{2^{5}}+1=4294967297=641\cdot 6700417}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>4294967297</mn> <mo>=</mo> <mn>641</mn> <mo>⋅</mo> <mn>6700417</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{5}=2^{2^{5}}+1=4294967297=641\cdot 6700417}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcbc7b5ea97a519b6aee30faa51c8946e81d5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.404ex; height:3.343ex;" alt="{\displaystyle \ F_{5}=2^{2^{5}}+1=4294967297=641\cdot 6700417}"></span>. יתרה מזו, אוילר הראה שכל גורם ראשוני של <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15212d03bd24e8bfc015b8657b70d30381eb9a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.294ex; height:2.509ex;" alt="{\displaystyle \ F_{n}}"></span> מוכרח להיות מהצורה <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}k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2^{n+1}k+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff20867309261a136d6dc53cc8b36b085abe647a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.276ex; height:2.843ex;" alt="{\displaystyle \ 2^{n+1}k+1}"></span> (הוכחה: אם <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^{n}}+1\equiv 0{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <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> <mo>≡</mo> <mn>0</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^{2^{n}}+1\equiv 0{\pmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0a6d4286045fb48ef9e415d8197f168b7175ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.879ex; height:3.176ex;" alt="{\displaystyle \ 2^{2^{n}}+1\equiv 0{\pmod {p}}}"></span> אז הסדר של 2 ב<a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8" title="חבורת אוילר">חבורת אוילר</a> של p הוא <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfe572ffe7aa1802100b9f749cef82cf76f2e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.062ex; height:2.676ex;" alt="{\displaystyle \ 2^{n+1}}"></span>, ו<a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%9C%D7%92%D7%A8%D7%A0%D7%96%27_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" class="mw-redirect" title="משפט לגרנז' (תורת החבורות)">לכן</a> מספר זה מחלק את סדר החבורה שהוא p-1). הגורם 641 הוא הראשוני החמישי בלבד מהצורה <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 \ 64k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>64</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 64k+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e429c840e9d227ad2e1523763afeeb8dece02b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.12ex; height:2.343ex;" alt="{\displaystyle \ 64k+1}"></span>. מאז התגלו גורמים ראשוניים של מספרי פרמה רבים, כגון: <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_{6}=2^{2^{6}}+1=18446744073709551617=274177\cdot 67280421310721}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>18446744073709551617</mn> <mo>=</mo> <mn>274177</mn> <mo>⋅</mo> <mn>67280421310721</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{6}=2^{2^{6}}+1=18446744073709551617=274177\cdot 67280421310721}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1612bf3818a28dcfd882c60444352b6baa9ffe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:67.653ex; height:3.343ex;" alt="{\displaystyle \ F_{6}=2^{2^{6}}+1=18446744073709551617=274177\cdot 67280421310721}"></span>, ולא נמצא אף ראשוני פרמה נוסף. </p><p>אחת התכונות המעניינות של מספרי פרמה הוא הקשר שלהם <a href="/wiki/%D7%A9%D7%9C%D7%95%D7%A9_%D7%94%D7%91%D7%A2%D7%99%D7%95%D7%AA_%D7%A9%D7%9C_%D7%99%D7%9E%D7%99_%D7%A7%D7%93%D7%9D" class="mw-redirect" title="שלוש הבעיות של ימי קדם">לבעיות של ימי קדם</a>. <a href="/wiki/%D7%A7%D7%A8%D7%9C_%D7%A4%D7%A8%D7%99%D7%93%D7%A8%D7%99%D7%9A_%D7%92%D7%90%D7%95%D7%A1" title="קרל פרידריך גאוס">גאוס</a> הוכיח שניתן <a href="/wiki/%D7%91%D7%A0%D7%99%D7%99%D7%94_%D7%91%D7%A1%D7%A8%D7%92%D7%9C_%D7%95%D7%91%D7%9E%D7%97%D7%95%D7%92%D7%94" title="בנייה בסרגל ובמחוגה">לבנות עם סרגל ומחוגה</a> <a href="/wiki/%D7%9E%D7%A6%D7%95%D7%9C%D7%A2_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מצולע משוכלל">מצולע משוכלל</a> בן <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f9cce76f4a3b0a80f475e5b344a0cc0bc00c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.557ex; height:2.009ex;" alt="{\displaystyle \,p}"></span> צלעות, כאשר <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f9cce76f4a3b0a80f475e5b344a0cc0bc00c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.557ex; height:2.009ex;" alt="{\displaystyle \,p}"></span> ראשוני, <a href="/wiki/%D7%90%D7%9D_%D7%95%D7%A8%D7%A7_%D7%90%D7%9D" title="אם ורק אם">אם ורק אם</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f9cce76f4a3b0a80f475e5b344a0cc0bc00c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.557ex; height:2.009ex;" alt="{\displaystyle \,p}"></span> הוא ראשוני פרמה. ובאופן כללי ניתן לבנות מצולע משוכלל בן <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"> <mspace width="thinmathspace" /> <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/a798350c07bbc4b91b4190498a18535149176d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle \,n}"></span> צלעות (כאשר <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>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82965de9446e5dde2874a18aaf997057d51f03c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle \,n>2}"></span> ) אם ורק אם n הוא מכפלה של ראשוניי פרמה שונים זה מזה וחזקה כלשהי של 2. </p><p>מספר <a href="/wiki/%D7%91%D7%A2%D7%99%D7%94_%D7%A4%D7%AA%D7%95%D7%97%D7%94_%D7%91%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" class="mw-redirect" title="בעיה פתוחה במתמטיקה">בעיות פתוחות</a> בתורת המספרים עוסקות במספרי פרמה: </p> <ul><li>האם <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6690b43000e0cc7f19033f863525d3afb45f69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.1ex; height:2.509ex;" alt="{\displaystyle \,F_{n}}"></span> הוא <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%99%D7%A7" title="מספר פריק">מספר פריק</a> לכל <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>4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>n</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,n>4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/685395c1584fb6acc069ec6ecf63a18780235436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle \,n>4}"></span> ?</li> <li>האם יש מספר אינסופי של ראשוני פרמה?</li> <li>האם יש מספר אינסופי של מספרי פרמה פריקים?</li></ul> <p>עד כה פורקו פירוק מלא לגורמים 12 מספרי פרמה הראשונים; המספר הראשון שטבעו (ראשוני או פריק) אינו ידוע הוא <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_{33}=2^{2^{33}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{33}=2^{2^{33}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa226cedde5570920740e3004102db00ccad2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.768ex; height:3.343ex;" alt="{\displaystyle \ F_{33}=2^{2^{33}}+1}"></span>; ונמצאו עוד כ-250 מספרי פרמה שהוכחו כפריקים.<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> </p><p>הפירוק המלא של 12 מספרי פרמה הראשונים הוא: </p> <dl><dd><table> <tbody><tr valign="top"> <td><i>F</i><sub>0</sub></td> <td>=</td> <td>2<sup>1</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/3_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="3 (מספר)">3</a> ראשוני</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>1</sub></td> <td>=</td> <td>2<sup>2</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/5_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="5 (מספר)">5</a> ראשוני</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>2</sub></td> <td>=</td> <td>2<sup>4</sup></td> <td>+</td> <td>1</td> <td>=</td> <td><a href="/wiki/17_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="17 (מספר)">17</a> ראשוני</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>3</sub></td> <td>=</td> <td>2<sup>8</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>257 ראשוני</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>4</sub></td> <td>=</td> <td>2<sup>16</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>65,537 מספר פרמה הראשוני המוּכָּר הגדול ביותר</td> <td> </td></tr> <tr valign="top"> <td><i>F</i><sub>5</sub></td> <td>=</td> <td>2<sup>32</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>4,294,967,297</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>641 × 6,700,417 (פורק באופן מלא בשנת 1732) </td></tr> <tr valign="top"> <td><i>F</i><sub>6</sub></td> <td>=</td> <td>2<sup>64</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>18,446,744,073,709,551,617 (20 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>274,177 × 67,280,421,310,721 (14 ספרות) (פורק באופן מלא בשנת 1855) </td></tr> <tr valign="top"> <td><i>F</i><sub>7</sub></td> <td>=</td> <td>2<sup>128</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>340,282,366,920,938,463,463,374,607,431,768,211,457 (39 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>59,649,589,127,497,217 (17 ספרות) × 5,704,689,200,685,129,054,721 (22 ספרות) (פורק באופן מלא בשנת 1970) </td></tr> <tr valign="top"> <td><i>F</i><sub>8</sub></td> <td>=</td> <td>2<sup>256</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,<br />639,937 (78 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>1,238,926,361,552,897 (16 ספרות) × <br />93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 ספרות) (פורק באופן מלא בשנת 1980) </td></tr> <tr valign="top"> <td><i>F</i><sub>9</sub></td> <td>=</td> <td>2<sup>512</sup></td> <td>+</td> <td>1</td> <td>=</td> <td>13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0<br />30,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,6<br />49,006,084,097 (155 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 ספרות) × <br />741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759,<br />504,705,008,092,818,711,693,940,737 (99 ספרות) (פורק באופן מלא בשנת 1990) </td></tr> <tr valign="top"> <td><i>F</i><sub>10</sub></td> <td>=</td> <td>2<sup>1024</sup></td> <td>+</td> <td>1 </td> <td>=</td> <td>179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 ספרות) × <br />130,439,874,405,488,189,727,484...806,217,820,753,127,014,424,577 (252 ספרות) (פורק באופן מלא בשנת 1995) </td></tr> <tr valign="top"> <td><i>F</i><sub>11</sub></td> <td>=</td> <td>2<sup>2048</sup></td> <td>+</td> <td>1 </td> <td>=</td> <td>32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 ספרות)</td> <td> </td></tr> <tr style="background:white; color:#700000; vertical-align:top"> <td></td> <td></td> <td></td> <td></td> <td></td> <td>=</td> <td>319,489 × 974,849 × 167,988,556,341,760,475,137 (21 ספרות) × 3,560,841,906,445,833,920,513 (22 ספרות) × <br />173,462,447,179,147,555,430,258...491,382,441,723,306,598,834,177 (564 ספרות) (פורק באופן מלא בשנת 1988) </td></tr> </tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="מקורות"><span id=".D7.9E.D7.A7.D7.95.D7.A8.D7.95.D7.AA"></span>מקורות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&action=edit&section=1" title="עריכת קוד המקור של הפרק: מקורות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&veaction=edit&section=1" title="עריכת פסקה: "מקורות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div dir="ltr" class="mw-content-ltr"> <ul><li>L.E. Dickson, <b>Theory of Numbers</b>, Vol. I, Chapter XV.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&action=edit&section=2" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&veaction=edit&section=2" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FermatNumber.html">מספר פרמה</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/Fermat-prime">מספר פרמה</a>, באתר <a href="/wiki/%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="אנציקלופדיה בריטניקה">אנציקלופדיה בריטניקה</a> <span dir="rtl" class="languageicon">(באנגלית)</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36549940"></li> <li><a rel="nofollow" class="external text" href="http://oeis.org/A000215">סדרת מספרי פרמה</a> באתר <a href="/wiki/%D7%94%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%94%D7%9E%D7%A7%D7%95%D7%95%D7%A0%D7%AA_%D7%9C%D7%A1%D7%93%D7%A8%D7%95%D7%AA_%D7%A9%D7%9C_%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%A9%D7%9C%D7%9E%D7%99%D7%9D" title="האנציקלופדיה המקוונת לסדרות של מספרים שלמים">OEIS – האנציקלופדיה המקוונת לסדרות של מספרים שלמים</a></li></ul> <div dir="ltr"> <ul><li>R. D. Carmichael, <a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2971748">Fermat Numbers</a>, <b>American Mathematical Monthly</b>, April 1919, in <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="הערות_שוליים"><span id=".D7.94.D7.A2.D7.A8.D7.95.D7.AA_.D7.A9.D7.95.D7.9C.D7.99.D7.99.D7.9D"></span>הערות שוליים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&action=edit&section=3" title="עריכת קוד המקור של הפרק: הערות שוליים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%A8%D7%9E%D7%94&veaction=edit&section=3" title="עריכת פסקה: "הערות שוליים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist references-small"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">^</a></span> <span class="reference-text">פירוט מלא ב: <a rel="nofollow" class="external free" href="http://www.prothsearch.net/fermat.html">http://www.prothsearch.net/fermat.html</a></span> </li> </ol></div></div> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q207264?uselang=he&#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q207264?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q207264?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/15px-Blue_pencil_RTL.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/23px-Blue_pencil_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/30px-Blue_pencil_RTL.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-right-width:2px;border-right-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div dir="ltr"> <ul><li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">NLI</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007532614505171">987007532614505171</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5_%D7%91%D7%A7%D7%A8%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%91" title="קובץ בקרה משולב">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4672709-7">4672709-7</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA_%D7%A9%D7%9C_%D7%A6%27%D7%9B%D7%99%D7%94" title="הספרייה הלאומית של צ'כיה">NKC</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph195830&CON_LNG=ENG">ph195830</a></span></span></li></ul> </div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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