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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</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"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</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/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP này</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Trang Web"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Nội dung" 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">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</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">Đầu</div> </a> </li> <li id="toc-Các_ví_dụ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Các_ví_dụ"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Các ví dụ</span> </div> </a> <ul id="toc-Các_ví_dụ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Các_tính_chất" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Các_tính_chất"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Các tính chất</span> </div> </a> <button aria-controls="toc-Các_tính_chất-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Hiện/ẩn mục Các tính chất</span> </button> <ul id="toc-Các_tính_chất-sublist" class="vector-toc-list"> <li id="toc-Số_lạ_nguyên_thủy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Số_lạ_nguyên_thủy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Số lạ nguyên thủy</span> </div> </a> <ul id="toc-Số_lạ_nguyên_thủy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Xem_thêm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Xem_thêm"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Xem thêm</span> </div> </a> <ul id="toc-Xem_thêm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tham_khảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tham_khảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tham khảo</span> </div> </a> <ul id="toc-Tham_khảo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liên_kết_ngoài" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Liên_kết_ngoài"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Liên kết ngoài</span> </div> </a> <ul id="toc-Liên_kết_ngoài-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="Nội dung" 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="Đóng mở mục lục" > <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">Đóng mở mục lục</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">Số lạ</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="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 20 ngôn ngữ" > <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-20" 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">20 ngôn ngữ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Podivn%C3%A9_%C4%8D%C3%ADslo" title="Podivné číslo – Tiếng Séc" lang="cs" hreflang="cs" data-title="Podivné číslo" data-language-autonym="Čeština" data-language-local-name="Tiếng Séc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Merkw%C3%BCrdige_Zahl" title="Merkwürdige Zahl – Tiếng Đức" lang="de" hreflang="de" data-title="Merkwürdige Zahl" data-language-autonym="Deutsch" data-language-local-name="Tiếng Đức" 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%A0%CE%B5%CF%81%CE%AF%CE%B5%CF%81%CE%B3%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Περίεργος αριθμός – Tiếng Hy Lạp" lang="el" hreflang="el" data-title="Περίεργος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Tiếng Hy Lạp" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Weird_number" title="Weird number – Tiếng Anh" lang="en" hreflang="en" data-title="Weird number" data-language-autonym="English" data-language-local-name="Tiếng Anh" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_extra%C3%B1o" title="Número extraño – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Número extraño" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Bizara_nombro" title="Bizara nombro – Tiếng Quốc Tế Ngữ" lang="eo" hreflang="eo" data-title="Bizara nombro" data-language-autonym="Esperanto" data-language-local-name="Tiếng Quốc Tế Ngữ" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_%C3%A9trange" title="Nombre étrange – Tiếng Pháp" lang="fr" hreflang="fr" data-title="Nombre étrange" data-language-autonym="Français" data-language-local-name="Tiếng Pháp" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수 – Tiếng Hàn" lang="ko" hreflang="ko" data-title="기묘수" data-language-autonym="한국어" data-language-local-name="Tiếng Hàn" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_bizzarro" title="Numero bizzarro – Tiếng Italy" lang="it" hreflang="it" data-title="Numero bizzarro" data-language-autonym="Italiano" data-language-local-name="Tiếng Italy" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%95%D7%96%D7%A8" title="מספר מוזר – Tiếng Do Thái" lang="he" hreflang="he" data-title="מספר מוזר" data-language-autonym="עברית" data-language-local-name="Tiếng Do Thái" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Furcsa_sz%C3%A1mok" title="Furcsa számok – Tiếng Hungary" lang="hu" hreflang="hu" data-title="Furcsa számok" data-language-autonym="Magyar" data-language-local-name="Tiếng Hungary" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vreemd_getal" title="Vreemd getal – Tiếng Hà Lan" lang="nl" hreflang="nl" data-title="Vreemd getal" data-language-autonym="Nederlands" data-language-local-name="Tiếng Hà Lan" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%8D%E6%80%9D%E8%AD%B0%E6%95%B0" title="不思議数 – Tiếng Nhật" lang="ja" hreflang="ja" data-title="不思議数" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_estranho" title="Número estranho – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Número estranho" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" 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%A1%D1%82%D1%80%D0%B0%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Странное число – Tiếng Nga" lang="ru" hreflang="ru" data-title="Странное число" data-language-autonym="Русский" data-language-local-name="Tiếng Nga" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_strammu" title="Nùmmuru strammu – Tiếng Sicilia" lang="scn" hreflang="scn" data-title="Nùmmuru strammu" data-language-autonym="Sicilianu" data-language-local-name="Tiếng Sicilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Outo_luku" title="Outo luku – Tiếng Phần Lan" lang="fi" hreflang="fi" data-title="Outo luku" data-language-autonym="Suomi" data-language-local-name="Tiếng Phần Lan" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/%C3%96vernaturligt_tal" title="Övernaturligt tal – Tiếng Thụy Điển" lang="sv" hreflang="sv" data-title="Övernaturligt tal" data-language-autonym="Svenska" data-language-local-name="Tiếng Thụy Điển" 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%B5%E0%AE%BF%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AF%88_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="விந்தை எண் – Tiếng Tamil" lang="ta" hreflang="ta" data-title="விந்தை எண்" data-language-autonym="தமிழ்" data-language-local-name="Tiếng Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A5%87%E7%95%B0%E6%95%B8_(%E6%95%B8%E8%AB%96)" title="奇異數 (數論) – Tiếng Trung" lang="zh" hreflang="zh" data-title="奇異數 (數論)" data-language-autonym="中文" data-language-local-name="Tiếng Trung" class="interlanguage-link-target"><span>中文</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/Q2145900#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Không gian tên"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/S%E1%BB%91_l%E1%BA%A1" title="Xem bài viết [c]" accesskey="c"><span>Bài viết</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Th%E1%BA%A3o_lu%E1%BA%ADn:S%E1%BB%91_l%E1%BA%A1" rel="discussion" title="Thảo luận về trang này [t]" accesskey="t"><span>Thảo luận</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" 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data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Công cụ" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Công cụ</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Công cụ</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ẩn</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Thêm tùy chọn" > <div class="vector-menu-heading"> Tác vụ </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/S%E1%BB%91_l%E1%BA%A1"><span>Đọc</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit" title="Sửa đổi trang này [v]" accesskey="v"><span>Sửa đổi</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit" title="Sửa đổi mã nguồn của trang này [e]" accesskey="e"><span>Sửa mã nguồn</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=history"><span>Xem lịch sử</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Chung </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Li%C3%AAn_k%E1%BA%BFt_%C4%91%E1%BA%BFn_%C4%91%C3%A2y/S%E1%BB%91_l%E1%BA%A1" title="Các trang liên kết đến đây [j]" accesskey="j"><span>Các liên kết đến đây</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Thay_%C4%91%E1%BB%95i_li%C3%AAn_quan/S%E1%BB%91_l%E1%BA%A1" 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href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:DownloadAsPdf&page=S%E1%BB%91_l%E1%BA%A1&action=show-download-screen"><span>Tải dưới dạng PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&printable=yes" title="Bản để in ra của trang [p]" accesskey="p"><span>Bản để in ra</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Tại dự án khác </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2145900" title="Liên kết đến khoản mục kết nối trong kho dữ liệu [g]" accesskey="g"><span>Khoản mục Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Công cụ trang"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Bách khoa toàn thư mở Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="vi" dir="ltr"><p>Trong <a href="/wiki/L%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91" title="Lý thuyết số">lý thuyết số</a>, <b>số lạ</b> là <a href="/wiki/S%E1%BB%91_t%E1%BB%B1_nhi%C3%AAn" title="Số tự nhiên">số tự nhiên</a> <a href="/wiki/S%E1%BB%91_phong_ph%C3%BA" title="Số phong phú">phong phú</a> nhưng không <a href="/w/index.php?title=S%E1%BB%91_n%E1%BB%ADa_ho%C3%A0n_thi%E1%BB%87n&action=edit&redlink=1" class="new" title="Số nửa hoàn thiện (trang không tồn tại)">nửa hoàn hảo</a>.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Nói cách khác, tổng các <a href="/wiki/%C6%AF%E1%BB%9Bc_th%E1%BB%B1c_s%E1%BB%B1" class="mw-redirect" title="Ước thực sự">ước thực sự</a> (các ước số bao gồm 1 nhưng không phải chính nó) của số lạ lớn hơn số lạ đó, nhưng không có tập con nào của các ước số đó mà tổng của nó bằng với chính số đó. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Các_ví_dụ"><span id="C.C3.A1c_v.C3.AD_d.E1.BB.A5"></span>Các ví dụ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=1" title="Sửa đổi phần “Các ví dụ”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=1" title="Sửa mã nguồn tại đề mục: Các ví dụ"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Số lạ nhỏ nhất là 70. Các ước số thực sự của nó là: 1, 2, 5, 7, 10, 14 và 35; tổng của các ước số này là 74, nhưng không có tập con nào của các ước số mà tổng của nó bằng 70. Số 12 được gọi là <a href="/wiki/S%E1%BB%91_phong_ph%C3%BA" title="Số phong phú">số phong phú</a> nhưng không phải là số lạ, bởi vì các ước số chia hết của 12 là 1, 2, 3, 4 và 6 có tổng là 16, tuy nhiên 2 + 4 + 6 = 12 (tổng của một tập con trong các ước số thực sự). </p><p>Dãy số các số lạ là: </p> <dl><dd>70, 836, 4030, 5830, 7192, 7912, 9272, 10430,... (dãy số <span class="nowrap"><a href="//oeis.org/A006037" class="extiw" title="oeis:A006037">A006037</a></span> trong bảng <a href="/wiki/B%E1%BA%A3ng_tra_c%E1%BB%A9u_d%C3%A3y_s%E1%BB%91_nguy%C3%AAn_tr%E1%BB%B1c_tuy%E1%BA%BFn" title="Bảng tra cứu dãy số nguyên trực tuyến">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Các_tính_chất"><span id="C.C3.A1c_t.C3.ADnh_ch.E1.BA.A5t"></span>Các tính chất</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=2" title="Sửa đổi phần “Các tính chất”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=2" title="Sửa mã nguồn tại đề mục: Các tính chất"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right; width:30%; margin:0em 1em 1em 1em; background:#eee; border:#ccc solid"> <tbody><tr> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/20px-Question%2C_Web_Fundamentals.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/30px-Question%2C_Web_Fundamentals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Question%2C_Web_Fundamentals.svg/40px-Question%2C_Web_Fundamentals.svg.png 2x" data-file-width="44" data-file-height="44" /></span></span></td> <td><b>Vấn đề mở trong toán học</b>:<br /><div style="padding:5px; font-style:italic;">Liệu có tồn tại số lạ lẻ?</div><small>(<a href="/wiki/Danh_s%C3%A1ch_v%E1%BA%A5n_%C4%91%E1%BB%81_m%E1%BB%9F_trong_to%C3%A1n_h%E1%BB%8Dc" title="Danh sách vấn đề mở trong toán học">các vấn đề mở khác trong toán học</a>)</small> </td></tr></tbody></table> <p>Có vô số số lạ.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Ví dụ như, 70<i>p</i> là lạ với mọi <a href="/wiki/S%E1%BB%91_nguy%C3%AAn_t%E1%BB%91" title="Số nguyên tố">số nguyên tố</a> <i>p</i> ≥ 149. Thậm chí, <a href="/wiki/T%E1%BA%ADp_h%E1%BB%A3p_(to%C3%A1n_h%E1%BB%8Dc)" title="Tập hợp (toán học)">tập</a> các số lạ có <a href="/w/index.php?title=M%E1%BA%ADt_%C4%91%E1%BB%99_ti%E1%BB%87m_c%E1%BA%ADn&action=edit&redlink=1" class="new" title="Mật độ tiệm cận (trang không tồn tại)">mật độ tiệm cận</a> dương.<sup id="cite_ref-benk1_4-0" class="reference"><a href="#cite_note-benk1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Hiện nay ta vẫn chưa biết liệu có tồn tại số lạ <a href="/wiki/S%E1%BB%91_l%E1%BA%BB" class="mw-redirect" title="Số lẻ">lẻ</a>. Nếu có tồn tại thì số đó phải lớn hơn 10<sup>21</sup>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Sidney Kravitz chứng minh rằng với <i>k</i> là <a href="/wiki/S%E1%BB%91_nguy%C3%AAn" title="Số nguyên">số nguyên</a> dương, <i>Q</i> là số nguyên tố lớn hơn 2<sup><i>k</i></sup>, và </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {2^{k}Q-(Q+1)}{(Q+1)-2^{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>Q</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {2^{k}Q-(Q+1)}{(Q+1)-2^{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dfae3e971e487781603c668fa8e7c897d1c781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.279ex; height:7.176ex;" alt="{\displaystyle R={\frac {2^{k}Q-(Q+1)}{(Q+1)-2^{k}}}}"></span></dd></dl> <p>cũng là số nguyên tố và lớn hơn 2<sup><i>k</i></sup>, thì </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{k-1}QR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>Q</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{k-1}QR}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a39c704719dbfaa1d60829878c4e2843521a6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.447ex; height:3.009ex;" alt="{\displaystyle n=2^{k-1}QR}"></span></dd></dl> <p>là số lạ.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Với công thức này ông tìm ra số lạ lớn sau: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\ \approx \ 2\cdot 10^{52}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>61</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>153722867280912929</mn> <mtext> </mtext> <mo>≈</mo> <mtext> </mtext> <mn>2</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>52</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\ \approx \ 2\cdot 10^{52}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02443c1da0f59cb93b345305a4b83e3bb76fb94c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.614ex; height:3.176ex;" alt="{\displaystyle n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\ \approx \ 2\cdot 10^{52}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Số_lạ_nguyên_thủy"><span id="S.E1.BB.91_l.E1.BA.A1_nguy.C3.AAn_th.E1.BB.A7y"></span>Số lạ nguyên thủy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=3" title="Sửa đổi phần “Số lạ nguyên thủy”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=3" title="Sửa mã nguồn tại đề mục: Số lạ nguyên thủy"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Một tính chất của số lạ đó là nếu <i>n</i> lạ và <i>p</i> là số nguyên tố lớn hơn <a href="/w/index.php?title=T%E1%BB%95ng_c%C3%A1c_%C6%B0%E1%BB%9Bc&action=edit&redlink=1" class="new" title="Tổng các ước (trang không tồn tại)">tổng các ước</a> σ(<i>n</i>), thì <i>pn</i> cũng là số lạ.<sup id="cite_ref-benk1_4-1" class="reference"><a href="#cite_note-benk1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Tính chất dẫn tới định nghĩa <i>số lạ nguyên thủy</i>, tức số lạ không phải <a href="/wiki/B%E1%BB%99i_s%E1%BB%91" title="Bội số">bội</a> của các số lạ khác (dãy số <span class="nowrap"><a href="//oeis.org/A002975" class="extiw" title="oeis:A002975">A002975</a></span> trong bảng <a href="/wiki/B%E1%BA%A3ng_tra_c%E1%BB%A9u_d%C3%A3y_s%E1%BB%91_nguy%C3%AAn_tr%E1%BB%B1c_tuy%E1%BA%BFn" title="Bảng tra cứu dãy số nguyên trực tuyến">OEIS</a>). Chỉ có 24 số lạ nguyên thủy nhỏ hơn 1 triệu so với 1765 số lạ nói chung dưới 1 triệu. Cách sinh của Kravitz đưa ra số lạ nguyên thủy bởi mọi số <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^{k}pq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}pq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6526a572f2b5bde9113421df4b8c2ec4d276187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.49ex; height:3.009ex;" alt="{\displaystyle 2^{k}pq}"></span> đều nguyên thủy, nhưng việc tồn tại vô số <i>k</i> và <i>Q</i> sao cho <i>R</i> là nguyên tố không được đảm bảo. Do đó người ta giả thuyết rằng có vô số số lạ nguyên thủy, <a href="/w/index.php?title=Giuseppe_Melfi&action=edit&redlink=1" class="new" title="Giuseppe Melfi (trang không tồn tại)">Melfi</a> đã chứng minh rằng sự vô hạn của số lạ nguyên thủy là hệ quả của <a href="/w/index.php?title=Gi%E1%BA%A3_thuy%E1%BA%BFt_Cramer&action=edit&redlink=1" class="new" title="Giả thuyết Cramer (trang không tồn tại)">giả thuyết Cramer</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Số lạ nguyên thủy có lên tới 16 ước nguyên tố và 14712 chữ số đã được tìm thấy.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Xem_thêm"><span id="Xem_th.C3.AAm"></span>Xem thêm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=4" title="Sửa đổi phần “Xem thêm”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=4" title="Sửa mã nguồn tại đề mục: Xem thêm"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=S%E1%BB%91_kh%C3%B4ng_ch%E1%BA%A1m_t%E1%BB%9Bi_%C4%91%C6%B0%E1%BB%A3c&action=edit&redlink=1" class="new" title="Số không chạm tới được (trang không tồn tại)">Số không chạm tới được</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Tham_khảo"><span id="Tham_kh.E1.BA.A3o"></span>Tham khảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=5" title="Sửa đổi phần “Tham khảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=5" title="Sửa mã nguồn tại đề mục: Tham khảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71728118">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r67233549">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"“""”""‘""’"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFBenkoski1972" class="citation journal cs1">Benkoski, Stan (1972). “E2308 (in Problems and Solutions)”. <i>The American Mathematical Monthly</i>. <b>79</b> (7): 774. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2316276">10.2307/2316276</a>. <a href="/wiki/JSTOR_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="JSTOR (định danh)">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2316276">2316276</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=E2308+%28in+Problems+and+Solutions%29&rft.volume=79&rft.issue=7&rft.pages=774&rft.date=1972&rft_id=info%3Adoi%2F10.2307%2F2316276&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2316276%23id-name%3DJSTOR&rft.aulast=Benkoski&rft.aufirst=Stan&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFRichard_K._Guy2004" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Richard K. Guy</a> (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/unsolvedproblems0003guyr"><i>Unsolved Problems in Number Theory</i></a>. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-387-20860-7" title="Đặc biệt:Nguồn sách/0-387-20860-7"><bdi>0-387-20860-7</bdi></a>. <a href="/wiki/OCLC_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="OCLC (định danh)">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/54611248">54611248</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.pub=Springer-Verlag&rft.date=2004&rft_id=info%3Aoclcnum%2F54611248&rft.isbn=0-387-20860-7&rft.au=Richard+K.+Guy&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funsolvedproblems0003guyr&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span> Section B2.</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFSándorMitrinovićCrstici2006" class="citation book cs1">Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav biên tập (2006). <i>Handbook of number theory I</i>. Dordrecht: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. tr. 113–114. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/1-4020-4215-9" title="Đặc biệt:Nguồn sách/1-4020-4215-9"><bdi>1-4020-4215-9</bdi></a>. <a href="/w/index.php?title=Zbl_(%C4%91%E1%BB%8Bnh_danh)&action=edit&redlink=1" class="new" title="Zbl (định danh) (trang không tồn tại)">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1151.11300">1151.11300</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+number+theory+I&rft.place=Dordrecht&rft.pages=113-114&rft.pub=Springer-Verlag&rft.date=2006&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1151.11300%23id-name%3DZbl&rft.isbn=1-4020-4215-9&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> <li id="cite_note-benk1-4">^ <a href="#cite_ref-benk1_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-benk1_4-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFBenkoskiErdős1974" class="citation journal cs1">Benkoski, Stan; <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős, Paul</a> (tháng 4 năm 1974). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-of-computation_1974-04_28_126/page/617">“On Weird and Pseudoperfect Numbers”</a>. <i><a href="/w/index.php?title=Mathematics_of_Computation&action=edit&redlink=1" class="new" title="Mathematics of Computation (trang không tồn tại)">Mathematics of Computation</a></i>. <b>28</b> (126): 617–623. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2005938">10.2307/2005938</a></span>. <a href="/wiki/MR_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="MR (định danh)">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0347726">0347726</a>. <a href="/w/index.php?title=Zbl_(%C4%91%E1%BB%8Bnh_danh)&action=edit&redlink=1" class="new" title="Zbl (định danh) (trang không tồn tại)">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0279.10005">0279.10005</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=On+Weird+and+Pseudoperfect+Numbers&rft.volume=28&rft.issue=126&rft.pages=617-623&rft.date=1974-04&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0279.10005%23id-name%3DZbl&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D347726%23id-name%3DMR&rft_id=info%3Adoi%2F10.2307%2F2005938&rft.aulast=Benkoski&rft.aufirst=Stan&rft.au=Erd%C5%91s%2C+Paul&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_mathematics-of-computation_1974-04_28_126%2Fpage%2F617&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFSloane_"A006037"" class="citation web cs1"><a href="/w/index.php?title=Neil_Sloane&action=edit&redlink=1" class="new" title="Neil Sloane (trang không tồn tại)">Sloane, N. J. A.</a> (biên tập). <a rel="nofollow" class="external text" href="https://oeis.org/A006037">“Dãy  A006037 (Weird numbers: abundant (A005101) but not pseudoperfect (A005835))”</a>. <i><a href="/wiki/B%E1%BA%A3ng_tra_c%E1%BB%A9u_d%C3%A3y_s%E1%BB%91_nguy%C3%AAn_tr%E1%BB%B1c_tuy%E1%BA%BFn" title="Bảng tra cứu dãy số nguyên trực tuyến">Bảng tra cứu dãy số nguyên trực tuyến</a></i>. Tổ chức OEIS.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=B%E1%BA%A3ng+tra+c%E1%BB%A9u+d%C3%A3y+s%E1%BB%91+nguy%C3%AAn+tr%E1%BB%B1c+tuy%E1%BA%BFn&rft.atitle=D%C3%A3y+%26%23x20%3BA006037%26%23x20%3B%28Weird+numbers%3A+abundant+%28A005101%29+but+not+pseudoperfect+%28A005835%29%29&rft_id=https%3A%2F%2Foeis.org%2FA006037&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span> -- comments concerning odd weird numbers</span> </li> <li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFKravitz1976" class="citation journal cs1">Kravitz, Sidney (1976). “A search for large weird numbers”. <i>Journal of Recreational Mathematics</i>. Baywood Publishing. <b>9</b> (2): 82–85. <a href="/w/index.php?title=Zbl_(%C4%91%E1%BB%8Bnh_danh)&action=edit&redlink=1" class="new" title="Zbl (định danh) (trang không tồn tại)">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0365.10003">0365.10003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Recreational+Mathematics&rft.atitle=A+search+for+large+weird+numbers&rft.volume=9&rft.issue=2&rft.pages=82-85&rft.date=1976&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0365.10003%23id-name%3DZbl&rft.aulast=Kravitz&rft.aufirst=Sidney&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFMelfi2015" class="citation journal cs1">Melfi, Giuseppe (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2014.07.024">“On the conditional infiniteness of primitive weird numbers”</a>. <i>Journal of Number Theory</i>. Elsevier. <b>147</b>: 508–514. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2014.07.024">10.1016/j.jnt.2014.07.024</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=On+the+conditional+infiniteness+of+primitive+weird+numbers&rft.volume=147&rft.pages=508-514&rft.date=2015&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2014.07.024&rft.aulast=Melfi&rft.aufirst=Giuseppe&rft_id=%2F%2Fdoi.org%2F10.1016%252Fj.jnt.2014.07.024&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFAmatoHaslerMelfiParton2019" class="citation journal cs1">Amato, Gianluca; Hasler, Maximilian; Melfi, Giuseppe; Parton, Maurizio (2019). “Primitive abundant and weird numbers with many prime factors”. <i>Journal of Number Theory</i>. Elsevier. <b>201</b>: 436–459. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<span class="cs1-lock-free" title="Truy cập mở"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1802.07178">1802.07178</a></span>. <a href="/wiki/%C4%90%E1%BB%8Bnh_danh_%C4%91%E1%BB%91i_t%C6%B0%E1%BB%A3ng_s%E1%BB%91" class="mw-redirect" title="Định danh đối tượng số">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2019.02.027">10.1016/j.jnt.2019.02.027</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Primitive+abundant+and+weird+numbers+with+many+prime+factors&rft.volume=201&rft.pages=436-459&rft.date=2019&rft_id=info%3Aarxiv%2F1802.07178&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2019.02.027&rft.aulast=Amato&rft.aufirst=Gianluca&rft.au=Hasler%2C+Maximilian&rft.au=Melfi%2C+Giuseppe&rft.au=Parton%2C+Maurizio&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AS%E1%BB%91+l%E1%BA%A1" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Liên_kết_ngoài"><span id="Li.C3.AAn_k.E1.BA.BFt_ngo.C3.A0i"></span>Liên kết ngoài</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&veaction=edit&section=6" title="Sửa đổi phần “Liên kết ngoài”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=S%E1%BB%91_l%E1%BA%A1&action=edit&section=6" title="Sửa mã nguồn tại đề mục: Liên kết ngoài"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71922045">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Cổng thông tin" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/T%E1%BA%ADp_tin:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/C%E1%BB%95ng_th%C3%B4ng_tin:To%C3%A1n_h%E1%BB%8Dc" title="Cổng thông tin:Toán học">Cổng thông tin Toán học</a></span></li></ul> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Weird_number"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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