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id="toc-The_algorithm-sublist" class="vector-toc-list"> <li id="toc-Proof_of_validity_outline" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_of_validity_outline"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Proof of validity outline</span> </div> </a> <ul id="toc-Proof_of_validity_outline-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_1:_n_=_31_is_prime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_1:_n_=_31_is_prime"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Example 1: <i>n</i> = 31 is prime</span> </div> </a> <ul id="toc-Example_1:_n_=_31_is_prime-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">AKS primality test</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" 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">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%AE%D8%AA%D8%A8%D8%A7%D8%B1_%D8%A3.%D9%83.%D8%A3%D8%B3_%D9%84%D8%A3%D9%88%D9%84%D9%8A%D8%A9_%D8%B9%D8%AF%D8%AF_%D9%85%D8%A7" title="اختبار أ.ك.أس لأولية عدد ما – Arabic" lang="ar" hreflang="ar" data-title="اختبار أ.ك.أس لأولية عدد ما" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/AKS-Primzahltest" title="AKS-Primzahltest – German" lang="de" hreflang="de" data-title="AKS-Primzahltest" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Test_de_primalidad_AKS" title="Test de primalidad AKS – Spanish" lang="es" hreflang="es" data-title="Test de primalidad AKS" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Primeco-testo_AKS" title="Primeco-testo AKS – Esperanto" lang="eo" hreflang="eo" data-title="Primeco-testo AKS" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D8%B2%D9%85%D9%88%D9%86_%D8%A7%D9%88%D9%84_%D8%A8%D9%88%D8%AF%D9%86_%D8%A7%DB%8C%E2%80%8C%DA%A9%DB%8C%E2%80%8C%D8%A7%D8%B3" title="آزمون اول بودن ای‌کی‌اس – Persian" lang="fa" hreflang="fa" data-title="آزمون اول بودن ای‌کی‌اس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Test_de_primalit%C3%A9_AKS" title="Test de primalité AKS – French" lang="fr" hreflang="fr" data-title="Test de primalité AKS" data-language-autonym="Français" data-language-local-name="French" 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/AKS_%EC%86%8C%EC%88%98%ED%8C%90%EB%B3%84%EB%B2%95" title="AKS 소수판별법 – Korean" lang="ko" hreflang="ko" data-title="AKS 소수판별법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%8F%E0%A5%B0_%E0%A4%95%E0%A5%87%E0%A5%B0_%E0%A4%90%E0%A4%B8%E0%A5%B0_%E0%A4%A8%E0%A4%82%E0%A4%AC%E0%A4%B0_%E0%A4%85%E0%A4%AD%E0%A4%BE%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BE_%E0%A4%9F%E0%A5%87%E0%A4%B8%E0%A5%8D%E0%A4%9F" title="ए॰ के॰ ऐस॰ नंबर अभाज्यता टेस्ट – Hindi" lang="hi" hreflang="hi" data-title="ए॰ के॰ ऐस॰ नंबर अभाज्यता टेस्ट" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algoritmo_AKS" title="Algoritmo AKS – Italian" lang="it" hreflang="it" data-title="Algoritmo AKS" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%91%D7%97%D7%9F_AKS_%D7%9C%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99%D7%95%D7%AA" title="מבחן AKS לראשוניות – Hebrew" lang="he" hreflang="he" data-title="מבחן AKS לראשוניות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%8E.%E0%B4%95%E0%B5%86.%E0%B4%8E%E0%B4%B8%E0%B5%8D._%E0%B4%85%E0%B4%AD%E0%B4%BE%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%A4%E0%B4%BE%E0%B4%AA%E0%B4%B0%E0%B4%BF%E0%B4%B6%E0%B5%8B%E0%B4%A7%E0%B4%A8" title="എ.കെ.എസ്. അഭാജ്യതാപരിശോധന – Malayalam" lang="ml" hreflang="ml" data-title="എ.കെ.എസ്. അഭാജ്യതാപരിശോധന" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/AKS-test" title="AKS-test – Dutch" lang="nl" hreflang="nl" data-title="AKS-test" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/AKS%E7%B4%A0%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95" title="AKS素数判定法 – Japanese" lang="ja" hreflang="ja" data-title="AKS素数判定法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Test_pierwszo%C5%9Bci_AKS" title="Test pierwszości AKS – Polish" lang="pl" hreflang="pl" data-title="Test pierwszości AKS" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teste_de_primalidade_AKS" title="Teste de primalidade AKS – Portuguese" lang="pt" hreflang="pt" data-title="Teste de primalidade AKS" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D1%81%D1%82_%D0%90%D0%B3%D1%80%D0%B0%D0%B2%D0%B0%D0%BB%D0%B0_%E2%80%94_%D0%9A%D0%B0%D1%8F%D0%BB%D0%B0_%E2%80%94_%D0%A1%D0%B0%D0%BA%D1%81%D0%B5%D0%BD%D1%8B" title="Тест Агравала — Каяла — Саксены – Russian" lang="ru" hreflang="ru" data-title="Тест Агравала — Каяла — Саксены" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%81%D1%82_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%82%D0%B8_AKS" title="Тест простоти AKS – Ukrainian" lang="uk" hreflang="uk" data-title="Тест простоти AKS" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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/Ph%C3%A9p_ki%E1%BB%83m_tra_t%C3%ADnh_nguy%C3%AAn_t%E1%BB%91_AKS" title="Phép kiểm tra tính nguyên tố AKS – Vietnamese" lang="vi" hreflang="vi" data-title="Phép kiểm tra tính nguyên tố AKS" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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/AKS%E8%B3%AA%E6%95%B8%E6%B8%AC%E8%A9%A6" title="AKS質數測試 – Chinese" lang="zh" hreflang="zh" data-title="AKS質數測試" data-language-autonym="中文" data-language-local-name="Chinese" 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/Q294284#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <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/AKS_primality_test" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm checking for prime numbers</div> <p>The <b>AKS primality test</b> (also known as <b>Agrawal–Kayal–Saxena primality test</b> and <b>cyclotomic AKS test</b>) is a <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministic</a> <a href="/wiki/Primality_test" title="Primality test">primality-proving</a> <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> created and published by <a href="/wiki/Manindra_Agrawal" title="Manindra Agrawal">Manindra Agrawal</a>, <a href="/wiki/Neeraj_Kayal" title="Neeraj Kayal">Neeraj Kayal</a>, and <a href="/wiki/Nitin_Saxena" title="Nitin Saxena">Nitin Saxena</a>, computer scientists at the <a href="/wiki/Indian_Institute_of_Technology_Kanpur" class="mw-redirect" title="Indian Institute of Technology Kanpur">Indian Institute of Technology Kanpur</a>, on August 6, 2002, in an article titled "PRIMES is in P".<sup id="cite_ref-AKS_1-0" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The algorithm was the first one which is able to determine in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, whether a given number is <a href="/wiki/Prime_number" title="Prime number">prime</a> or <a href="/wiki/Composite_number" title="Composite number">composite</a> without relying on <a href="/wiki/Mathematical_conjecture" class="mw-redirect" title="Mathematical conjecture">mathematical conjectures</a> such as the <a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalized Riemann hypothesis</a>. The proof is also notable for not relying on the field of <a href="/wiki/Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analysis</a>.<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> In 2006 the authors received both the <a href="/wiki/G%C3%B6del_Prize" title="Gödel Prize">Gödel Prize</a> and <a href="/wiki/Fulkerson_Prize" title="Fulkerson Prize">Fulkerson Prize</a> for their work. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Importance">Importance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=1" title="Edit section: Importance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>AKS is the first primality-proving algorithm to be simultaneously <i>general</i>, <i>polynomial-time</i>, <i>deterministic</i>, and <i>unconditionally correct</i>. Previous algorithms had been developed for centuries and achieved three of these properties at most, but not all four. </p> <ul><li>The AKS algorithm can be used to verify the primality of any <b>general</b> number given. Many fast primality tests are known that work only for numbers with certain properties. For example, the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer test</a> works only for <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne numbers</a>, while <a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's test</a> can be applied to <a href="/wiki/Fermat_number" title="Fermat number">Fermat numbers</a> only.</li> <li>The maximum running time of the algorithm can be bounded by a <b><a href="/wiki/Polynomial_time#Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial</a></b> over the number of digits in the target number. <a href="/wiki/Elliptic_curve_primality_proving" class="mw-redirect" title="Elliptic curve primality proving">ECPP</a> and <a href="/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test" title="Adleman–Pomerance–Rumely primality test">APR</a> conclusively prove or disprove that a given number is prime, but are not known to have polynomial time bounds for all inputs.</li> <li>The algorithm is guaranteed to distinguish <b><a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministically</a></b> whether the target number is prime or composite. Randomized tests, such as <a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin</a> and <a href="/wiki/Baillie%E2%80%93PSW_primality_test" title="Baillie–PSW primality test">Baillie–PSW</a>, can test any given number for primality in polynomial time, but are known to produce only a probabilistic result.</li> <li>The correctness of AKS is <b>not conditional</b> on any subsidiary unproven <a href="/wiki/Hypothesis" title="Hypothesis">hypothesis</a>. In contrast, Miller's version of the <a href="/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants" title="Miller–Rabin primality test">Miller–Rabin test</a> is fully deterministic and runs in polynomial time over all inputs, but its correctness depends on the truth of the yet-unproven <a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalized Riemann hypothesis</a>.</li></ul> <p>While the algorithm is of immense theoretical importance, it is not used in practice, rendering it a <a href="/wiki/Galactic_algorithm" title="Galactic algorithm">galactic algorithm</a>. For 64-bit inputs, the <a href="/wiki/Baillie%E2%80%93PSW_primality_test" title="Baillie–PSW primality test">Baillie–PSW test</a> is deterministic and runs many orders of magnitude faster. For larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is <i>far</i> superior to AKS. Additionally, ECPP can output a <a href="/wiki/Primality_certificate" title="Primality certificate">primality certificate</a> that allows independent and rapid verification of the results, which is not possible with the AKS algorithm. </p> <div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=2" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The AKS primality test is based upon the following theorem: Given an integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span> and integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is prime if and only if the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial</a> <a href="/wiki/Congruence_relation#Modular_arithmetic" title="Congruence relation">congruence relation</a> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X+a)^{n}\equiv X^{n}+a{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>≡</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X+a)^{n}\equiv X^{n}+a{\pmod {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08199a5a1adbc0d07e1b7538567df3fdd94047c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.541ex; height:2.843ex;" alt="{\displaystyle (X+a)^{n}\equiv X^{n}+a{\pmod {n}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>holds within the polynomial ring <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 (\mathbb {Z} /n\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1089692d39a21e3384b0c84500376e7b9efde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}"></span>.<sup id="cite_ref-AKS_1-1" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> denotes the <a href="/wiki/Polynomial_ring#Definition_(univariate_case)" title="Polynomial ring">indeterminate</a> which generates this polynomial ring. </p><p>This theorem is a generalization to polynomials of <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>. In one direction it can easily be proven using the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> together with the following property of the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}\equiv 0{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose k}\equiv 0{\pmod {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2e85d1d9cca8fe5e1bd2a610b9db777f66d5c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.156ex; height:6.176ex;" alt="{\displaystyle {n \choose k}\equiv 0{\pmod {n}}}"></span> for all <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 0<k<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<k<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9346654d73329403fdfffb51d8002b65cb4c8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.965ex; height:2.176ex;" alt="{\displaystyle 0<k<n}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is prime.</dd></dl> <p>While the relation (<b><a href="#math_1">1</a></b>) constitutes a primality test in itself, verifying it takes <a href="/wiki/Exponential_time" class="mw-redirect" title="Exponential time">exponential time</a>: the <a href="/wiki/Brute_force_method" class="mw-redirect" title="Brute force method">brute force</a> approach would require the expansion of the <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 (X+a)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X+a)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a064611ce1de66137bf69d5f6d891b14433f3592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle (X+a)^{n}}"></span> polynomial and a reduction <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 {\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\pmod {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1296775902b168a3d79709d582fe6b544004bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.079ex; height:2.843ex;" alt="{\displaystyle {\pmod {n}}}"></span> of the resulting <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+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> coefficients. </p><p>The congruence is an equality in the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</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 (\mathbb {Z} /n\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1089692d39a21e3384b0c84500376e7b9efde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}"></span>. Evaluating in a quotient ring of <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 (\mathbb {Z} /n\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1089692d39a21e3384b0c84500376e7b9efde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}"></span> creates an upper bound for the degree of the polynomials involved. The AKS evaluates the equality in <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 (\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32164f537c96deb3b24f34e389ad75e30cab2c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.686ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}"></span>, making the <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> dependent on the size of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. For clarity,<sup id="cite_ref-AKS_1-2" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> this is expressed as the congruence </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X+a)^{n}\equiv (X^{n}+a){\pmod {X^{r}-1,n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X+a)^{n}\equiv (X^{n}+a){\pmod {X^{r}-1,n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d155e4d600deb86b8268a3a355f9694f6853a0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.357ex; height:2.843ex;" alt="{\displaystyle (X+a)^{n}\equiv (X^{n}+a){\pmod {X^{r}-1,n}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>which is the same as: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X+a)^{n}-(X^{n}+a)=(X^{r}-1)g+nf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>g</mi> <mo>+</mo> <mi>n</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X+a)^{n}-(X^{n}+a)=(X^{r}-1)g+nf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2932beb5c5c7efe204d20e5ac1e739c2e7e4a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.524ex; height:2.843ex;" alt="{\displaystyle (X+a)^{n}-(X^{n}+a)=(X^{r}-1)g+nf}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>for some polynomials <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>. </p><p>Note that all primes satisfy this relation (choosing <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 g=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0607b4f563220901d455691e4a1705d598fdbaeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=0}"></span> in (<b><a href="#math_3">3</a></b>) gives (<b><a href="#math_1">1</a></b>), which holds for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> prime). This congruence can be checked in polynomial time when <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is polynomial to the digits of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. The AKS algorithm evaluates this congruence for a large set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> values, whose size is polynomial to the digits of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. The proof of validity of the AKS algorithm shows that one can find an <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and a set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> values with the above properties such that if the congruences hold then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is a power of a prime.<sup id="cite_ref-AKS_1-3" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History_and_running_time">History and running time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=3" title="Edit section: History and running time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the first version of the above-cited paper, the authors proved the asymptotic time complexity of the algorithm to be <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 {\tilde {O}}(\log(n)^{12})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{12})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5c5df608ea55d58d333418f151244172c27775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.635ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{12})}"></span> (using <a href="/wiki/Big_O_notation#Extensions_to_the_Bachmann–Landau_notations" title="Big O notation">Õ</a> from <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>)—the twelfth power of the number of digits in <i>n</i> times a factor that is polylogarithmic in the number of digits. However, this upper bound was rather loose; a widely-held conjecture about the distribution of the <a href="/wiki/Sophie_Germain_prime" class="mw-redirect" title="Sophie Germain prime">Sophie Germain primes</a> would, if true, immediately cut the worst case down to <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 {\tilde {O}}(\log(n)^{6})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{6})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6da5249f5d5255416a4c9e2873400ca8331ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.813ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{6})}"></span>. </p><p>In the months following the discovery, new variants appeared (Lenstra 2002, Pomerance 2002, Berrizbeitia 2002, Cheng 2003, Bernstein 2003a/b, Lenstra and Pomerance 2003), which improved the speed of computation greatly. Owing to the existence of the many variants, Crandall and Papadopoulos refer to the "AKS-class" of algorithms in their scientific paper "On the implementation of AKS-class primality tests", published in March 2003. </p><p>In response to some of these variants, and to other feedback, the paper "PRIMES is in P" was updated with a new formulation of the AKS algorithm and of its proof of correctness. (This version was eventually published in <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>.) While the basic idea remained the same, <i>r</i> was chosen in a new manner, and the proof of correctness was more coherently organized. The new proof relied almost exclusively on the behavior of <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a> over <a href="/wiki/Finite_fields" class="mw-redirect" title="Finite fields">finite fields</a>. The new upper bound on time complexity was <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 {\tilde {O}}(\log(n)^{10.5})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>10.5</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{10.5})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e42041cb20852a99231de0b85781418351260235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.914ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{10.5})}"></span>, later reduced using additional results from <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a> to <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 {\tilde {O}}(\log(n)^{7.5})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7.5</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{7.5})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542153a6de501c139e9e815545495eb9a131ae0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.092ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{7.5})}"></span>. </p><p>In 2005, <a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Pomerance</a> and <a href="/wiki/Hendrik_Lenstra" title="Hendrik Lenstra">Lenstra</a> demonstrated a variant of AKS that runs in <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 {\tilde {O}}(\log(n)^{6})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{6})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6da5249f5d5255416a4c9e2873400ca8331ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.813ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{6})}"></span> operations,<sup id="cite_ref-lenstra_pomerance_2005_3-0" class="reference"><a href="#cite_note-lenstra_pomerance_2005-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> leading to another updated version of the paper.<sup id="cite_ref-lenstra_pomerance_2011_4-0" class="reference"><a href="#cite_note-lenstra_pomerance_2011-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Agrawal, Kayal and Saxena proposed a variant which would run in <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 {\tilde {O}}(\log(n)^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(\log(n)^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398bd87bd466a6f43ea9ae0e44dc8299723362da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.813ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}(\log(n)^{3})}"></span> if <a href="/wiki/Agrawal%27s_conjecture" title="Agrawal's conjecture">Agrawal's conjecture</a> were true; however, a heuristic argument by Pomerance and Lenstra suggested that it is probably false. </p> <div class="mw-heading mw-heading2"><h2 id="The_algorithm">The algorithm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=4" title="Edit section: The algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The algorithm is as follows:<sup id="cite_ref-AKS_1-4" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>Input: integer <span class="texhtml"><i>n</i> > 1</span>.</dd></dl> <ol><li>Check if <i>n</i> is a <a href="/wiki/Perfect_power" title="Perfect power">perfect power</a>: if <span class="texhtml"><i>n</i> = <i>a</i><sup><i>b</i></sup></span> for integers <span class="texhtml"><i>a</i> > 1</span> and <span class="texhtml"><i>b</i> > 1</span>, then output <i>composite</i>.</li> <li>Find the smallest <i>r</i> such that <span class="texhtml"><a href="/wiki/Multiplicative_order" title="Multiplicative order">ord</a><sub><i>r</i></sub>(<i>n</i>) > (log<sub>2</sub> <i>n</i>)<sup>2</sup></span>. If <i>r</i> and <i>n</i> are not coprime, then output <i>composite</i>.</li> <li>For all 2 ≤ <i>a</i> ≤ min (<i>r</i>, <i>n</i>−1), check that <i>a</i> does not divide <i>n</i>: If <i>a</i>|<i>n</i> for some 2 ≤ <i>a</i> ≤ min (<i>r</i>, <i>n</i>−1), then output <i>composite</i>.</li> <li>If <i>n</i> ≤ <i>r</i>, then output <i>prime</i>.</li> <li>For <span class="texhtml"><i>a</i> = 1</span> to <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 \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e8f8e1f842f93eeafb29da46bd297fd6accdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.774ex; height:4.843ex;" alt="{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"></span> do <dl><dd>if (<i>X</i>+<i>a</i>)<sup><i>n</i></sup> ≠ <i>X</i><sup><i>n</i></sup>+<i>a</i> (mod <i>X</i><sup><i>r</i></sup> − 1,<i>n</i>), then output <i>composite</i>;</dd></dl></li> <li>Output <i>prime</i>.</li></ol> <p>Here <a href="/wiki/Multiplicative_order" title="Multiplicative order">ord</a><sub><i>r</i></sub>(<i>n</i>) is the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of <i>n</i> <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <i>r</i>, log<sub>2</sub> is the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a>, and <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 \varphi (r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1a1b480a264e4a38135d0ae2589bfc49984669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle \varphi (r)}"></span> is <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a> of <i>r</i>. </p><p>Step 3 is shown in the paper as checking 1 < (<i>a</i>,<i>n</i>) < <i>n</i> for all <i>a</i> ≤ <i>r</i>. It can be seen this is equivalent to trial division up to <i>r</i>, which can be done very efficiently without using <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>. Similarly the comparison in step 4 can be replaced by having the trial division return <i>prime</i> once it has checked all values up to and including <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 \left\lfloor {\sqrt {n}}\right\rfloor .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo>⌋</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\sqrt {n}}\right\rfloor .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b6a48012686ce433dc52d29137361c689d3612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.429ex; height:3.009ex;" alt="{\displaystyle \left\lfloor {\sqrt {n}}\right\rfloor .}"></span> </p><p>Once beyond very small inputs, step 5 dominates the time taken. The essential reduction in complexity (from exponential to polynomial) is achieved by performing all calculations in the finite ring </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=(\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6873ec3978b478e588031f4d01c4685c7482a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.548ex; height:2.843ex;" alt="{\displaystyle R=(\mathbb {Z} /n\mathbb {Z} )[X]/(X^{r}-1)}"></span></dd></dl> <p>consisting of <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^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b016a3a6c7f94bf3b634260c24bf8cd81bbf51d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.368ex; height:2.343ex;" alt="{\displaystyle n^{r}}"></span> elements. This ring contains only the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> <a href="/wiki/Monomial" title="Monomial">monomials</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 \{X^{0},X^{1},\ldots ,X^{r-1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X^{0},X^{1},\ldots ,X^{r-1}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fc9acbc4870c2444846e51b60a355cff8f8f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.71ex; height:3.176ex;" alt="{\displaystyle \{X^{0},X^{1},\ldots ,X^{r-1}\}}"></span>, and the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> are in <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 \mathbb {Z} /n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2120ebbc85f91df66c6de5446367bf9fd620844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"></span> which has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> elements, all of them codable within <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 \log _{2}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e7727d9ed8daee1d80207aba949782dccf9547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.23ex; height:2.843ex;" alt="{\displaystyle \log _{2}(n)}"></span> bits. </p><p>Most later improvements made to the algorithm have concentrated on reducing the size of <i>r,</i> which makes the core operation in step 5 faster, and in reducing the size of <i>s</i>, the number of loops performed in step 5.<sup id="cite_ref-bernstein03_5-0" class="reference"><a href="#cite_note-bernstein03-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Typically these changes do not change the computational complexity, but can lead to many orders of magnitude less time taken; for example, Bernstein's final version has a theoretical speedup by a factor of over 2 million. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_of_validity_outline">Proof of validity outline</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=5" title="Edit section: Proof of validity outline"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the algorithm to be correct, all steps that identify <i>n</i> must be correct. Steps 1, 3, and 4 are trivially correct, since they are based on direct tests of the divisibility of <i>n</i>. Step 5 is also correct: since (2) is true for any choice of <i>a</i> coprime to <i>n</i> and <i>r</i> if <i>n</i> is prime, an inequality means that <i>n</i> must be composite. </p><p>The difficult part of the proof is showing that step 6 is true. Its proof of correctness is based on the upper and lower bounds of a <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a> in <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 \mathbb {Z} _{n}[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4995fef9354139d887121f41794989ca2502e54c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.392ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{n}[x]}"></span> constructed from the (<i>X</i> + <i>a</i>) binomials that are tested in step 5. Step 4 guarantees that these binomials are <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 \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e8f8e1f842f93eeafb29da46bd297fd6accdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.774ex; height:4.843ex;" alt="{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"></span> distinct elements of <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 \mathbb {Z} _{n}[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4995fef9354139d887121f41794989ca2502e54c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.392ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{n}[x]}"></span>. For the particular choice of <i>r</i>, the bounds produce a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">contradiction</a> unless <i>n</i> is prime or a power of a prime. Together with the test of step 1, this implies that <i>n</i> is always prime at step 6.<sup id="cite_ref-AKS_1-5" class="reference"><a href="#cite_note-AKS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example_1:_n_=_31_is_prime"><span id="Example_1:_n_.3D_31_is_prime"></span>Example 1: <i>n</i> = 31 is prime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=6" title="Edit section: Example 1: n = 31 is prime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1195917819">.mw-parser-output .pre-borderless{border:none}</style><pre class="pre" style="white-space:pre;overflow:auto;"> <b>Input</b>: integer <i>n</i> = 31 > 1. <u>(* Step 1 *)</u> <b>If</b> (<i>n</i> = <i>a</i><sup><i>b</i></sup> for integers <i>a</i> > 1 and <i>b</i> > 1), <b>output</b> <i>composite</i>. <b>For</b> ( b = 2; b <= log<sub>2</sub>(n); b++) { a = n<sup>1/b</sup>; <b>If</b> (a is integer), <b>Return</b>[Composite] } a = n<sup>1/2</sup>...n<sup>1/4</sup> = {5.568, 3.141, 2.360} <u>(* Step 2 *)</u> Find the smallest <i>r</i> such that <i>O</i><sub><i>r</i></sub>(<i>n</i>) > (log<sub><i>2</i></sub> <i>n</i>)<sup>2</sup>. maxk = ⌊(log<sub><i>2</i></sub> n)<sup>2</sup>⌋; maxr = Max[3, ⌈(Log<sub><i>2</i></sub> n)<sup>5</sup>⌉]; <u>(* maxr really isn't needed *)</u> nextR = True; <b>For</b> (r = 2; nextR && r < maxr; r++) { nextR = False; <b>For</b> (k = 1; (!nextR) && k ≤ maxk; k++) { nextR = (Mod[n<sup>k</sup>, r] == 1 || Mod[n<sup>k</sup>, r]==0) } } r--; <u>(*the loop over increments by one*)</u> r = 29 <u>(* Step 3 *)</u> <b>If</b> (1 < <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>(<i>a</i>,<i>n</i>) < <i>n</i> for some <i>a</i> ≤ <i>r</i>), <b>output</b> <i>composite</i>. <b>For</b> (a = r; a > 1; a--) { <b>If</b> ((gcd = GCD[a,n]) > 1 && gcd < n), <b>Return</b>[Composite] } gcd = {GCD(29,31)=1, GCD(28,31)=1, ..., GCD(2,31)=1} ≯ 1 <u>(* Step 4 *)</u> <b>If</b> (<i>n</i> ≤ <i>r</i>), <b>output</b> <i>prime</i>. <b>If</b> (n ≤ r), <b>Return</b>[Prime] <u>(* this step may be omitted if n > 5690034 *)</u> 31 > 29 <u>(* Step 5 *)</u> <b>For</b> <i>a</i> = 1 <b>to</b> <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 \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e8f8e1f842f93eeafb29da46bd297fd6accdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.774ex; height:4.843ex;" alt="{\displaystyle \left\lfloor {\sqrt {\varphi (r)}}\log _{2}(n)\right\rfloor }"></span> <b>do</b> <b>If</b> ((<i>X</i>+<i>a</i>)<sup><i>n</i></sup> ≠ <i>X</i><sup><i>n</i></sup> + <i>a</i> (mod <i>X</i><sup><i>r</i></sup> − 1,<i>n</i>)), <b>output</b> <i>composite</i>; φ[x_] := EulerPhi[x]; PolyModulo[f_] := PolynomialMod[<a href="/wiki/Polynomial_remainder" class="mw-redirect" title="Polynomial remainder">PolynomialRemainder</a>[f, x<sup>r</sup>-1, x], n]; max = Floor[Log[2, n]<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">φ[r]</span></span>]; <b>For</b> (a = 1; a ≤ max; a++) { <b>If</b> (PolyModulo[(x+a)<sup>n</sup> - PolynomialRemainder[x<sup>n</sup>+a, x<sup>r</sup>-1], x] ≠ 0) { <b>Return</b>[Composite] { } (x+a)<sup>31</sup> = a<sup>31</sup> +31a<sup>30</sup>x +465a<sup>29</sup>x<sup>2</sup> +4495a<sup>28</sup>x<sup>3</sup> +31465a<sup>27</sup>x<sup>4</sup> +169911a<sup>26</sup>x<sup>5</sup> +736281a<sup>25</sup>x<sup>6</sup> +2629575a<sup>24</sup>x<sup>7</sup> +7888725a<sup>23</sup>x<sup>8</sup> +20160075a<sup>22</sup>x<sup>9</sup> +44352165a<sup>21</sup>x<sup>10</sup> +84672315a<sup>20</sup>x<sup>11</sup> +141120525a<sup>19</sup>x<sup>12</sup> +206253075a<sup>18</sup>x<sup>13</sup> +265182525a<sup>17</sup>x<sup>14</sup> +300540195a<sup>16</sup>x<sup>15</sup> +300540195a<sup>15</sup>x<sup>16</sup> +265182525a<sup>14</sup>x<sup>17</sup> +206253075a<sup>13</sup>x<sup>18</sup> +141120525a<sup>12</sup>x<sup>19</sup> +84672315a<sup>11</sup>x<sup>20</sup> +44352165a<sup>10</sup>x<sup>21</sup> +20160075a<sup>9</sup>x<sup>22</sup> +7888725a<sup>8</sup>x<sup>23</sup> +2629575a<sup>7</sup>x<sup>24</sup> +736281a<sup>6</sup>x<sup>25</sup> +169911a<sup>5</sup>x<sup>26</sup> +31465a<sup>4</sup>x<sup>27</sup> +4495a<sup>3</sup>x<sup>28</sup> +465a<sup>2</sup>x<sup>29</sup> +31ax<sup>30</sup> +x<sup>31</sup> PolynomialRemainder [(x+a)<sup>31</sup>, x<sup>29</sup>-1] = 465a<sup>2</sup> +a<sup>31</sup> +(31a+31a<sup>30</sup>)x +(1+465a<sup>29</sup>)x<sup>2</sup> +4495a<sup>28</sup>x<sup>3</sup> +31465a<sup>27</sup>x<sup>4</sup> +169911a<sup>26</sup>x<sup>5</sup> +736281a<sup>25</sup>x<sup>6</sup> +2629575a<sup>24</sup>x<sup>7</sup> +7888725a<sup>23</sup>x<sup>8</sup> +20160075a<sup>22</sup>x<sup>9</sup> +44352165a<sup>21</sup>x<sup>10</sup> +84672315a<sup>20</sup>x<sup>11</sup> +141120525a<sup>19</sup>x<sup>12</sup> +206253075a<sup>18</sup>x<sup>13</sup> +265182525a<sup>17</sup>x<sup>14</sup> +300540195a<sup>16</sup>x<sup>15</sup> +300540195a<sup>15</sup>x<sup>16</sup> +265182525a<sup>14</sup>x<sup>17</sup> +206253075a<sup>13</sup>x<sup>18</sup> +141120525a<sup>12</sup>x<sup>19</sup> +84672315a<sup>11</sup>x<sup>20</sup> +44352165a<sup>10</sup>x<sup>21</sup> +20160075a<sup>9</sup>x<sup>22</sup> +7888725a<sup>8</sup>x<sup>23</sup> +2629575a<sup>7</sup>x<sup>24</sup> +736281a<sup>6</sup>x<sup>25</sup> +169911a<sup>5</sup>x<sup>26</sup> +31465a<sup>4</sup>x<sup>27</sup> +4495a<sup>3</sup>x<sup>28</sup> (<span id="math_A" class="reference nourlexpansion" style="font-weight:bold;">A</span>) PolynomialMod [PolynomialRemainder [(x+a)<sup>31</sup>, x<sup>29</sup>-1], 31] = a<sup>31</sup>+x<sup>2</sup> (<span id="math_B" class="reference nourlexpansion" style="font-weight:bold;">B</span>) PolynomialRemainder [x<sup>31</sup>+a, x<sup>29</sup>-1] = a+x<sup>2</sup> (<b><a href="#math_A">A</a></b>) - (<b><a href="#math_B">B</a></b>) = a<sup>31</sup>+x<sup>2</sup> - (a+x<sup>2</sup>) = a<sup>31</sup>-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 \max =\left\lfloor \log _{2}(31){\sqrt {\varphi (29)}}\right\rfloor =26}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>31</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>29</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mn>26</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max =\left\lfloor \log _{2}(31){\sqrt {\varphi (29)}}\right\rfloor =26}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5288562d460227c1e22bb4c65e0e7a55ade4aeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.441ex; height:4.843ex;" alt="{\displaystyle \max =\left\lfloor \log _{2}(31){\sqrt {\varphi (29)}}\right\rfloor =26}"></span> {1<sup>31</sup>-1 = 0 (mod 31), 2<sup>31</sup>-2 = 0 (mod 31), 3<sup>31</sup>-3 = 0 (mod 31), ..., 26<sup>31</sup>-26 = 0 (mod 31)} <u>(* Step 6 *)</u> <b>Output</b> <i>prime</i>. <samp style="padding-left:0.4em; padding-right:0.4em; color:var( --color-subtle, #666666);">31 Must be Prime</samp></pre> <p>Where PolynomialMod is a term-wise modulo reduction of the polynomial. e.g. PolynomialMod[x+2x<sup>2</sup>+3x<sup>3</sup>, 3] = x+2x<sup>2</sup>+0x<sup>3</sup> </p><p><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> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-AKS-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-AKS_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AKS_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-AKS_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-AKS_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-AKS_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-AKS_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAgrawalKayalSaxena2004" class="citation journal cs1">Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004). <a rel="nofollow" class="external text" href="http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf">"PRIMES is in P"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. <b>160</b> (2): 781–793. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2004.160.781">10.4007/annals.2004.160.781</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3597229">3597229</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=PRIMES+is+in+P&rft.volume=160&rft.issue=2&rft.pages=781-793&rft.date=2004&rft_id=info%3Adoi%2F10.4007%2Fannals.2004.160.781&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3597229%23id-name%3DJSTOR&rft.aulast=Agrawal&rft.aufirst=Manindra&rft.au=Kayal%2C+Neeraj&rft.au=Saxena%2C+Nitin&rft_id=http%3A%2F%2Fwww.cse.iitk.ac.in%2Fusers%2Fmanindra%2Falgebra%2Fprimality_v6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAKS+primality+test" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGranville2005" class="citation journal cs1">Granville, Andrew (2005). <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/home.html">"It is easy to determine whether a given integer is prime"</a>. <i>Bull. Amer. Math. Soc</i>. <b>42</b>: 3–38. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-04-01037-7">10.1090/S0273-0979-04-01037-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=It+is+easy+to+determine+whether+a+given+integer+is+prime&rft.volume=42&rft.pages=3-38&rft.date=2005&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-04-01037-7&rft.aulast=Granville&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fwww.ams.org%2Fbull%2F2005-42-01%2FS0273-0979-04-01037-7%2Fhome.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAKS+primality+test" class="Z3988"></span></span> </li> <li id="cite_note-lenstra_pomerance_2005-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-lenstra_pomerance_2005_3-0">^</a></b></span> <span class="reference-text">H. W. Lenstra Jr. and Carl Pomerance, "<a rel="nofollow" class="external text" href="http://www.math.dartmouth.edu/~carlp/PDF/complexity12.pdf">Primality testing with Gaussian periods</a>", preliminary version July 20, 2005.</span> </li> <li id="cite_note-lenstra_pomerance_2011-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-lenstra_pomerance_2011_4-0">^</a></b></span> <span class="reference-text">H. W. Lenstra Jr. and Carl Pomerance, "<a rel="nofollow" class="external text" href="http://www.math.dartmouth.edu/~carlp/aks041411.pdf">Primality testing with Gaussian periods</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120225052810/http://www.math.dartmouth.edu/~carlp/aks041411.pdf">Archived</a> 2012-02-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", version of April 12, 2011.</span> </li> <li id="cite_note-bernstein03-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-bernstein03_5-0">^</a></b></span> <span class="reference-text">Daniel J. Bernstein, "<a rel="nofollow" class="external text" href="https://cr.yp.to/papers/aks.pdf">Proving Primality After Agrawal-Kayal-Saxena</a>", version of January 25, 2003.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See <a href="/wiki/Talk:AKS_primality_test#Worked_Example" title="Talk:AKS primality test">AKS Talk</a> page for a discussion on why 'Example 2: n is not Prime past Step 4' is missing.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDietzfelbinger2004" class="citation book cs1">Dietzfelbinger, Martin (2004). <i>Primality testing in polynomial time. From randomized algorithms to </i>PRIMES is in P<i><span></span></i>. Lecture Notes in Computer Science. Vol. 3000. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-40344-2" title="Special:BookSources/3-540-40344-2"><bdi>3-540-40344-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1058.11070">1058.11070</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primality+testing+in+polynomial+time.+From+randomized+algorithms+to+PRIMES+is+in+P&rft.place=Berlin&rft.series=Lecture+Notes+in+Computer+Science&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11070%23id-name%3DZbl&rft.isbn=3-540-40344-2&rft.aulast=Dietzfelbinger&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAKS+primality+test" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AKS_primality_test&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-AKS_Primality_Test"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AKSPrimalityTest.html">"AKS Primality Test"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=AKS+Primality+Test&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAKSPrimalityTest.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAKS+primality+test" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140219064936/http://www.dm.unito.it/~cerruti/ac/aks-crandall.pdf">R. Crandall, Apple ACG, and J. Papadopoulos (March 18, 2003): On the implementation of AKS-class primality tests</a> (PDF)</li> <li><a rel="nofollow" class="external text" href="https://www.ams.org/notices/200305/fea-bornemann.pdf">Article by Bornemann, containing photos and information about the three Indian scientists</a> (PDF)</li> <li><a rel="nofollow" class="external text" href="https://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/home.html">Andrew Granville: It is easy to determine whether a given integer is prime</a></li> <li><a rel="nofollow" class="external text" href="http://www.scottaaronson.com/writings/prime.pdf">The Prime Facts: From Euclid to AKS</a>, by <a href="/wiki/Scott_Aaronson" title="Scott Aaronson">Scott Aaronson</a> (PDF)</li> <li><a rel="nofollow" class="external text" href="http://www.instantlogic.net/publications/PRIMES%20is%20in%20P%20little%20FAQ.htm">The PRIMES is in P little FAQ</a> by Anton Stiglic</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150327071905/http://www.sigact.org/Prizes/Godel/2006.html">2006 Gödel Prize Citation</a></li> <li><a rel="nofollow" class="external text" href="https://www.ams.org/notices/200611/comm-fulkerson.pdf">2006 Fulkerson Prize Citation</a></li> <li><a rel="nofollow" class="external text" href="http://fatphil.org/maths/AKS">The AKS "PRIMES in P" Algorithm Resource</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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test">Baillie–PSW</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve</a></li> <li><a href="/wiki/Pocklington_primality_test" title="Pocklington primality test">Pocklington</a></li> <li><a href="/wiki/Fermat_primality_test" title="Fermat primality test">Fermat</a></li> <li><a href="/wiki/Lucas_primality_test" title="Lucas primality test">Lucas</a></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer</a></i></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test" title="Lucas–Lehmer–Riesel test">Lucas–Lehmer–Riesel</a></i></li> <li><i><a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a></i></li> <li><i><a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's</a></i></li> <li><a href="/wiki/Quadratic_Frobenius_test" title="Quadratic Frobenius test">Quadratic Frobenius</a></li> <li><a href="/wiki/Solovay%E2%80%93Strassen_primality_test" title="Solovay–Strassen primality test">Solovay–Strassen</a></li> <li><a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Generating_primes" class="mw-redirect" title="Generating primes">Prime-generating</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sieve_of_Atkin" title="Sieve of Atkin">Sieve of Atkin</a></li> <li><a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a></li> <li><a href="/wiki/Sieve_of_Pritchard" title="Sieve of Pritchard">Sieve of Pritchard</a></li> <li><a href="/wiki/Sieve_of_Sundaram" title="Sieve of Sundaram">Sieve of Sundaram</a></li> <li><a href="/wiki/Wheel_factorization" title="Wheel factorization">Wheel factorization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Continued_fraction_factorization" title="Continued fraction factorization">Continued fraction (CFRAC)</a></li> <li><a href="/wiki/Dixon%27s_factorization_method" title="Dixon's factorization method">Dixon's</a></li> <li><a href="/wiki/Lenstra_elliptic-curve_factorization" title="Lenstra elliptic-curve factorization">Lenstra elliptic curve (ECM)</a></li> <li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho</a></li> <li><a href="/wiki/Pollard%27s_p_%E2%88%92_1_algorithm" title="Pollard's p − 1 algorithm"><i>p</i> − 1</a></li> <li><a href="/wiki/Williams%27s_p_%2B_1_algorithm" title="Williams's p + 1 algorithm"><i>p</i> + 1</a></li> <li><a href="/wiki/Quadratic_sieve" title="Quadratic sieve">Quadratic sieve (QS)</a></li> <li><a href="/wiki/General_number_field_sieve" title="General number field sieve">General number field sieve (GNFS)</a></li> <li><i><a href="/wiki/Special_number_field_sieve" title="Special number field sieve">Special number field sieve (SNFS)</a></i></li> <li><a href="/wiki/Rational_sieve" title="Rational sieve">Rational sieve</a></li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's</a></li> <li><a href="/wiki/Shanks%27s_square_forms_factorization" title="Shanks's square forms factorization">Shanks's square forms</a></li> <li><a href="/wiki/Trial_division" title="Trial division">Trial division</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_Egyptian_multiplication" title="Ancient Egyptian multiplication">Ancient Egyptian</a></li> <li><a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">Long</a></li> <li><a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba</a></li> <li><a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook</a></li> <li><a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen</a></li> <li><a href="/wiki/F%C3%BCrer%27s_algorithm" class="mw-redirect" title="Fürer's algorithm">Fürer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean</a> <a href="/wiki/Division_algorithm" title="Division algorithm">division</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_division" class="mw-redirect" title="Binary division">Binary</a></li> <li><a href="/wiki/Chunking_(division)" title="Chunking (division)">Chunking</a></li> <li><a href="/wiki/Fourier_division" title="Fourier division">Fourier</a></li> <li><a href="/wiki/Goldschmidt_division" class="mw-redirect" title="Goldschmidt division">Goldschmidt</a></li> <li><a href="/wiki/Newton%E2%80%93Raphson_division" class="mw-redirect" title="Newton–Raphson division">Newton-Raphson</a></li> <li><a href="/wiki/Long_division" title="Long division">Long</a></li> <li><a href="/wiki/Short_division" title="Short division">Short</a></li> <li><a href="/wiki/SRT_division" class="mw-redirect" title="SRT division">SRT</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_logarithm" title="Discrete logarithm">Discrete logarithm</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baby-step_giant-step" title="Baby-step giant-step">Baby-step giant-step</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm_for_logarithms" title="Pollard's rho algorithm for logarithms">Pollard rho</a></li> <li><a href="/wiki/Pollard%27s_kangaroo_algorithm" title="Pollard's kangaroo algorithm">Pollard kangaroo</a></li> <li><a href="/wiki/Pohlig%E2%80%93Hellman_algorithm" title="Pohlig–Hellman algorithm">Pohlig–Hellman</a></li> <li><a href="/wiki/Index_calculus_algorithm" title="Index calculus algorithm">Index calculus</a></li> <li><a href="/wiki/Function_field_sieve" title="Function field sieve">Function field sieve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_GCD_algorithm" title="Binary GCD algorithm">Binary</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean</a></li> <li><a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean</a></li> <li><a href="/wiki/Lehmer%27s_GCD_algorithm" title="Lehmer's GCD algorithm">Lehmer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratic_residue" title="Quadratic residue">Modular square root</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cipolla%27s_algorithm" title="Cipolla's algorithm">Cipolla</a></li> <li><a href="/wiki/Pocklington%27s_algorithm" title="Pocklington's algorithm">Pocklington's</a></li> <li><a href="/wiki/Tonelli%E2%80%93Shanks_algorithm" title="Tonelli–Shanks algorithm">Tonelli–Shanks</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Rabin_algorithm" title="Berlekamp–Rabin algorithm">Berlekamp</a></li> <li><a href="/wiki/Kunerth%27s_algorithm" title="Kunerth's algorithm">Kunerth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chakravala_method" title="Chakravala method">Chakravala</a></li> <li><a href="/wiki/Cornacchia%27s_algorithm" title="Cornacchia's algorithm">Cornacchia</a></li> <li><a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">Exponentiation by squaring</a></li> <li><a href="/wiki/Integer_square_root" title="Integer square root">Integer square root</a></li> <li><a href="/wiki/Integer_relation_algorithm" title="Integer relation algorithm">Integer relation</a> (<a href="/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm" title="Lenstra–Lenstra–Lovász lattice basis reduction algorithm">LLL</a>; <a href="/wiki/Korkine%E2%80%93Zolotarev_lattice_basis_reduction_algorithm" title="Korkine–Zolotarev lattice basis reduction algorithm">KZ</a>)</li> <li><a href="/wiki/Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li> <li><a href="/wiki/Montgomery_reduction" class="mw-redirect" title="Montgomery reduction">Montgomery reduction</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof</a></li> <li><a href="/wiki/Trachtenberg_system" title="Trachtenberg system">Trachtenberg system</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><i>Italics</i> indicate that algorithm is for numbers of special forms</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=AKS_primality_test&oldid=1258583787">https://en.wikipedia.org/w/index.php?title=AKS_primality_test&oldid=1258583787</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Primality_tests" title="Category:Primality tests">Primality tests</a></li><li><a href="/wiki/Category:Finite_fields" title="Category:Finite fields">Finite fields</a></li></ul></div><div 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