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id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 3 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Connection_with_Fermat's_Last_Theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Fermat's_Last_Theorem"> <div class="vector-toc-text"> 3.1 Connection with Fermat's Last Theorem </div> </a> <ul id="toc-Connection_with_Fermat's_Last_Theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_the_abc_conjecture_and_non-Wieferich_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_the_abc_conjecture_and_non-Wieferich_primes"> <div class="vector-toc-text"> 3.2 Connection with the abc conjecture and non-Wieferich primes </div> </a> <ul id="toc-Connection_with_the_abc_conjecture_and_non-Wieferich_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_Mersenne_and_Fermat_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Mersenne_and_Fermat_primes"> <div class="vector-toc-text"> 3.3 Connection with Mersenne and Fermat primes </div> </a> <ul id="toc-Connection_with_Mersenne_and_Fermat_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_other_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_other_equations"> <div class="vector-toc-text"> 3.4 Connection with other equations </div> </a> <ul id="toc-Connection_with_other_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Binary_periodicity_of_p_−_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_periodicity_of_p_−_1"> <div class="vector-toc-text"> 3.5 Binary periodicity of p − 1 </div> </a> <ul id="toc-Binary_periodicity_of_p_−_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abundancy_of_p_−_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abundancy_of_p_−_1"> <div class="vector-toc-text"> 3.6 Abundancy of p − 1 </div> </a> <ul id="toc-Abundancy_of_p_−_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_pseudoprimes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_pseudoprimes"> <div class="vector-toc-text"> 3.7 Connection with pseudoprimes </div> </a> <ul id="toc-Connection_with_pseudoprimes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_directed_graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_directed_graphs"> <div class="vector-toc-text"> 3.8 Connection with directed graphs </div> </a> <ul id="toc-Connection_with_directed_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_related_to_number_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_related_to_number_fields"> <div class="vector-toc-text"> 3.9 Properties related to number fields </div> </a> <ul id="toc-Properties_related_to_number_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 4 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Near-Wieferich_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Near-Wieferich_primes"> <div class="vector-toc-text"> 4.1 Near-Wieferich primes </div> </a> <ul id="toc-Near-Wieferich_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Base-a_Wieferich_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Base-a_Wieferich_primes"> <div class="vector-toc-text"> 4.2 Base-a Wieferich primes </div> </a> <ul id="toc-Base-a_Wieferich_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wieferich_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wieferich_pairs"> <div class="vector-toc-text"> 4.3 Wieferich pairs </div> </a> <ul id="toc-Wieferich_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wieferich_sequence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wieferich_sequence"> <div class="vector-toc-text"> 4.4 Wieferich sequence </div> </a> <ul id="toc-Wieferich_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wieferich_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wieferich_numbers"> <div class="vector-toc-text"> 4.5 Wieferich numbers </div> </a> <ul id="toc-Wieferich_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_Wieferich_prime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_Wieferich_prime"> <div class="vector-toc-text"> 4.6 Weak Wieferich prime </div> </a> <ul id="toc-Weak_Wieferich_prime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wieferich_prime_with_order_n" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wieferich_prime_with_order_n"> <div class="vector-toc-text"> 4.7 Wieferich prime with order n </div> </a> <ul id="toc-Wieferich_prime_with_order_n-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lucas–Wieferich_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lucas–Wieferich_primes"> <div class="vector-toc-text"> 4.8 Lucas–Wieferich primes </div> </a> <ul id="toc-Lucas–Wieferich_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wieferich_places" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wieferich_places"> <div class="vector-toc-text"> 4.9 Wieferich places </div> </a> <ul id="toc-Wieferich_places-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 5 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 6 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 7 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 8 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_primo_de_Wieferich" title="Número primo de Wieferich – Spanish" lang="es" hreflang="es" data-title="Número primo de Wieferich" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_premier_de_Wieferich" title="Nombre premier de Wieferich – French" lang="fr" hreflang="fr" data-title="Nombre premier de Wieferich" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_primo_di_Wieferich" title="Numero primo di Wieferich – Italian" lang="it" hreflang="it" data-title="Numero primo di Wieferich" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Wieferich-pr%C3%ADmek" title="Wieferich-prímek – Hungarian" lang="hu" hreflang="hu" data-title="Wieferich-prímek" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_pierwsze_Wiefericha" title="Liczby pierwsze Wiefericha – Polish" lang="pl" hreflang="pl" data-title="Liczby pierwsze Wiefericha" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a 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.infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title fn org">Wieferich prime</caption><tbody><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="/wiki/Arthur_Wieferich" title="Arthur Wieferich">Arthur Wieferich</a></td></tr><tr><th scope="row" class="infobox-label">Publication year</th><td class="infobox-data">1909</td></tr><tr><th scope="row" class="infobox-label">Author of publication</th><td class="infobox-data"><a href="/wiki/Arthur_Wieferich" title="Arthur Wieferich">Wieferich, A.</a></td></tr><tr><th scope="row" class="infobox-label"><abbr title="Number">No.</abbr> of known terms</th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label">Conjectured <abbr title="number">no.</abbr> of terms</th><td class="infobox-data">Infinite</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Subsequence" title="Subsequence">Subsequence</a> of</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li>Crandall numbers<a href="#cite_note-1">[1]</a></li> <li><a class="mw-selflink-fragment" href="#Wieferich_numbers">Wieferich numbers</a><a href="#cite_note-Banks,_Luca-2">[2]</a></li> <li><a class="mw-selflink-fragment" href="#Lucas-Wieferich_primes">Lucas–Wieferich primes</a><a href="#cite_note-McIntosh,_2007-3">[3]</a></li> <li><a class="mw-selflink-fragment" href="#Near-Wieferich_primes">near-Wieferich primes</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">First terms</th><td class="infobox-data"><a href="/wiki/1093_(number)" title="1093 (number)">1093</a>, <a href="/wiki/3511_(number)" class="mw-redirect" title="3511 (number)">3511</a></td></tr><tr><th scope="row" class="infobox-label">Largest known term</th><td class="infobox-data"><a href="/wiki/3511_(number)" class="mw-redirect" title="3511 (number)">3511</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> index</th><td class="infobox-data"><a rel="nofollow" class="external text" href="//oeis.org/A001220">A001220</a></td></tr></tbody></table> In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, a Wieferich prime is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> p such that p2 divides 2p − 1 − 1,<a href="#cite_note-The_Prime_Glossary-4">[4]</a> therefore connecting these primes with <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by <a href="/wiki/Arthur_Wieferich" title="Arthur Wieferich">Arthur Wieferich</a> in 1909 in works pertaining to <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>, at which time both of Fermat's theorems were already well known to mathematicians.<a href="#cite_note-Kleiner-5">[5]</a><a href="#cite_note-Euler-6">[6]</a> Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne</a> and <a href="/wiki/Fermat_number" title="Fermat number">Fermat</a> numbers, specific types of <a href="/wiki/Pseudoprime" title="Pseudoprime">pseudoprimes</a> and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number fields</a> and the <a href="/wiki/Abc_conjecture" title="Abc conjecture">abc conjecture</a>. As of 2024<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Wieferich_prime&action=edit">[update]</a>, the only known Wieferich primes are 1093 and 3511 (sequence <a href="//oeis.org/A001220" class="extiw" title="oeis:A001220">A001220</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions">Equivalent definitions</h2></div> The stronger version of <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>, which a Wieferich prime satisfies, is usually expressed as a <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a> 2p -1 ≡ 1 (mod p2). From the definition of the <a href="/wiki/Modular_arithmetic#Congruence" title="Modular arithmetic">congruence relation on integers</a>, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime p satisfies this congruence, this prime divides the <a href="/wiki/Fermat_quotient" title="Fermat quotient">Fermat quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{p-1}-1}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{p-1}-1}{p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd23e0dc8312c56c0e3584017637f9fc4ebb014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.3ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2^{p-1}-1}{p}}}">. The following are two illustrative examples using the primes 11 and 1093: <dl><dd>For p = 11, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{10}-1}{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>11</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{10}-1}{11}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf365249f31e5796969af39c0abf2b1815cf83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.258ex; height:4.009ex;" alt="{\displaystyle {\tfrac {2^{10}-1}{11}}}"> which is 93 and leaves a <a href="/wiki/Remainder" title="Remainder">remainder</a> of 5 after division by 11, hence 11 is not a Wieferich prime. For p = 1093, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{1092}-1}{1093}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1092</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>1093</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{1092}-1}{1093}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3006637b946dc3320653ccba89a8bd362a88cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.592ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2^{1092}-1}{1093}}}"> or 485439490310...852893958515 (302 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.</dd></dl> Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence 2p−1 ≡ 1 (mod p2) by 2 to get 2p ≡ 2 (mod p2). Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies 2p2 ≡2p ≡ 2 (mod p2), and hence 2pk ≡ 2 (mod p2) for all k ≥ 1. The converse is also true: 2pk ≡ 2 (mod p2) for some k ≥ 1 implies that the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of 2 modulo p2 divides <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>(pk − 1, <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">φ</a>(p2)) = p − 1, that is, 2p−1 ≡ 1 (mod p2) and thus p is a Wieferich prime. This also implies that Wieferich primes can be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p2 coincide: ordp2 2 = ordp 2, (By the way, ord10932 = 364, and ord35112 = 1755). <a href="/wiki/Harry_Vandiver" title="Harry Vandiver">H. S. Vandiver</a> proved that 2p−1 ≡ 1 (mod p3) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\tfrac {1}{3}}+\dots +{\tfrac {1}{p-2}}\equiv 0{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </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" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\tfrac {1}{3}}+\dots +{\tfrac {1}{p-2}}\equiv 0{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87c320d80ad0063e3f3a41ae1d6addbe1d69fb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:33.997ex; height:3.843ex;" alt="{\displaystyle 1+{\tfrac {1}{3}}+\dots +{\tfrac {1}{p-2}}\equiv 0{\pmod {p^{2}}}}">.<a href="#cite_note-7">[7]</a>: 187  <div class="mw-heading mw-heading2"><h2 id="History_and_search_status">History and search status</h2></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div>Unsolved problem in mathematics:</div> <div class="unsolved-body">Are there infinitely many Wieferich primes?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div>In 1902, <a href="/wiki/Wilhelm_Franz_Meyer" title="Wilhelm Franz Meyer">Meyer</a> proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr).<a href="#cite_note-Solutions-8">[8]</a>: 930 <a href="#cite_note-9">[9]</a> Later in that decade <a href="/wiki/Arthur_Wieferich" title="Arthur Wieferich">Arthur Wieferich</a> showed specifically that if the <a href="/wiki/First_case_of_Fermat%27s_last_theorem" class="mw-redirect" title="First case of Fermat's last theorem">first case of Fermat's last theorem</a> has solutions for an odd prime exponent, then that prime must satisfy that congruence for a = 2 and r = 2.<a href="#cite_note-Wieferich,_Arthur-10">[10]</a> In other words, if there exist solutions to xp + yp + zp = 0 in integers x, y, z and p an <a href="/wiki/Odd_prime" class="mw-redirect" title="Odd prime">odd prime</a> with p <a href="/wiki/List_of_mathematical_symbols#notdivide" class="mw-redirect" title="List of mathematical symbols">∤</a> xyz, then p satisfies 2p − 1 ≡ 1 (mod p2). In 1913, <a href="/wiki/Paul_Gustav_Heinrich_Bachmann" title="Paul Gustav Heinrich Bachmann">Bachmann</a> examined the <a href="/wiki/Remainder" title="Remainder">residues</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> </mrow> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6af7f0eaa3f04b1bcbc32c0cf380ac66493003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.924ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}">. He asked the question when this residue <a href="/wiki/Zero_of_a_function" title="Zero of a function">vanishes</a> and tried to find expressions for answering this question.<a href="#cite_note-11">[11]</a> The prime 1093 was found to be a Wieferich prime by <a href="/w/index.php?title=Waldemar_Meissner&action=edit&redlink=1" class="new" title="Waldemar Meissner (page does not exist)">W. Meissner</a> [<a href="https://cs.wikipedia.org/wiki/Waldemar_Meissner" class="extiw" title="cs:Waldemar Meissner">cs</a>] in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> </mrow> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507b57c2e7e03119f32965d9b9b93ad63cb85084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.03ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p}"> for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a <a href="/wiki/Conjecture" title="Conjecture">conjecture</a> by <a href="/wiki/Dmitry_Grave" title="Dmitry Grave">Grave</a> about the impossibility of the Wieferich congruence.<a href="#cite_note-Meissner-12">[12]</a> <a href="/w/index.php?title=Emil_Haentzschel&action=edit&redlink=1" class="new" title="Emil Haentzschel (page does not exist)">E. Haentzschel</a> [<a href="https://de.wikipedia.org/wiki/Emil_Haentzschel" class="extiw" title="de:Emil Haentzschel">de</a>] later ordered verification of the correctness of Meissner's congruence via only elementary calculations.<a href="#cite_note-13">[13]</a>: 664  Inspired by an earlier work of <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>, he simplified Meissner's proof by showing that 10932 | (2182 + 1) and remarked that (2182 + 1) is a factor of (2364 − 1).<a href="#cite_note-14">[14]</a> It was also shown that it is possible to prove that 1093 is a Wieferich prime without using <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> contrary to the method used by Meissner,<a href="#cite_note-15">[15]</a> although Meissner himself hinted at that he was aware of a proof without complex values.<a href="#cite_note-Meissner-12">[12]</a>: 665  The prime <a href="/wiki/3511_(number)" class="mw-redirect" title="3511 (number)">3511</a> was first found to be a Wieferich prime by <a href="/wiki/N._G._W._H._Beeger" title="N. G. W. H. Beeger">N. G. W. H. Beeger</a> in 1922<a href="#cite_note-16">[16]</a> and another proof of it being a Wieferich prime was published in 1965 by <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy</a>.<a href="#cite_note-17">[17]</a> In 1960, Kravitz<a href="#cite_note-18">[18]</a> doubled a previous record set by <a href="/w/index.php?title=Carl-Erik_Fr%C3%B6berg&action=edit&redlink=1" class="new" title="Carl-Erik Fröberg (page does not exist)">Fröberg</a> [<a href="https://sv.wikipedia.org/wiki/Carl-Erik_Fr%C3%B6berg" class="extiw" title="sv:Carl-Erik Fröberg">sv</a>]<a href="#cite_note-19">[19]</a> and in 1961 <a href="/wiki/Hans_Riesel" title="Hans Riesel">Riesel</a> extended the search to 500000 with the aid of <a href="/wiki/BESK" title="BESK">BESK</a>.<a href="#cite_note-20">[20]</a> Around 1980, <a href="/wiki/Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">Lehmer</a> was able to reach the search limit of 6×109.<a href="#cite_note-21">[21]</a> This limit was extended to over 2.5×1015 in 2006,<a href="#cite_note-Ribenboim,_2004-22">[22]</a> finally reaching 3×1015. Eventually, it was shown that if any other Wieferich primes exist, they must be greater than 6.7×1015.<a href="#cite_note-Dorais-23">[23]</a> In 2007–2016, a search for Wieferich primes was performed by the <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project Wieferich@Home.<a href="#cite_note-24">[24]</a> In 2011–2017, another search was performed by the <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> project, although later the work done in this project was claimed wasted.<a href="#cite_note-25">[25]</a> While these projects reached search bounds above 1×1017, neither of them reported any sustainable results. In 2020, PrimeGrid started another project that searched for Wieferich and <a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun primes</a> simultaneously. The new project used checksums to enable independent double-checking of each subinterval, thus minimizing the risk of missing an instance because of faulty hardware.<a href="#cite_note-26">[26]</a> The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 264 (about 18×1018).<a href="#cite_note-27">[27]</a> It has been conjectured (as for <a href="/wiki/Wilson_prime" title="Wilson prime">Wilson primes</a>) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log(log(x)), which is a <a href="/wiki/Heuristic_argument" title="Heuristic argument">heuristic result</a> that follows from the plausible assumption that for a prime p, the (p − 1)-th degree <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> modulo p2 are <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniformly distributed</a> in the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo p2</a>.<a href="#cite_note-Crandall-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div> <div class="mw-heading mw-heading3"><h3 id="Connection_with_Fermat's_Last_Theorem">Connection with Fermat's Last Theorem</h3></div> The following theorem connecting Wieferich primes and <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> was proven by Wieferich in 1909:<a href="#cite_note-Wieferich,_Arthur-10">[10]</a> <dl><dd>Let p be prime, and let x, y, z be <a href="/wiki/Integer" title="Integer">integers</a> such that xp + yp + zp = 0. Furthermore, assume that p does not divide the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> xyz. Then p is a Wieferich prime.</dd></dl> The above case (where p does not divide any of x, y or z) is commonly known as the <a href="/wiki/First_case_of_Fermat%27s_Last_Theorem" class="mw-redirect" title="First case of Fermat's Last Theorem">first case of Fermat's Last Theorem</a> (FLTI)<a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a> and FLTI is said to fail for a prime p, if solutions to the Fermat equation exist for that p, otherwise FLTI holds for p.<a href="#cite_note-Dilcher,_Skula-31">[31]</a> In 1910, <a href="/wiki/Mirimanoff" class="mw-redirect" title="Mirimanoff">Mirimanoff</a> expanded<a href="#cite_note-32">[32]</a> the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p − 1 − 1. Granville and Monagan further proved that p2 must actually divide mp − 1 − 1 for every prime m ≤ 89.<a href="#cite_note-Granville,_Monagan-33">[33]</a> Suzuki extended the proof to all primes m ≤ 113.<a href="#cite_note-34">[34]</a> Let Hp be a set of pairs of integers with 1 as their <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a>, p being prime to x, y and x + y, (x + y)p−1 ≡ 1 (mod p2), (x + ξy) being the pth power of an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the <a href="/wiki/Field_extension" title="Field extension">field extension</a> obtained by adjoining all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in the <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic number</a> ξ to the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (such an extension is known as a <a href="/wiki/Algebraic_number_field" title="Algebraic number field">number field</a> or in this particular case, where ξ is a <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a>, a <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic number field</a>).<a href="#cite_note-Granville,_Monagan-33">[33]</a>: 332  From <a href="/wiki/Fundamental_theorem_of_ideal_theory_in_number_fields" title="Fundamental theorem of ideal theory in number fields">uniqueness of factorization of ideals in Q(ξ)</a> it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and (x, y), (y, z) and (z, x) are elements of Hp.<a href="#cite_note-Granville,_Monagan-33">[33]</a>: 333  Granville and Monagan showed that (1, 1) ∈ Hp if and only if p is a Wieferich prime.<a href="#cite_note-Granville,_Monagan-33">[33]</a>: 333  <div class="mw-heading mw-heading3"><h3 id="Connection_with_the_abc_conjecture_and_non-Wieferich_primes">Connection with the abc conjecture and non-Wieferich primes</h3></div> A non-Wieferich prime is a prime p satisfying 2p − 1 ≢ 1 (mod p2). <a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">J. H. Silverman</a> showed in 1988 that if the <a href="/wiki/Abc_conjecture" title="Abc conjecture">abc conjecture</a> holds, then there exist infinitely many non-Wieferich primes.<a href="#cite_note-35">[35]</a> More precisely he showed that the abc conjecture implies the existence of a constant only depending on α such that the number of non-Wieferich primes to base α with p less than or equal to a variable X is greater than log(X) as X goes to infinity.<a href="#cite_note-36">[36]</a>: 227  Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by W2 and W2c respectively,<a href="#cite_note-DeKoninckDoyon-37">[37]</a> are <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complementary sets</a>, so if one of them is shown to be finite, the other one would necessarily have to be infinite. It was later shown that the existence of infinitely many non-Wieferich primes already follows from a weaker version of the abc conjecture, called the ABC-(k, ε) conjecture.<a href="#cite_note-38">[38]</a> Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers<a href="#cite_note-39">[39]</a> as well as if there exists a real number ξ such that the set {n ∈ N : λ(2n − 1) < 2 − ξ} is of <a href="/wiki/Natural_density" title="Natural density">density</a> one, where the index of composition λ(n) of an integer n is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\log n}{\log \gamma (n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\log n}{\log \gamma (n)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d443bfb5f290c0505ca6744a8feebcbc61e684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.483ex; height:4.676ex;" alt="{\displaystyle {\tfrac {\log n}{\log \gamma (n)}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (n)=\prod _{p\mid n}p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (n)=\prod _{p\mid n}p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80978e37734b7a8d67b14879b63ba30e89f8fa78" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -3.505ex; width:12.091ex; height:6.009ex;" aria-hidden="true" alt="{\displaystyle \gamma (n)=\prod _{p\mid n}p}">, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb45f2b3e867439b3baeb6060a98e2b62980f912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.466ex; height:2.843ex;" alt="{\displaystyle \gamma (n)}"> gives the product of all <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> of n.<a href="#cite_note-DeKoninckDoyon-37">[37]</a>: 4  <div class="mw-heading mw-heading3"><h3 id="Connection_with_Mersenne_and_Fermat_primes">Connection with Mersenne and Fermat primes</h3></div> It is known that the nth <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne number</a> Mn = 2n − 1 is prime only if n is prime. <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> implies that if p > 2 is prime, then Mp−1 (= 2p − 1 − 1) is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime, <dl><dd><dl><dd>A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if p2 divides Mq.<a href="#cite_note-40">[40]</a></dd></dl></dd></dl> Thus, a Mersenne prime cannot also be a Wieferich prime. A notable <a href="/wiki/Unsolved_problems_in_mathematics" class="mw-redirect" title="Unsolved problems in mathematics">open problem</a> is to determine whether or not all Mersenne numbers of prime index are <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>. If q is prime and the Mersenne number Mq is not square-free, that is, there exists a prime p for which p2 divides Mq, then p is a Wieferich prime. Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.<a href="#cite_note-41">[41]</a> Similarly, if p is prime and p2 divides some <a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a> Fn = 22n + 1, then p must be a Wieferich prime.<a href="#cite_note-42">[42]</a> In fact, there exists a natural number n and a prime p that p2 divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <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 \Phi _{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9f7ff4345961ece5d5cc83f956d4670bcc36fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.036ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(x)}"> is the n-th <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomial</a>) <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> p is a Wieferich prime. For example, 10932 divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{364}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>364</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{364}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff2a80605294464458a23cb9442c7d1f48bac6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.348ex; height:2.843ex;" alt="{\displaystyle \Phi _{364}(2)}">, 35112 divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{1755}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1755</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{1755}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01199ce56db3b7cbcc304df32578baede18a2ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle \Phi _{1755}(2)}">. Mersenne and Fermat numbers are just special situations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}">. Thus, if 1093 and 3511 are only two Wieferich primes, then all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}"> are <a href="/wiki/Square-free_number" class="mw-redirect" title="Square-free number">square-free</a> except <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{364}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>364</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{364}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff2a80605294464458a23cb9442c7d1f48bac6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.348ex; height:2.843ex;" alt="{\displaystyle \Phi _{364}(2)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{1755}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1755</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{1755}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01199ce56db3b7cbcc304df32578baede18a2ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle \Phi _{1755}(2)}"> (In fact, when there exists a prime p which p2 divides some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}">, then it is a Wieferich prime); and clearly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}"> is a prime, then it cannot be Wieferich prime. (Any odd prime p divides only one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}"> and n divides p − 1, and if and only if the period length of 1/p in <a href="/wiki/Binary_number" title="Binary number">binary</a> is n, then p divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1fa46e986a529728b03fce5800b22a7fbe145e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle \Phi _{n}(2)}">. Besides, if and only if p is a Wieferich prime, then the period length of 1/p and 1/p2 are the same (in binary). Otherwise, this is p times than that.) For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because 364 and 1755 are neither prime nor powers of 2.<a href="#cite_note-43">[43]</a> <div class="mw-heading mw-heading3"><h3 id="Connection_with_other_equations">Connection with other equations</h3></div> Scott and Styer showed that the equation px – 2y = d has at most one solution in positive integers (x, y), unless when p4 | 2ordp 2 – 1 if p ≢ 65 (mod 192) or unconditionally when p2 | 2ordp 2 – 1, where ordp 2 denotes the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of 2 modulo p.<a href="#cite_note-44">[44]</a>: 215, 217–218  They also showed that a solution to the equation ±ax1 ± 2y1 = ±ax2 ± 2y2 = c must be from a specific set of equations but that this does not hold, if a is a Wieferich prime greater than 1.25 x 1015.<a href="#cite_note-45">[45]</a>: 258  <div class="mw-heading mw-heading3"><h3 id="Binary_periodicity_of_p_−_1">Binary periodicity of p − 1</h3></div> Johnson observed<a href="#cite_note-John-46">[46]</a> that the two known Wieferich primes are one greater than numbers with periodic <a href="/wiki/Binary_expansion" class="mw-redirect" title="Binary expansion">binary expansions</a> (1092 = 0100010001002=44416; 3510 = 1101101101102=66668). The Wieferich@Home project searched for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.<a href="#cite_note-DoKu-47">[47]</a> <div class="mw-heading mw-heading3"><h3 id="Abundancy_of_p_−_1">Abundancy of p − 1</h3></div> It has been noted (sequence <a href="//oeis.org/A239875" class="extiw" title="oeis:A239875">A239875</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) that the known Wieferich primes are one greater than mutually <a href="/wiki/Friendly_number" title="Friendly number">friendly numbers</a> (the shared abundancy index being 112/39). <div class="mw-heading mw-heading3"><h3 id="Connection_with_pseudoprimes">Connection with pseudoprimes</h3></div> It was observed that the two known Wieferich primes are the square factors of all <a href="/wiki/Square-free_integer" title="Square-free integer">non-square free</a> base-2 <a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprimes</a> up to 25×109.<a href="#cite_note-48">[48]</a> Later computations showed that the only repeated factors of the pseudoprimes up to 1012 are 1093 and 3511.<a href="#cite_note-49">[49]</a> In addition, the following connection exists: <dl><dd>Let n be a base 2 pseudoprime and p be a prime divisor of n. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{n-1}-1}{n}}\not \equiv 0{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </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>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{n-1}-1}{n}}\not \equiv 0{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc92b1dd5108af06da8b5a9dac3918b147dd4a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.544ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2^{n-1}-1}{n}}\not \equiv 0{\pmod {p}}}">, then also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{p-1}-1}{p}}\not \equiv 0{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mstyle> </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>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{p-1}-1}{p}}\not \equiv 0{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4a945aaaddd987c2a7a87ea5767b027a6303b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.415ex; height:4.176ex;" alt="{\displaystyle {\tfrac {2^{p-1}-1}{p}}\not \equiv 0{\pmod {p}}}">.<a href="#cite_note-Dilcher,_Skula-31">[31]</a>: 378  Furthermore, if p is a Wieferich prime, then p2 is a <a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Connection_with_directed_graphs">Connection with directed graphs</h3></div> For all primes p up to 100000, L(pn+1) = L(pn) only in two cases: L(10932) = L(1093) = 364 and L(35112) = L(3511) = 1755, where L(m) is the number of vertices in the cycle of 1 in the doubling diagram modulo m. Here the doubling diagram represents the <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m.<a href="#cite_note-Ehrlich-50">[50]</a>: 74  It was shown, that for all odd prime numbers either L(pn+1) = p · L(pn) or L(pn+1) = L(pn).<a href="#cite_note-Ehrlich-50">[50]</a>: 75  <div class="mw-heading mw-heading3"><h3 id="Properties_related_to_number_fields">Properties related to number fields</h3></div> It was shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732c290f4a29518ad021d52af6dd67fef2b634c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.44ex; height:3.176ex;" alt="{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mspace width="thinmathspace" /> <msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252154ad0139a6273710d98f28ce634451da21cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.045ex; height:3.509ex;" alt="{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}"> if and only if 2p − 1 ≢ 1 (mod p2) where p is an odd prime and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db01a602cd0d8eec69a9d94046509db7aaf5085d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle D_{0}<0}"> is the <a href="/wiki/Fundamental_discriminant" class="mw-redirect" title="Fundamental discriminant">fundamental discriminant</a> of the imaginary <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p^{2}}}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p^{2}}}{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75fca548ed5db2c994ef67f7d9cc649e1e7fe961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.488ex; height:4.843ex;" alt="{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p^{2}}}{\big )}}">. Furthermore, the following was shown: Let p be a Wieferich prime. If p ≡ 3 (mod 4), let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db01a602cd0d8eec69a9d94046509db7aaf5085d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle D_{0}<0}"> be the fundamental discriminant of the imaginary quadratic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p}}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mi>p</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p}}{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce90c2c2e49bd4b57da62b05961a8a4026feb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.434ex; height:3.509ex;" alt="{\displaystyle \mathbb {Q} {\big (}{\sqrt {1-p}}{\big )}}"> and if p ≡ 1 (mod 4), let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db01a602cd0d8eec69a9d94046509db7aaf5085d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle D_{0}<0}"> be the fundamental discriminant of the imaginary quadratic field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} {\big (}{\sqrt {4-p}}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mo>−</mo> <mi>p</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} {\big (}{\sqrt {4-p}}{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436d7100dafdfae9cd180facd2855a02644256fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.434ex; height:3.509ex;" alt="{\displaystyle \mathbb {Q} {\big (}{\sqrt {4-p}}{\big )}}">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732c290f4a29518ad021d52af6dd67fef2b634c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.44ex; height:3.176ex;" alt="{\displaystyle \chi _{D_{0}}{\big (}p{\big )}=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mspace width="thinmathspace" /> <msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252154ad0139a6273710d98f28ce634451da21cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.045ex; height:3.509ex;" alt="{\displaystyle \lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1}"> (χ and λ in this context denote Iwasawa <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a>).<a href="#cite_note-51">[51]</a>: 27  Furthermore, the following result was obtained: Let q be an odd prime number, k and p are primes such that p = 2k + 1, k ≡ 3 (mod 4), p ≡ −1 (mod q), p ≢ −1 (mod q3) and the order of q modulo k is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {k-1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {k-1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc981e84f5ee3097d48a6b426e0bb2cbb5ce9505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.793ex; height:3.676ex;" alt="{\displaystyle {\tfrac {k-1}{2}}}">. Assume that q divides h+, the <a href="/wiki/Ideal_class_group" title="Ideal class group">class number</a> of the real <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} {\big (}\zeta \,\!_{p}+\zeta \,\!_{p}^{-1}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ζ</mi> <mspace width="thinmathspace" /> <msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mi>ζ</mi> <mspace width="thinmathspace" /> <msubsup> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} {\big (}\zeta \,\!_{p}+\zeta \,\!_{p}^{-1}{\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640b7beba31418c7fcb77c91c242c0191bd09197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.36ex; height:3.343ex;" alt="{\displaystyle \mathbb {Q} {\big (}\zeta \,\!_{p}+\zeta \,\!_{p}^{-1}{\big )}}">, the cyclotomic field obtained by adjoining the sum of a p-th <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a> and its <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> to the field of rational numbers. Then q is a Wieferich prime.<a href="#cite_note-Jakubec-52">[52]</a>: 55  This also holds if the conditions p ≡ −1 (mod q) and p ≢ −1 (mod q3) are replaced by p ≡ −3 (mod q) and p ≢ −3 (mod q3) as well as when the condition p ≡ −1 (mod q) is replaced by p ≡ −5 (mod q) (in which case q is a <a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun prime</a>) and the incongruence condition replaced by p ≢ −5 (mod q3).<a href="#cite_note-53">[53]</a>: 376  <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div> <div class="mw-heading mw-heading3"><h3 id="Near-Wieferich_primes">Near-Wieferich primes</h3></div> A prime p satisfying the congruence 2(p−1)/2 ≡ ±1 + Ap (mod p2) with small |A| is commonly called a near-Wieferich prime (sequence <a href="//oeis.org/A195988" class="extiw" title="oeis:A195988">A195988</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).<a href="#cite_note-Crandall-28">[28]</a><a href="#cite_note-Knauer-54">[54]</a> Near-Wieferich primes with A = 0 represent Wieferich primes. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes.<a href="#cite_note-Dorais-23">[23]</a><a href="#cite_note-elmath-55">[55]</a> The following table lists all near-Wieferich primes with |A| ≤ 10 in the interval [1×109, 3×1015].<a href="#cite_note-56">[56]</a> This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.<a href="#cite_note-Ribenboim,_2004-22">[22]</a><a href="#cite_note-57">[57]</a> Bigger entries are by PrimeGrid. <table class="wikitable" style="width:40%; border:0; text-align:right;"> <tbody><tr> <th>p</th> <th>1 or −1</th> <th>A </th></tr> <tr> <td>3520624567</td> <td>+1</td> <td>−6 </td></tr> <tr> <td>46262476201</td> <td>+1</td> <td>+5 </td></tr> <tr> <td>47004625957</td> <td>−1</td> <td>+1 </td></tr> <tr> <td>58481216789</td> <td>−1</td> <td>+5 </td></tr> <tr> <td>76843523891</td> <td>−1</td> <td>+1 </td></tr> <tr> <td>1180032105761</td> <td>+1</td> <td>−6 </td></tr> <tr> <td>12456646902457</td> <td>+1</td> <td>+2 </td></tr> <tr> <td>134257821895921</td> <td>+1</td> <td>+10 </td></tr> <tr> <td>339258218134349</td> <td>−1</td> <td>+2 </td></tr> <tr> <td>2276306935816523</td> <td>−1</td> <td>−3 </td></tr> <tr> <td>82687771042557349</td> <td>-1</td> <td>-10 </td></tr> <tr> <td>3156824277937156367</td> <td>+1</td> <td>+7 </td></tr></tbody></table> The sign +1 or -1 above can be easily predicted by <a href="/wiki/Euler%27s_criterion" title="Euler's criterion">Euler's criterion</a> (and the second supplement to the law of <a href="/wiki/Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>). Dorais and Klyve<a href="#cite_note-Dorais-23">[23]</a> used a different definition of a near-Wieferich prime, defining it as a prime p with small value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23150644c25d9a2c6fc759fb707f2dafad1d2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.258ex; height:4.843ex;" alt="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> </mrow> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abda4c40aa3d0ff39c3ecbf1a1928a0f81bd499e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.447ex; height:4.176ex;" alt="{\displaystyle \omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p}"> is the <a href="/wiki/Fermat_quotient" title="Fermat quotient">Fermat quotient</a> of 2 with respect to p modulo p (the <a href="/wiki/Modulo_operation" class="mw-redirect" title="Modulo operation">modulo operation</a> here gives the residue with the smallest absolute value). The following table lists all primes p ≤ 6.7 × 1015 with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\leq 10^{-14}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\leq 10^{-14}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48df3c472dd78c8c69b3b5546840c2a031a1598e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.837ex; height:4.843ex;" alt="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\leq 10^{-14}}">. <table class="wikitable" style="width:40%; border:0; text-align:right;"> <tbody><tr> <th>p</th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c9e903e57a81e8964561485fe7ed2e14be3443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:2.843ex;" alt="{\displaystyle \omega (p)}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\times 10^{14}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>|</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\times 10^{14}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a41c92ad1059cde1934ea0e2f8fe3462f280acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.3ex; height:4.843ex;" alt="{\displaystyle \left|{\tfrac {\omega (p)}{p}}\right|\times 10^{14}}"> </th></tr> <tr> <td>1093</td> <td>0</td> <td>0 </td></tr> <tr> <td>3511</td> <td>0</td> <td>0 </td></tr> <tr> <td>2276306935816523</td> <td>+6</td> <td>0.264 </td></tr> <tr> <td>3167939147662997</td> <td>−17</td> <td>0.537 </td></tr> <tr> <td>3723113065138349</td> <td>−36</td> <td>0.967 </td></tr> <tr> <td>5131427559624857</td> <td>−36</td> <td>0.702 </td></tr> <tr> <td>5294488110626977</td> <td>−31</td> <td>0.586 </td></tr> <tr> <td>6517506365514181</td> <td>+58</td> <td>0.890 </td></tr></tbody></table> The two notions of nearness are related as follows. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{(p-1)/2}\equiv \pm 1+Ap{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mo>±</mo> <mn>1</mn> <mo>+</mo> <mi>A</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{(p-1)/2}\equiv \pm 1+Ap{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d764398acbc1a599ce87bee399dc7c9cea6dda2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.975ex; height:3.343ex;" alt="{\displaystyle 2^{(p-1)/2}\equiv \pm 1+Ap{\pmod {p^{2}}}}">, then by squaring, clearly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{p-1}\equiv 1\pm 2Ap{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mo>±</mo> <mn>2</mn> <mi>A</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p-1}\equiv 1\pm 2Ap{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cb7597d22da1543301ea6ca43f727156c37a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.406ex; height:3.176ex;" alt="{\displaystyle 2^{p-1}\equiv 1\pm 2Ap{\pmod {p^{2}}}}">. So if A had been chosen with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648fce92f29d925f04d39244ccfe435320dfc6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle |A|}"> small, then clearly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\!\pm 2A|=2|A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="negativethinmathspace" /> <mo>±</mo> <mn>2</mn> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\!\pm 2A|=2|A|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebae73262826b4aea1d756fd6c8539ba932a7647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.95ex; height:2.843ex;" alt="{\displaystyle |\!\pm 2A|=2|A|}"> is also (quite) small, and an even number. However, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c9e903e57a81e8964561485fe7ed2e14be3443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:2.843ex;" alt="{\displaystyle \omega (p)}"> is odd above, the related A from before the last squaring was not "small". For example, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=3167939147662997}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>3167939147662997</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=3167939147662997}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d1b87c17e0e066a6f3972f6d4954f0f7a01ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:22.957ex; height:2.509ex;" alt="{\displaystyle p=3167939147662997}">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{(p-1)/2}\equiv -1-1583969573831490p{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>1583969573831490</mn> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{(p-1)/2}\equiv -1-1583969573831490p{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/848ac4d7b5039a32dbd18020965065cd7771848d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.831ex; height:3.343ex;" alt="{\displaystyle 2^{(p-1)/2}\equiv -1-1583969573831490p{\pmod {p^{2}}}}"> which reads extremely non-near, but after squaring this is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{p-1}\equiv 1-17p{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mo>−</mo> <mn>17</mn> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p-1}\equiv 1-17p{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05593c065e83dc0661dcf14d233202ae376ca2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.825ex; height:3.176ex;" alt="{\displaystyle 2^{p-1}\equiv 1-17p{\pmod {p^{2}}}}"> which is a near-Wieferich by the second definition. <div class="mw-heading mw-heading3"><h3 id="Base-a_Wieferich_primes">Base-a Wieferich primes</h3></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fermat_quotient#Generalized_Wieferich_primes" title="Fermat quotient">Fermat quotient</a></div> A Wieferich prime base a is a prime p that satisfies <dl><dd>ap − 1 ≡ 1 (mod p2),<a href="#cite_note-Solutions-8">[8]</a> with a less than p but greater than 1.</dd></dl> Such a prime cannot divide a, since then it would also divide 1. It's a conjecture that for every natural number a, there are infinitely many Wieferich primes in base a. Bolyai showed that if p and q are primes, a is a positive integer not divisible by p and q such that ap−1 ≡ 1 (mod q), aq−1 ≡ 1 (mod p), then apq−1 ≡ 1 (mod pq). Setting p = q leads to ap2−1 ≡ 1 (mod p2).<a href="#cite_note-Kiss-58">[58]</a>: 284  It was shown that ap2−1 ≡ 1 (mod p2) if and only if ap−1 ≡ 1 (mod p2).<a href="#cite_note-Kiss-58">[58]</a>: 285–286  Known solutions of ap−1 ≡ 1 (mod p2) for small values of a are:<a href="#cite_note-59">[59]</a> (checked up to 5 × 1013) <dl><dd><table class="wikitable"> <tbody><tr> <th>a </th> <th>primes p such that ap − 1 = 1 (mod p2) </th> <th><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence </th></tr> <tr> <td>1</td> <td>2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) </td> <td><a href="//oeis.org/A000040" class="extiw" title="oeis:A000040">A000040</a> </td></tr> <tr> <td>2</td> <td>1093, 3511, ... </td> <td><a href="//oeis.org/A001220" class="extiw" title="oeis:A001220">A001220</a> </td></tr> <tr> <td>3</td> <td>11, 1006003, ... </td> <td><a href="//oeis.org/A014127" class="extiw" title="oeis:A014127">A014127</a> </td></tr> <tr> <td>4</td> <td>1093, 3511, ... </td> <td> </td></tr> <tr> <td>5</td> <td>2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... </td> <td><a href="//oeis.org/A123692" class="extiw" title="oeis:A123692">A123692</a> </td></tr> <tr> <td>6</td> <td>66161, 534851, 3152573, ... </td> <td><a href="//oeis.org/A212583" class="extiw" title="oeis:A212583">A212583</a> </td></tr> <tr> <td>7</td> <td>5, 491531, ... </td> <td><a href="//oeis.org/A123693" class="extiw" title="oeis:A123693">A123693</a> </td></tr> <tr> <td>8</td> <td>3, 1093, 3511, ... </td> <td> </td></tr> <tr> <td>9</td> <td>2, 11, 1006003, ... </td> <td> </td></tr> <tr> <td>10</td> <td>3, 487, 56598313, ... </td> <td><a href="//oeis.org/A045616" class="extiw" title="oeis:A045616">A045616</a> </td></tr> <tr> <td>11</td> <td>71, ... </td> <td> </td></tr> <tr> <td>12</td> <td>2693, 123653, ... </td> <td><a href="//oeis.org/A111027" class="extiw" title="oeis:A111027">A111027</a> </td></tr> <tr> <td>13</td> <td>2, 863, 1747591, ... </td> <td><a href="//oeis.org/A128667" class="extiw" title="oeis:A128667">A128667</a> </td></tr> <tr> <td>14</td> <td>29, 353, 7596952219, ... </td> <td><a href="//oeis.org/A234810" class="extiw" title="oeis:A234810">A234810</a> </td></tr> <tr> <td>15</td> <td>29131, 119327070011, ... </td> <td><a href="//oeis.org/A242741" class="extiw" title="oeis:A242741">A242741</a> </td></tr> <tr> <td>16</td> <td>1093, 3511, ... </td> <td> </td></tr> <tr> <td>17</td> <td>2, 3, 46021, 48947, 478225523351, ... </td> <td><a href="//oeis.org/A128668" class="extiw" title="oeis:A128668">A128668</a> </td></tr> <tr> <td>18</td> <td>5, 7, 37, 331, 33923, 1284043, ... </td> <td><a href="//oeis.org/A244260" class="extiw" title="oeis:A244260">A244260</a> </td></tr> <tr> <td>19</td> <td>3, 7, 13, 43, 137, 63061489, ... </td> <td><a href="//oeis.org/A090968" class="extiw" title="oeis:A090968">A090968</a> </td></tr> <tr> <td>20</td> <td>281, 46457, 9377747, 122959073, ... </td> <td><a href="//oeis.org/A242982" class="extiw" title="oeis:A242982">A242982</a> </td></tr> <tr> <td>21</td> <td>2, ... </td> <td> </td></tr> <tr> <td>22</td> <td>13, 673, 1595813, 492366587, 9809862296159, ... </td> <td><a href="//oeis.org/A298951" class="extiw" title="oeis:A298951">A298951</a> </td></tr> <tr> <td>23</td> <td>13, 2481757, 13703077, 15546404183, 2549536629329, ... </td> <td><a href="//oeis.org/A128669" class="extiw" title="oeis:A128669">A128669</a> </td></tr> <tr> <td>24</td> <td>5, 25633, ... </td> <td> </td></tr> <tr> <td>25</td> <td>2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... </td> <td> </td></tr> <tr> <td>26</td> <td>3, 5, 71, 486999673, 6695256707, ... </td> <td><a href="//oeis.org/A306255" class="extiw" title="oeis:A306255">A306255</a> </td></tr> <tr> <td>27</td> <td>11, 1006003, ... </td> <td> </td></tr> <tr> <td>28</td> <td>3, 19, 23, ... </td> <td> </td></tr> <tr> <td>29</td> <td>2, ... </td> <td> </td></tr> <tr> <td>30</td> <td>7, 160541, 94727075783, ... </td> <td><a href="//oeis.org/A306256" class="extiw" title="oeis:A306256">A306256</a> </td></tr> <tr> <td>31</td> <td>7, 79, 6451, 2806861, ... </td> <td><a href="//oeis.org/A331424" class="extiw" title="oeis:A331424">A331424</a> </td></tr> <tr> <td>32</td> <td>5, 1093, 3511, ... </td> <td> </td></tr> <tr> <td>33</td> <td>2, 233, 47441, 9639595369, ... </td> <td> </td></tr> <tr> <td>34</td> <td>46145917691, ... </td> <td> </td></tr> <tr> <td>35</td> <td>3, 1613, 3571, ... </td> <td> </td></tr> <tr> <td>36</td> <td>66161, 534851, 3152573, ... </td> <td> </td></tr> <tr> <td>37</td> <td>2, 3, 77867, 76407520781, ... </td> <td><a href="//oeis.org/A331426" class="extiw" title="oeis:A331426">A331426</a> </td></tr> <tr> <td>38</td> <td>17, 127, ... </td> <td> </td></tr> <tr> <td>39</td> <td>8039, ... </td> <td> </td></tr> <tr> <td>40</td> <td>11, 17, 307, 66431, 7036306088681, ... </td> <td> </td></tr> <tr> <td>41</td> <td>2, 29, 1025273, 138200401, ... </td> <td><a href="//oeis.org/A331427" class="extiw" title="oeis:A331427">A331427</a> </td></tr> <tr> <td>42</td> <td>23, 719867822369, ... </td> <td> </td></tr> <tr> <td>43</td> <td>5, 103, 13368932516573, ... </td> <td> </td></tr> <tr> <td>44</td> <td>3, 229, 5851, ... </td> <td> </td></tr> <tr> <td>45</td> <td>2, 1283, 131759, 157635607, ... </td> <td> </td></tr> <tr> <td>46</td> <td>3, 829, ... </td> <td> </td></tr> <tr> <td>47</td> <td>... </td> <td> </td></tr> <tr> <td>48</td> <td>7, 257, ... </td> <td> </td></tr> <tr> <td>49</td> <td>2, 5, 491531, ... </td> <td> </td></tr> <tr> <td>50</td> <td>7, ... </td> <td> </td></tr></tbody></table></dd></dl> For more information, see<a href="#cite_note-60">[60]</a><a href="#cite_note-61">[61]</a><a href="#cite_note-62">[62]</a> and.<a href="#cite_note-63">[63]</a> (Note that the solutions to a = bk is the union of the prime divisors of k which does not divide b and the solutions to a = b) The smallest solutions of np−1 ≡ 1 (mod p2) are <dl><dd>2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... (The next term > 4.9×1013) (sequence <a href="//oeis.org/A039951" class="extiw" title="oeis:A039951">A039951</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> There are no known solutions of np−1 ≡ 1 (mod p2) for n = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, 1002, 1023, 1130, 1136, 1138, .... It is a conjecture that there are infinitely many solutions of ap−1 ≡ 1 (mod p2) for every natural number a. The bases b < p2 which p is a Wieferich prime are (for b > p2, the solutions are just shifted by k·p2 for k > 0), and there are p − 1 solutions < p2 of p and the set of the solutions <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">congruent</a> to p are {1, 2, 3, ..., p − 1}) (sequence <a href="//oeis.org/A143548" class="extiw" title="oeis:A143548">A143548</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) <dl><dd><table class="wikitable"> <tbody><tr> <th>p </th> <th>values of b < p2 </th></tr> <tr> <td>2 </td> <td>1 </td></tr> <tr> <td>3 </td> <td>1, 8 </td></tr> <tr> <td>5 </td> <td>1, 7, 18, 24 </td></tr> <tr> <td>7 </td> <td>1, 18, 19, 30, 31, 48 </td></tr> <tr> <td>11 </td> <td>1, 3, 9, 27, 40, 81, 94, 112, 118, 120 </td></tr> <tr> <td>13 </td> <td>1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168 </td></tr> <tr> <td>17 </td> <td>1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288 </td></tr> <tr> <td>19 </td> <td>1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360 </td></tr> <tr> <td>23 </td> <td>1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528 </td></tr> <tr> <td>29 </td> <td>1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840 </td></tr></tbody></table></dd></dl> The least base b > 1 which prime(n) is a Wieferich prime are <dl><dd>5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, ... (sequence <a href="//oeis.org/A039678" class="extiw" title="oeis:A039678">A039678</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> We can also consider the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f5a73200b966d55dfb42b8b7bf820d98d17cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.6ex; height:3.176ex;" alt="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}">, (because of the generalized Fermat little theorem, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f5a73200b966d55dfb42b8b7bf820d98d17cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.6ex; height:3.176ex;" alt="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> is true for all prime p and all natural number a such that both a and a + 1 are not divisible by p). It's a conjecture that for every natural number a, there are infinitely many primes such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f5a73200b966d55dfb42b8b7bf820d98d17cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.6ex; height:3.176ex;" alt="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}">. Known solutions for small a are: (checked up to 4 × 1011) <a href="#cite_note-64">[64]</a> <dl><dd><table class="wikitable"> <tbody><tr> <th>⁠<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><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}">⁠ </th> <th>primes ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">⁠ such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f5a73200b966d55dfb42b8b7bf820d98d17cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.6ex; height:3.176ex;" alt="{\displaystyle (a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}}"> </th></tr> <tr> <td>1 </td> <td>1093, 3511, ... </td></tr> <tr> <td>2 </td> <td>23, 3842760169, 41975417117, ... </td></tr> <tr> <td>3 </td> <td>5, 250829, ... </td></tr> <tr> <td>4 </td> <td>3, 67, ... </td></tr> <tr> <td>5 </td> <td>3457, 893122907, ... </td></tr> <tr> <td>6 </td> <td>72673, 1108905403, 2375385997, ... </td></tr> <tr> <td>7 </td> <td>13, 819381943, ... </td></tr> <tr> <td>8 </td> <td>67, 139, 499, 26325777341, ... </td></tr> <tr> <td>9 </td> <td>67, 887, 9257, 83449, 111539, 31832131, ... </td></tr> <tr> <td>10 </td> <td>... </td></tr> <tr> <td>11 </td> <td>107, 4637, 239357, ... </td></tr> <tr> <td>12 </td> <td>5, 11, 51563, 363901, 224189011, ... </td></tr> <tr> <td>13 </td> <td>3, ... </td></tr> <tr> <td>14 </td> <td>11, 5749, 17733170113, 140328785783, ... </td></tr> <tr> <td>15 </td> <td>292381, ... </td></tr> <tr> <td>16 </td> <td>4157, ... </td></tr> <tr> <td>17 </td> <td>751, 46070159, ... </td></tr> <tr> <td>18 </td> <td>7, 142671309349, ... </td></tr> <tr> <td>19 </td> <td>17, 269, ... </td></tr> <tr> <td>20 </td> <td>29, 162703, ... </td></tr> <tr> <td>21 </td> <td>5, 2711, 104651, 112922981, 331325567, 13315963127, ... </td></tr> <tr> <td>22 </td> <td>3, 7, 13, 94447, 1198427, 23536243, ... </td></tr> <tr> <td>23 </td> <td>43, 179, 1637, 69073, ... </td></tr> <tr> <td>24 </td> <td>7, 353, 402153391, ... </td></tr> <tr> <td>25 </td> <td>43, 5399, 21107, 35879, ... </td></tr> <tr> <td>26 </td> <td>7, 131, 653, 5237, 97003, ... </td></tr> <tr> <td>27 </td> <td>2437, 1704732131, ... </td></tr> <tr> <td>28 </td> <td>5, 617, 677, 2273, 16243697, ... </td></tr> <tr> <td>29 </td> <td>73, 101, 6217, ... </td></tr> <tr> <td>30 </td> <td>7, 11, 23, 3301, 48589, 549667, ... </td></tr> <tr> <td>31 </td> <td>3, 41, 416797, ... </td></tr> <tr> <td>32 </td> <td>95989, 2276682269, ... </td></tr> <tr> <td>33 </td> <td>139, 1341678275933, ... </td></tr> <tr> <td>34 </td> <td>83, 139, ... </td></tr> <tr> <td>35 </td> <td>... </td></tr> <tr> <td>36 </td> <td>107, 137, 613, 2423, 74304856177, ... </td></tr> <tr> <td>37 </td> <td>5, ... </td></tr> <tr> <td>38 </td> <td>167, 2039, ... </td></tr> <tr> <td>39 </td> <td>659, 9413, ... </td></tr> <tr> <td>40 </td> <td>3, 23, 21029249, ... </td></tr> <tr> <td>41 </td> <td>31, 71, 1934399021, 474528373843, ... </td></tr> <tr> <td>42 </td> <td>4639, 1672609, ... </td></tr> <tr> <td>43 </td> <td>31, 4962186419, ... </td></tr> <tr> <td>44 </td> <td>36677, 17786501, ... </td></tr> <tr> <td>45 </td> <td>241, 26120375473, ... </td></tr> <tr> <td>46 </td> <td>5, 13877, ... </td></tr> <tr> <td>47 </td> <td>13, 311, 797, 906165497, ... </td></tr> <tr> <td>48 </td> <td>... </td></tr> <tr> <td>49 </td> <td>3, 13, 2141, 281833, 1703287, 4805298913, ... </td></tr> <tr> <td>50 </td> <td>2953, 22409, 99241, 5427425917, ... </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wieferich_pairs">Wieferich pairs</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Wieferich_pair" title="Wieferich pair">Wieferich pair</a></div> A <a href="/wiki/Wieferich_pair" title="Wieferich pair">Wieferich pair</a> is a pair of primes p and q that satisfy <dl><dd>pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)</dd></dl> so that a Wieferich prime p ≡ 1 (mod 4) will form such a pair (p, 2): the only known instance in this case is p = 1093. There are only 7 known Wieferich pairs.<a href="#cite_note-65">[65]</a> <dl><dd>(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A282293" class="extiw" title="oeis:A282293">A282293</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wieferich_sequence">Wieferich sequence</h3></div> Start with a(1) any natural number (>1), a(n) = the smallest prime p such that (a(n − 1))p − 1 = 1 (mod p2) but p2 does not divide a(n − 1) − 1 or a(n − 1) + 1. (If p2 divides a(n − 1) − 1 or a(n − 1) + 1, then the solution is a <a href="/wiki/Triviality_(mathematics)" title="Triviality (mathematics)">trivial solution</a>) It is a conjecture that every natural number k = a(1) > 1 makes this sequence become periodic, for example, let a(1) = 2: <dl><dd>2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.</dd> <dd>(sequence <a href="//oeis.org/A359952" class="extiw" title="oeis:A359952">A359952</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> Let a(1) = 83: <dl><dd>83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}.</dd></dl> Let a(1) = 59 (a longer sequence): <dl><dd>59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ..., it also gets 5.</dd></dl> However, there are many values of a(1) with unknown status, for example, let a(1) = 3: <dl><dd>3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).</dd></dl> Let a(1) = 14: <dl><dd>14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 22 = 4 divides 29 − 1 = 28)</dd></dl> Let a(1) = 39 (a longer sequence): <dl><dd>39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)</dd></dl> It is unknown that values for a(1) > 1 exist such that the resulting sequence does not eventually become periodic. When a(n − 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown) <div class="mw-heading mw-heading3"><h3 id="Wieferich_numbers">Wieferich numbers</h3></div> A Wieferich number is an odd natural number n satisfying the congruence 2φ(n) ≡ 1 (mod n2), where φ denotes the <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a> (according to <a href="/wiki/Euler%27s_theorem" title="Euler's theorem">Euler's theorem</a>, 2φ(n) ≡ 1 (mod n) for every odd natural number n). If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are: <dl><dd>1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... (sequence <a href="//oeis.org/A077816" class="extiw" title="oeis:A077816">A077816</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.<a href="#cite_note-Banks,_Luca-2">[2]</a> More generally, a natural number n is a Wieferich number to base a, if aφ(n) ≡ 1 (mod n2).<a href="#cite_note-66">[66]</a>: 31  Another definition specifies a Wieferich number as odd natural number n such that n and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2^{m}-1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2^{m}-1}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000725ee097bda39e94dda780a8275248deef28d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.094ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2^{m}-1}{n}}}"> are not <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>, where m is the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of 2 modulo n. The first of these numbers are:<a href="#cite_note-67">[67]</a> <dl><dd>21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, ... (sequence <a href="//oeis.org/A182297" class="extiw" title="oeis:A182297">A182297</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> As above, if Wieferich number q is prime, then it is a Wieferich prime. <div class="mw-heading mw-heading3"><h3 id="Weak_Wieferich_prime">Weak Wieferich prime</h3></div> A weak Wieferich prime to base a is a prime p satisfies the condition <dl><dd>ap ≡ a (mod p2)</dd></dl> Every Wieferich prime to base a is also a weak Wieferich prime to base a. If the base a is <a href="/wiki/Squarefree_number" class="mw-redirect" title="Squarefree number">squarefree</a>, then a prime p is a weak Wieferich prime to base a if and only if p is a Wieferich prime to base a. Smallest weak Wieferich prime to base n are (start with n = 0) <dl><dd>2, 2, 1093, 11, 2, 2, 66161, 5, 2, 2, 3, 71, 2, 2, 29, 29131, 2, 2, 3, 3, 2, 2, 13, 13, 2, 2, 3, 3, 2, 2, 7, 7, 2, 2, 46145917691, 3, 2, 2, 17, 8039, 2, 2, 23, 5, 2, 2, 3, ...</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wieferich_prime_with_order_n">Wieferich prime with order n</h3></div> For integer n ≥2, a Wieferich prime to base a with order n is a prime p satisfies the condition <dl><dd>ap−1 ≡ 1 (mod pn)</dd></dl> Clearly, a Wieferich prime to base a with order n is also a Wieferich prime to base a with order m for all 2 ≤ m ≤ n, and Wieferich prime to base a with order 2 is equivalent to Wieferich prime to base a, so we can only consider the n ≥ 3 case. However, there are no known Wieferich prime to base 2 with order 3. The first base with known Wieferich prime with order 3 is 9, where 2 is a Wieferich prime to base 9 with order 3. Besides, both 5 and 113 are Wieferich prime to base 68 with order 3. <div class="mw-heading mw-heading3"><h3 id="Lucas–Wieferich_primes">Lucas–Wieferich primes</h3></div> Let P and Q be integers. The <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas sequence</a> <a href="/wiki/Lucas_sequence#Recurrence_relations" title="Lucas sequence">of the first kind</a> associated with the <a href="/wiki/Ordered_pair" title="Ordered pair">pair</a> (P, Q) is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mo>⋅</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Q</mi> <mo>⋅</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce10d52aa8db4c5727ec36003b56ebe385dba4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.532ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\end{aligned}}}"></dd></dl> for all <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><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}">. A Lucas–Wieferich prime associated with (P, Q) is a prime p such that Up−ε(P, Q) ≡ 0 (mod p2), where ε equals the <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {P^{2}-4Q}{p}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>Q</mi> </mrow> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {P^{2}-4Q}{p}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c072c5ed08411ae0f8e174167279e2d7e5ce9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.132ex; height:5.009ex;" alt="{\displaystyle \left({\tfrac {P^{2}-4Q}{p}}\right)}">. All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).<a href="#cite_note-McIntosh,_2007-3">[3]</a>: 2088  <div class="mw-heading mw-heading3"><h3 id="Wieferich_places">Wieferich places</h3></div> Let K be a <a href="/wiki/Global_field" title="Global field">global field</a>, i.e. a <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a> or a <a href="/wiki/Algebraic_function_field" title="Algebraic function field">function field</a> in one variable over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> and let E be an <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curve</a>. If v is a <a href="/wiki/Algebraic_number_field#Nonarchimedian_or_ultrametric_places" title="Algebraic number field">non-archimedean place</a> of <a href="/wiki/Field_norm" title="Field norm">norm</a> qv of K and a ∈ K, with v(a) = 0 then v(aqv − 1 − 1) ≥ 1. v is called a Wieferich place for base a, if v(aqv − 1 − 1) > 1, an elliptic Wieferich place for base P ∈ E, if NvP ∈ E2 and a strong elliptic Wieferich place for base P ∈ E if nvP ∈ E2, where nv is the order of P modulo v and Nv gives the number of <a href="/wiki/Rational_point" title="Rational point">rational points</a> (over the <a href="/wiki/Residue_field" title="Residue field">residue field</a> of v) of the reduction of E at v.<a href="#cite_note-68">[68]</a>: 206  <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun prime</a> – another type of prime number which in the broadest sense also resulted from the study of FLT</li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a> – another type of prime number which in the broadest sense also resulted from the study of FLT</li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson prime</a></li> <li><a href="/wiki/Table_of_congruences" title="Table of congruences">Table of congruences</a> – lists other congruences satisfied by prime numbers</li> <li><a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> – primes search project</li> <li><a href="/wiki/BOINC" class="mw-redirect" title="BOINC">BOINC</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output 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(1998), <a rel="nofollow" class="external text" href="http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00916-8/S0025-5718-98-00916-8.pdf">"On divisibility of the class number h+ of the real cyclotomic fields of prime degree l"</a> (PDF), Mathematics of Computation, 67 (221): 369–398, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-98-00916-8">10.1090/s0025-5718-98-00916-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=On+divisibility+of+the+class+number+h%3Csup%3E%2B%3C%2Fsup%3E+of+the+real+cyclotomic+fields+of+prime+degree+l&rft.volume=67&rft.issue=221&rft.pages=369-398&rft.date=1998&rft_id=info%3Adoi%2F10.1090%2Fs0025-5718-98-00916-8&rft.aulast=Jakubec&rft.aufirst=S.&rft_id=http%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1998-67-221%2FS0025-5718-98-00916-8%2FS0025-5718-98-00916-8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-Knauer-54"><a href="#cite_ref-Knauer_54-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoshua_KnauerJörg_Richstein2005" class="citation cs2">Joshua Knauer; Jörg Richstein (2005), <a rel="nofollow" class="external text" href="http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-05-01723-0/S0025-5718-05-01723-0.pdf">"The continuing search for Wieferich primes"</a> (PDF), Mathematics of Computation, 74 (251): 1559–1563, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005MaCom..74.1559K">2005MaCom..74.1559K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-05-01723-0">10.1090/S0025-5718-05-01723-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=The+continuing+search+for+Wieferich+primes&rft.volume=74&rft.issue=251&rft.pages=1559-1563&rft.date=2005&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-05-01723-0&rft_id=info%3Abibcode%2F2005MaCom..74.1559K&rft.au=Joshua+Knauer&rft.au=J%C3%B6rg+Richstein&rft_id=http%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F2005-74-251%2FS0025-5718-05-01723-0%2FS0025-5718-05-01723-0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-elmath-55"><a href="#cite_ref-elmath_55-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120322140402/http://www.elmath.org/index.php?id=main">"About project Wieferich@Home"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.elmath.org/index.php?id=main">the original</a> on 2012-03-22. Retrieved 2010-07-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=About+project+Wieferich%40Home&rft_id=http%3A%2F%2Fwww.elmath.org%2Findex.php%3Fid%3Dmain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> PrimeGrid, <a rel="nofollow" class="external text" href="http://www.primegrid.com/download/wieferich_list.pdf">Wieferich & near Wieferich primes p < 11e15</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018120338/http://www.primegrid.com/download/wieferich_list.pdf">Archived</a> 2012-10-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim2000" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (2000), My numbers, my friends: popular lectures on number theory, New York: Springer, pp. 213–229, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98911-2" title="Special:BookSources/978-0-387-98911-2"><bdi>978-0-387-98911-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=My+numbers%2C+my+friends%3A+popular+lectures+on+number+theory&rft.place=New+York&rft.pages=213-229&rft.pub=Springer&rft.date=2000&rft.isbn=978-0-387-98911-2&rft.aulast=Ribenboim&rft.aufirst=Paulo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-Kiss-58">^ <a href="#cite_ref-Kiss_58-0">a</a> <a href="#cite_ref-Kiss_58-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKissSándor,_J.2004" class="citation journal cs1">Kiss, E.; Sándor, J. 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Mathematica Pannonica. 15 (2): 283–288.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematica+Pannonica&rft.atitle=On+a+congruence+by+J%C3%A1nos+Bolyai%2C+connected+with+pseudoprimes&rft.volume=15&rft.issue=2&rft.pages=283-288&rft.date=2004&rft.aulast=Kiss&rft.aufirst=E.&rft.au=S%C3%A1ndor%2C+J.&rft_id=http%3A%2F%2Fttk.pte.hu%2Fmii%2Fhtml%2Fpannonica%2Findex_elemei%2Fmp15-2%2Fmp15-2-283-288.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/xpage/FermatQuotient.html">Fermat Quotient</a> at The Prime Glossary </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150911144520/http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort">"Wieferich primes to base 1052"</a>. 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Retrieved 2019-09-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www1.uni-hamburg.de&rft.atitle=Fermat+quotients+q%3Csub%3Ep%3C%2Fsub%3E%28a%29+that+are+divisible+by+p&rft.date=2014-08-09&rft_id=http%3A%2F%2Fwww1.uni-hamburg.de%2FRRZ%2FW.Keller%2FFermatQuotient.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/FermatQuotienten/FermatQ3">"Wieferich primes with level ≥ 3"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Wieferich+primes+with+level+%E2%89%A5+3&rft_id=http%3A%2F%2Fwww.fermatquotient.com%2FFermatQuotienten%2FFermatQ3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.fermatquotient.com/FermatQuotienten/PQuotient_Sort.txt">"Solution of (a + 1)p−1 − ap−1 ≡ 0 (mod p2)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Solution+of+%3Cspan+class%3D%22texhtml+%22+%3E%28a+%2B+1%29%3Csup%3Ep%E2%88%921%3C%2Fsup%3E+%E2%88%92+a%3Csup%3Ep%E2%88%921%3C%2Fsup%3E+%E2%89%A1+0+%28mod+p%3Csup%3E2%3C%2Fsup%3E%29%3C%2Fspan%3E&rft_id=http%3A%2F%2Fwww.fermatquotient.com%2FFermatQuotienten%2FPQuotient_Sort.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <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/DoubleWieferichPrimePair.html">"Double Wieferich Prime Pair"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=Double+Wieferich+Prime+Pair&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDoubleWieferichPrimePair.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgohDilcherSkula1997" class="citation cs2">Agoh, T.; Dilcher, K.; Skula, L. (1997), "Fermat Quotients for Composite Moduli", Journal of Number Theory, 66 (1): 29–50, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.1997.2162">10.1006/jnth.1997.2162</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Fermat+Quotients+for+Composite+Moduli&rft.volume=66&rft.issue=1&rft.pages=29-50&rft.date=1997&rft_id=info%3Adoi%2F10.1006%2Fjnth.1997.2162&rft.aulast=Agoh&rft.aufirst=T.&rft.au=Dilcher%2C+K.&rft.au=Skula%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller2009" class="citation journal cs1 cs1-prop-foreign-lang-source">Müller, H. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hgYTAQAAMAAJ&q=21%2C+39%2C+55%2C+57%2C+105%2C+111%2C+147%2C+155%2C+165%2C+171%2C+183">"Über Periodenlängen und die Vermutungen von Collatz und Crandall"</a>. 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(2000), "Elliptic Wieferich Primes", Journal of Number Theory, 81 (2): 205–209, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.1999.2471">10.1006/jnth.1999.2471</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Number+Theory&rft.atitle=Elliptic+Wieferich+Primes&rft.volume=81&rft.issue=2&rft.pages=205-209&rft.date=2000&rft_id=info%3Adoi%2F10.1006%2Fjnth.1999.2471&rft.aulast=Voloch&rft.aufirst=J.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaussner1926" class="citation cs2 cs1-prop-foreign-lang-source">Haussner, R. (1926), <a rel="nofollow" class="external text" href="http://jfm.sub.uni-goettingen.de/cgi-bin/jfmen/JFM/en/quick.html?first=1&maxdocs=20&type=html&an=JFM%2052.0141.06&format=complete">"Über die Kongruenzen 2p−1 − 1 ≡ 0 (mod p2) für die Primzahlen p=1093 und 3511"</a>, Archiv for Mathematik og Naturvidenskab (in German), 39 (5): 7, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:52.0141.06">52.0141.06</a>, <a href="/wiki/German_National_Library" title="German National Library">DNB</a> <a rel="nofollow" class="external text" href="http://d-nb.info/363953469/about/html">363953469</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+for+Mathematik+og+Naturvidenskab&rft.atitle=%C3%9Cber+die+Kongruenzen+%3Cspan+class%3D%22texhtml+%22+%3E2%3Csup%3Ep%E2%88%921%3C%2Fsup%3E+%E2%88%92+1%3C%2Fspan%3E+%E2%89%A1+0+%28mod+p%3Csup%3E2%3C%2Fsup%3E%29+f%C3%BCr+die+Primzahlen+p%3D1093+und+3511&rft.volume=39&rft.issue=5&rft.pages=7&rft.date=1926&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A52.0141.06%23id-name%3DJFM&rft.aulast=Haussner&rft.aufirst=R.&rft_id=http%3A%2F%2Fjfm.sub.uni-goettingen.de%2Fcgi-bin%2Fjfmen%2FJFM%2Fen%2Fquick.html%3Ffirst%3D1%26maxdocs%3D20%26type%3Dhtml%26an%3DJFM%252052.0141.06%26format%3Dcomplete&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaussner1927" class="citation cs2 cs1-prop-foreign-lang-source">Haussner, R. (1927), <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002169924">"Über numerische Lösungen der Kongruenz up−1 − 1 ≡ 0 (mod p2)"</a>, Journal für die Reine und Angewandte Mathematik (in German), 1927 (156): 223–226, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1927.156.223">10.1515/crll.1927.156.223</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117969297">117969297</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=%C3%9Cber+numerische+L%C3%B6sungen+der+Kongruenz+%3Cspan+class%3D%22texhtml+%22+%3Eu%3Csup%3Ep%E2%88%921%3C%2Fsup%3E+%E2%88%92+1%3C%2Fspan%3E+%E2%89%A1+0+%28mod+p%3Csup%3E2%3C%2Fsup%3E%29&rft.volume=1927&rft.issue=156&rft.pages=223-226&rft.date=1927&rft_id=info%3Adoi%2F10.1515%2Fcrll.1927.156.223&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117969297%23id-name%3DS2CID&rft.aulast=Haussner&rft.aufirst=R.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN002169924&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1979" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, P.</a> (1979), Thirteen lectures on Fermat's Last Theorem, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. 139, 151, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90432-0" title="Special:BookSources/978-0-387-90432-0"><bdi>978-0-387-90432-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thirteen+lectures+on+Fermat%27s+Last+Theorem&rft.pages=139%2C+151&rft.pub=Springer-Verlag&rft.date=1979&rft.isbn=978-0-387-90432-0&rft.aulast=Ribenboim&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation cs2"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004), Unsolved Problems in Number Theory (3rd ed.), <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>, p. 14, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20860-2" title="Special:BookSources/978-0-387-20860-2"><bdi>978-0-387-20860-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.pages=14&rft.edition=3rd&rft.pub=Springer+Verlag&rft.date=2004&rft.isbn=978-0-387-20860-2&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrandallPomerance2005" class="citation cs2">Crandall, R. E.; Pomerance, C. (2005), <a rel="nofollow" class="external text" href="http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf">Prime numbers: a computational perspective</a> (PDF), Springer Science+Business Media, pp. 31–32, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-25282-7" title="Special:BookSources/978-0-387-25282-7"><bdi>978-0-387-25282-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+numbers%3A+a+computational+perspective&rft.pages=31-32&rft.pub=Springer+Science%2BBusiness+Media&rft.date=2005&rft.isbn=978-0-387-25282-7&rft.aulast=Crandall&rft.aufirst=R.+E.&rft.au=Pomerance%2C+C.&rft_id=http%3A%2F%2Fthales.doa.fmph.uniba.sk%2Fmacaj%2Fskola%2Fteoriapoli%2Fprimes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation cs2">Ribenboim, P. (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=72eg8bFw40kC&pg=PA333">The new book of prime number records</a>, New York: Springer-Verlag, pp. 333–346, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94457-9" title="Special:BookSources/978-0-387-94457-9"><bdi>978-0-387-94457-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+new+book+of+prime+number+records&rft.place=New+York&rft.pages=333-346&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=978-0-387-94457-9&rft.aulast=Ribenboim&rft.aufirst=P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D72eg8bFw40kC%26pg%3DPA333&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><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/WieferichPrime.html">"Wieferich prime"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=Wieferich+prime&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FWieferichPrime.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWieferich+prime" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://go.helms-net.de/math/expdioph/fermatquotients.pdf">Fermat-/Euler-quotients (ap−1 − 1)/pk with arbitrary k</a></li> <li><a rel="nofollow" class="external text" href="http://library.uwinnipeg.ca/people/dobson/mathematics/Wieferich_primes.html">A note on the two known Wieferich primes</a></li> <li>PrimeGrid's <a rel="nofollow" class="external text" href="https://www.primegrid.com/forum_thread.php?id=9436">Wieferich Prime Search project</a> page</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist 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.navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Prime_number_classes" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fermat_number" title="Fermat number">Fermat (22n + 1)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (2p − 1)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (22p−1 − 1)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff (2p + 1)/3</a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (k·2n + 1)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (n! ± 1)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (pn# ± 1)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (pn# + 1)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (4n + 1)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (2m·3n + 1)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (x4 + y4)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (2m ± 2n ± 1)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (n·2n + 1)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (n·2n − 1)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (x3 − y3)/(x − y)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (xy + yx)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (3·2n − 1)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams ((b−1)·bn − 1)</a></li> <li><a href="/wiki/Mills%27_constant" title="Mills' constant">Mills (⌊A3n⌋)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</a>)</li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit (10n − 1)/9</a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple">k-tuples</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twin_prime" title="Twin prime">Twin (p, p + 2)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (p, p + 2 or p + 4, p + 6)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (p, p + 2, p + 6, p + 8)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (p, p + 4)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (p, p + 6)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (consecutive p − n, p, p + n)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)</a></li> <li><a href="/wiki/Chen_prime" title="Chen prime">Chen</a></li> <li><a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">Sophie Germain/Safe (p, 2p + 1)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By size</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Megaprime" title="Megaprime">Mega (1,000,000+ digits)</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known</a> <ul><li><a href="/wiki/List_of_largest_known_primes_and_probable_primes" title="List of largest known primes and probable primes">list</a></li></ul></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Gaussian_integer#Gaussian_primes" title="Gaussian integer">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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