View source for Wieferich prime

{{Short description|Prime such that p^2 divides 2^(p-1)-1}}
{{pp-semi-indef|small=yes}}
{{Good article}}
{{Infobox integer sequence
| named_after           = [[Arthur Wieferich]]
| publication_year      = 1909
| author                = [[Arthur Wieferich|Wieferich, A.]]
| terms_number          = 2
| con_number            = Infinite
| parentsequence        = {{plainlist|1=
*Crandall numbers<ref>{{Citation |last1= Franco |first1= Z. |last2= Pomerance |first2= C. |author2-link = Carl Pomerance |title= On a conjecture of Crandall concerning the ''qx'' + 1 problem |journal= Mathematics of Computation |volume= 64 |issue=211 |pages= 1333–36 |year=1995 |url= http://www.math.dartmouth.edu/~carlp/PDF/paper101.pdf |doi= 10.2307/2153499 |postscript=. |jstor= 2153499 |bibcode= 1995MaCom..64.1333F }}</ref>
*[[Wieferich prime#Wieferich numbers|Wieferich numbers]]<ref name="Banks, Luca">{{Citation |last1= Banks |first1= W.D. |last2= Luca |first2= F. |last3= Shparlinski |first3= I.E. |title= Estimates for Wieferich numbers |journal= The Ramanujan Journal |volume= 14 |issue= 3 |pages= 361–378 |year= 2007 |url= http://web.science.mq.edu.au/~igor/Wieferich.pdf |doi= 10.1007/s11139-007-9030-z |s2cid= 39279379 |postscript= . |access-date= 2011-03-12 |archive-date= 2013-05-03 |archive-url= https://web.archive.org/web/20130503091812/http://web.science.mq.edu.au/~igor/Wieferich.pdf |url-status= dead }}</ref>
*[[Wieferich prime#Lucas-Wieferich primes|Lucas–Wieferich primes]]<ref name="McIntosh, 2007">{{Citation |last1= McIntosh |first1= R.J. |last2= Roettger |first2= E.L. |title= A search for Fibonacci–Wieferich and Wolstenholme primes |journal= Mathematics of Computation |volume= 76 |issue= 260 |pages= 2087–2094 |year= 2007 |url= http://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01955-2/S0025-5718-07-01955-2.pdf |citeseerx= 10.1.1.105.9393 |doi= 10.1090/S0025-5718-07-01955-2|bibcode= 2007MaCom..76.2087M }}</ref>
*[[Wieferich prime#Near-Wieferich primes|near-Wieferich primes]]
}}
| first_terms           = [[1093 (number)|1093]], [[3511 (number)|3511]]
| largest_known_term    = [[3511 (number)|3511]]
| OEIS                  = A001220
}}
In [[number theory]], a '''Wieferich prime''' is a [[prime number]] ''p'' such that ''p''<sup>2</sup> divides {{math|1=2<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1}},<ref name="The Prime Glossary">{{Citation |url= http://primes.utm.edu/glossary/xpage/WieferichPrime.html |title=The Prime Glossary: Wieferich prime }}</ref> therefore connecting these primes with [[Fermat's little theorem]], which states that every odd prime ''p'' divides {{math|1=2<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1}}. Wieferich primes were first described by [[Arthur Wieferich]] in 1909 in works pertaining to [[Fermat's Last Theorem]], at which time both of Fermat's theorems were already well known to mathematicians.<ref name=Kleiner>{{Citation |author=Israel Kleiner |title= From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem |journal= [[Elemente der Mathematik]] |volume=55 |year=2000 |page=21 |doi= 10.1007/PL00000079 |s2cid= 53319514 |postscript=. |doi-access= free }}</ref><ref name=Euler>{{Citation |author= Leonhard Euler |title= Theorematum quorundam ad numeros primos spectantium demonstratio |journal= Novi Comm. Acad. Sci. Petropol. |volume=8 |year=1736 |language=la |pages=33–37 |url= http://www.math.dartmouth.edu/~euler/docs/translations/E054tr.pdf|postscript=.}}</ref>

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as [[Mersenne number|Mersenne]] and [[Fermat number|Fermat]] numbers, specific types of [[pseudoprime]]s 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 [[number field]]s and the [[abc conjecture|''abc'' conjecture]].

{{Asof|2024}}, the only known Wieferich primes are 1093 and 3511 {{OEIS|id=A001220}}.

== Equivalent definitions ==
The stronger version of [[Fermat's little theorem]], which a Wieferich prime satisfies, is usually expressed as a [[congruence relation]] {{math|2<sup>''p'' -1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}. From the definition of the [[Modular arithmetic#Congruence|congruence relation on integers]], 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 [[Fermat quotient]] <math>\tfrac{2^{p-1}-1}{p}</math>. The following are two illustrative examples using the primes 11 and 1093:
: For ''p'' = 11, we get <math>\tfrac{2^{10}-1}{11}</math> which is 93 and leaves a [[remainder]] of 5 after division by 11, hence 11 is not a Wieferich prime. For ''p'' = 1093, we get <math>\tfrac{2^{1092}-1}{1093}</math> 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.

Wieferich primes can be defined by other equivalent congruences. If ''p'' is a Wieferich prime, one can multiply both sides of the congruence {{math|1=2<sup>''p''−1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)}} by 2 to get {{math|1=2<sup>''p''</sup>&nbsp;≡&nbsp;2&nbsp;(mod&nbsp;''p''<sup>2</sup>)}}. Raising both sides of the congruence to the power ''p'' shows that a Wieferich prime also satisfies {{math|1=2<sup>''p''<sup>2</sup></sup>&nbsp;≡2<sup>''p''</sup>&nbsp;≡&nbsp;2&nbsp;(mod&nbsp;''p''<sup>2</sup>)}}, and hence {{math|1=2<sup>''p''<sup>k</sup></sup>&nbsp;≡&nbsp;2&nbsp;(mod&nbsp;''p''<sup>2</sup>)}} for all {{math|1=''k''&nbsp;≥&nbsp;1}}. The converse is also true: {{math|1=2<sup>''p''<sup>k</sup></sup>&nbsp;≡&nbsp;2&nbsp;(mod&nbsp;''p''<sup>2</sup>)}} for some {{math|1=''k''&nbsp;≥&nbsp;1}} implies that the [[multiplicative order]] of 2 modulo ''p''<sup>2</sup> divides [[Greatest common divisor|gcd]]{{math|1=(''p''<sup>k</sup>&nbsp;−&nbsp;1}}, [[Euler's totient function|φ]]{{math|1=(''p''<sup>2</sup>))&nbsp;=&nbsp;''p''&nbsp;−&nbsp;1}}, that is, {{math|1=2<sup>''p''−1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)}} 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 ''p''<sup>2</sup> coincide: {{math|1=ord<sub>''p''<sup>2</sup></sub> 2 = ord<sub>''p''</sub> 2}}, (By the way, ord<sub>1093</sub>2 = 364, and ord<sub>3511</sub>2 = 1755).

[[Harry Vandiver|H. S. Vandiver]] proved that {{math|1=2<sup>''p''−1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>3</sup>)}} if and only if <math>1 + \tfrac{1}{3} + \dots + \tfrac{1}{p-2} \equiv 0 \pmod{p^2}</math>.<ref>{{citation | first=L. E. | last=Dickson | title=Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers | journal=Annals of Mathematics | volume=18 | issue=4 | year=1917 | pages=161–187| jstor=2007234 | doi=10.2307/2007234}}</ref>{{rp|187}}

== History and search status ==
{{Unsolved|mathematics|Are there infinitely many Wieferich primes?}}In 1902, [[Wilhelm Franz Meyer|Meyer]] proved a theorem about solutions of the congruence ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>''r''</sup>).<ref name="Solutions">{{Citation |author=Wilfrid Keller |author2=Jörg Richstein |title=Solutions of the congruence {{math|1=''a''<sup>''p''−1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>''r''</sup>)}} |journal=Mathematics of Computation |volume=74 |year=2005 |pages=927–936 |doi=10.1090/S0025-5718-04-01666-7 |url=http://www.ams.org/journals/mcom/2005-74-250/S0025-5718-04-01666-7/S0025-5718-04-01666-7.pdf |postscript=. |issue=250|doi-access=free }}</ref>{{rp|930}}<ref>{{cite journal |last=Meyer |first=W. Fr. |author-link=Wilhelm Franz Meyer |date=1902 |title=Ergänzungen zum Fermatschen und Wilsonschen Satze |url=https://archive.org/details/bub_gb_zaMKAAAAIAAJ/page/n540/mode/2up |journal=Arch. Math. Physik |series=3 |volume=2 |pages=141–146 |access-date=2020-09-02}}</ref> Later in that decade [[Arthur Wieferich]] showed specifically that if the [[first case of Fermat's last theorem]] has solutions for an odd prime exponent, then that prime must satisfy that congruence for ''a'' = 2 and ''r'' = 2.<ref name="Wieferich, Arthur">{{Citation | first=A. |last=Wieferich | author-link=Arthur Wieferich | title=Zum letzten Fermat'schen Theorem | journal=[[Journal für die reine und angewandte Mathematik]] | volume=1909 | year=1909 | language=de | pages=293–302 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002166968 | doi=10.1515/crll.1909.136.293 | issue=136 |s2cid=118715277 | postscript=.}}</ref> In other words, if there exist solutions to ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> = 0 in integers ''x'', ''y'', ''z'' and ''p'' an [[odd prime]] with ''p'' [[List of mathematical symbols#∤|∤]] ''xyz'', then ''p'' satisfies 2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>). In 1913, [[Paul Gustav Heinrich Bachmann|Bachmann]] examined the [[Remainder|residues]] of <math>\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math>. He asked the question when this residue [[Zero of a function|vanishes]] and tried to find expressions for answering this question.<ref>{{cite journal | last = Bachmann | first = P. | author-link = Paul Gustav Heinrich Bachmann | title = Über den Rest von <math>\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math> | journal = Journal für Mathematik | volume = 142 | issue = 1 | pages = 41–50 | language=de | year = 1913 | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167646}}</ref>

The prime 1093 was found to be a Wieferich prime by {{ill|Waldemar Meissner|cs|lt=W. Meissner}} in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of <math>\tfrac{2^{t}-1}{p}\,\bmod\,p</math> for all primes ''p'' < 2000 and found this residue to be zero for ''t'' = 364 and ''p'' = 1093, thereby providing a counterexample to a [[conjecture]] by [[Dmitry Grave|Grave]] about the impossibility of the Wieferich congruence.<ref name="Meissner">{{citation | first=W. |last=Meissner |title=Über die Teilbarkeit von 2<sup>''p''</sup> − 2 durch das Quadrat der Primzahl ''p''=1093 | journal=Sitzungsber. D. Königl. Preuss. Akad. D. Wiss. | volume=Zweiter Halbband. Juli bis Dezember | place=Berlin | language=de | year=1913 | pages=663–667 | url=https://oeis.org/A001917/a001917.pdf | jfm=44.0218.02}}</ref> {{ill|Emil Haentzschel|de|lt=E. Haentzschel}} later ordered verification of the correctness of Meissner's congruence via only elementary calculations.<ref>{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2<sup>1092</sup> ≡ 1 (mod 1093<sup>2</sup>) | journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume=25 | year=1916 | language=de | page=284 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00212534X}}</ref>{{rp|664}} Inspired by an earlier work of [[Leonhard Euler|Euler]], he simplified Meissner's proof by showing that 1093<sup>2</sup> | (2<sup>182</sup> + 1) and remarked that (2<sup>182</sup> + 1) is a factor of (2<sup>364</sup> − 1).<ref>{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2<sup>1092</sup> ≡ 1 (mod 1093<sup>2</sup>) | journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume=34 | year=1925 | language=de | page=184 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002127695}}</ref> It was also shown that it is possible to prove that 1093 is a Wieferich prime without using [[complex number]]s contrary to the method used by Meissner,<ref>{{citation | first=P. | last=Ribenboim | author-link=Paulo Ribenboim | title=1093 | journal=The Mathematical Intelligencer | volume=5 | issue=2 | year=1983 | pages=28–34 | doi=10.1007/BF03023623}}</ref> although Meissner himself hinted at that he was aware of a proof without complex values.<ref name="Meissner" />{{rp|665}}

The prime [[3511 (number)|3511]] was first found to be a Wieferich prime by [[N. G. W. H. Beeger]] in 1922<ref>{{citation | first=N. G. W. H. |last=Beeger | author-link=N. G. W. H. Beeger | title=On a new case of the congruence 2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>) | journal=[[Messenger of Mathematics]] | volume=51 | year=1922 | pages=149–150 |url=https://archive.org/stream/messengerofmathe5051cambuoft#page/148/mode/2up}}</ref> and another proof of it being a Wieferich prime was published in 1965 by [[Richard K. Guy|Guy]].<ref>{{citation | first=R. K. | last=Guy | author-link=Richard K. Guy | title=A property of the prime 3511 | journal=The Mathematical Gazette | volume=49 | issue=367 | year=1965 | pages=78–79 | jstor=3614249 | doi=10.2307/3614249}}</ref> In 1960, Kravitz<ref>{{cite journal |author=Kravitz, S. |url=http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0121334-7/S0025-5718-1960-0121334-7.pdf |title=The Congruence 2<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>) for ''p'' < 100,000 |journal=Mathematics of Computation |volume=14 |issue=72 |year=1960 |pages=378 |doi=10.1090/S0025-5718-1960-0121334-7|doi-access=free }}</ref> doubled a previous record set by {{ill|Carl-Erik Fröberg|sv|lt=Fröberg}}<ref>{{cite journal |author=Fröberg C. E. |url=http://www.ams.org/journals/mcom/1958-12-064/S0025-5718-58-99270-6/S0025-5718-58-99270-6.pdf |title=Some Computations of Wilson and Fermat Remainders |journal=Mathematics of Computation |volume=12 |issue=64 |year=1958 |pages=281 |doi=10.1090/S0025-5718-58-99270-6 |doi-access=free }}</ref> and in 1961 [[Hans Riesel|Riesel]] extended the search to 500000 with the aid of [[BESK]].<ref>{{cite journal |author=Riesel, H.|author-link=Hans Riesel  |url=http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0157928-6/S0025-5718-1964-0157928-6.pdf |title=Note on the Congruence ''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>) |journal=Mathematics of Computation |volume=18 |issue=85 |year=1964 |pages=149–150 |doi=10.1090/S0025-5718-1964-0157928-6|doi-access=free }}</ref> Around 1980, [[Derrick Henry Lehmer|Lehmer]] was able to reach the search limit of 6{{e|9}}.<ref>{{cite journal | last = Lehmer  | first = D. H. | author-link = Derrick Henry Lehmer | title = On Fermat's quotient, base two | journal = Mathematics of Computation | volume = 36 | issue = 153 | pages = 289–290 | url = http://www.ams.org/journals/mcom/1981-36-153/S0025-5718-1981-0595064-5/S0025-5718-1981-0595064-5.pdf |doi=10.1090/S0025-5718-1981-0595064-5| year = 1981 | doi-access = free }}</ref> This limit was extended to over 2.5{{e|15}} in 2006,<ref name="Ribenboim, 2004" /> finally reaching 3{{e|15}}. Eventually, it was shown that if any other Wieferich primes exist, they must be greater than 6.7{{e|15}}.<ref name=Dorais>{{cite journal | last = Dorais | first = F. G. |author2=Klyve, D.  | title = A Wieferich Prime Search Up to 6.7{{e|15}} | journal = Journal of Integer Sequences | volume = 14 | issue = 9 | year = 2011 | url = http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.pdf | zbl = 1278.11003 | access-date = 2011-10-23}}</ref>

In 2007–2016, a search for Wieferich primes was performed by the [[distributed computing]] project Wieferich@Home.<ref>{{cite web | title=statistics | website=elMath.org | date=2016-09-02 | url=http://www.elmath.org/index.php?id=statistics | archive-url=https://web.archive.org/web/20160902123101/http://www.elmath.org/index.php?id=statistics | archive-date=2016-09-02 | url-status=dead | access-date=2019-09-18}}</ref> In 2011–2017, another search was performed by the [[PrimeGrid]] project, although later the work done in this project was claimed wasted.<ref>{{cite web | url=http://www.primegrid.com/forum_thread.php?id=7436 | title=WSS and WFS are suspended | website=PrimeGrid Message Board|date=May 11, 2017}}</ref> While these projects reached search bounds above 1{{e|17}}, neither of them reported any sustainable results.

In 2020, PrimeGrid started another project that searched for Wieferich and [[Wall–Sun–Sun prime]]s 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.<ref>{{cite web|title=Message boards : Wieferich and Wall-Sun-Sun Prime Search|url=https://www.primegrid.com/forum_forum.php?id=127|website=PrimeGrid}}</ref> The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 2<sup>64</sup> (about 18{{e|18}}).<ref>{{cite web|title=WW Statistics|url=https://www.primegrid.com/stats_ww.php|website=PrimeGrid}}</ref>

It has been conjectured (as for [[Wilson prime]]s) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below ''x'' is approximately log(log(''x'')), which is a [[heuristic argument|heuristic result]] that follows from the plausible assumption that for a prime ''p'', the {{math|1=(''p''&nbsp;−&nbsp;1)-th}} degree [[roots of unity]] modulo ''p''<sup>2</sup> are [[uniform distribution (discrete)|uniformly distributed]] in the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''p''<sup>2</sup>]].<ref name=Crandall />

== Properties ==

=== Connection with Fermat's Last Theorem ===

The following theorem connecting Wieferich primes and [[Fermat's Last Theorem]] was proven by Wieferich in 1909:<ref name="Wieferich, Arthur" />

:Let ''p'' be prime, and let ''x'', ''y'', ''z'' be [[integer]]s such that {{math|1=''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>&nbsp;+&nbsp;''z''<sup>''p''</sup>&nbsp;=&nbsp;0}}. Furthermore, assume that ''p'' does not divide the [[product (mathematics)|product]]&nbsp;''xyz''. Then ''p'' is a Wieferich prime.

The above case (where ''p'' does not divide any of ''x'', ''y'' or ''z'') is commonly known as the [[first case of Fermat's Last Theorem]] (FLTI)<ref>{{Citation | last=Coppersmith | first=D. | author-link=Don Coppersmith | title=Fermat's Last Theorem (Case I) and the Wieferich Criterion | journal=Mathematics of Computation  | volume=54 | issue=190  | pages=895–902 | year=1990  | jstor=2008518 |url=http://www.ams.org/journals/mcom/1990-54-190/S0025-5718-1990-1010598-2/S0025-5718-1990-1010598-2.pdf | postscript=. | doi=10.1090/s0025-5718-1990-1010598-2| bibcode=1990MaCom..54..895C | doi-access=free }}</ref><ref>{{Citation | last=Cikánek | first=P. | title=A Special Extension of Wieferich's Criterion  | journal=Mathematics of Computation  | volume=62  | issue=206 | pages=923–930 | jstor=3562296 | year=1994 | url=http://www.ams.org/journals/mcom/1994-62-206/S0025-5718-1994-1216257-9/S0025-5718-1994-1216257-9.pdf | postscript=. | doi=10.2307/2153550| bibcode=1994MaCom..62..923C }}</ref> 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''.<ref name="Dilcher, Skula">{{Citation | last1=Dilcher | first1=K. | last2=Skula | first2=L. | title=A new criterion for the first case of Fermat's last theorem | journal=Mathematics of Computation | volume=64 | issue=209 | pages=363–392 | jstor=2153341 | year=1995 | url=http://www.ams.org/journals/mcom/1995-64-209/S0025-5718-1995-1248969-6/S0025-5718-1995-1248969-6.pdf | doi=10.1090/s0025-5718-1995-1248969-6| bibcode=1995MaCom..64..363D | doi-access=free }}</ref>
In 1910, [[Mirimanoff]] expanded<ref>{{Citation | first=D. |last=Mirimanoff |title=Sur le dernier théorème de Fermat | language=fr | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences | volume=150 | year=1910 | pages=204–206 | postscript=. }}</ref> the theorem by showing that, if the preconditions of the theorem hold true for some prime ''p'', then ''p''<sup>2</sup> must also divide {{math|1=3<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1}}. Granville and Monagan further proved that ''p''<sup>2</sup> must actually divide {{math|1=''m''<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1}} for every prime ''m'' ≤ 89.<ref name="Granville, Monagan">{{Citation | first1=A. | last1=Granville | first2=M. B. | last2=Monagan | title=The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389 | journal=Transactions of the American Mathematical Society | volume=306 | issue=1 | year=1988 | pages=329–359 | doi=10.1090/S0002-9947-1988-0927694-5 | postscript=.| doi-access=free }}</ref> Suzuki extended the proof to all primes ''m'' ≤ 113.<ref>{{citation | first=Jiro |last=Suzuki | title=On the generalized Wieferich criteria | journal=Proceedings of the Japan Academy, Series A| volume=70 |issue=7 | pages=230–234 | year=1994 | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195510946 | doi=10.3792/pjaa.70.230| doi-access=free }}</ref>

Let ''H<sub>p</sub>'' be a set of pairs of integers with 1 as their [[greatest common divisor]], ''p'' being prime to ''x'', ''y'' and ''x'' + ''y'', (''x'' + ''y'')<sup>''p''−1</sup> ≡ 1 (mod p<sup>2</sup>),  (''x'' + ''ξy'') being the ''p''th power of an [[ideal (ring theory)|ideal]] of ''K'' with ''ξ'' defined as cos 2''π''/''p'' + ''i'' sin 2''π''/''p''. ''K'' = '''Q'''(''ξ'') is the [[field extension]] obtained by adjoining all [[polynomial]]s in the [[algebraic number]] ''ξ'' to the [[Field (mathematics)|field]] of [[rational number]]s (such an extension is known as a [[Algebraic number field|number field]] or in this particular case, where ''ξ'' is a [[root of unity]], a [[cyclotomic field|cyclotomic number field]]).<ref name="Granville, Monagan" />{{rp|332}}
From [[Fundamental theorem of ideal theory in number fields|uniqueness of factorization of ideals in '''Q'''(ξ)]] 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 ''H<sub>p</sub>''.<ref name="Granville, Monagan" />{{rp|333}}
Granville and Monagan showed that (1, 1) ∈ ''H<sub>p</sub>'' if and only if ''p'' is a Wieferich prime.<ref name="Granville, Monagan" />{{rp|333}}

=== Connection with the ''abc'' conjecture and non-Wieferich primes ===

A non-Wieferich prime is a prime ''p'' satisfying {{math|1=2<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;≢&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)}}. [[Joseph H. Silverman|J. H. Silverman]] showed in 1988 that if the [[abc conjecture|''abc'' conjecture]] holds, then there exist infinitely many non-Wieferich primes.<ref>{{cite web | author=Charles, D. X. | url=http://pages.cs.wisc.edu/~cdx/Criterion.pdf | title=On Wieferich primes |website=wisc.edu}}</ref> 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.<ref>{{citation | first=J. H. |last=Silverman | author-link=Joseph H. Silverman | title=Wieferich's criterion and the abc-conjecture | journal=[[Journal of Number Theory]] | volume=30 | issue=2 | year=1988 | pages=226–237 | doi=10.1016/0022-314X(88)90019-4 | doi-access=free }}</ref>{{rp|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 ''W<sub>2</sub>'' and ''W<sub>2</sub><sup>c</sup>'' respectively,<ref name=DeKoninckDoyon>{{citation | first1=J.-M. | last1=DeKoninck | first2=N. | last2=Doyon | title=On the set of Wieferich primes and of its complement | journal=Annales Univ. Sci. Budapest., Sect. Comp. | volume=27 | year=2007 | pages=3–13 | url=http://ac.inf.elte.hu/Vol_027_2007/003.pdf}}</ref> are [[Complement (set theory)|complementary sets]], 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''.<ref>{{citation | first=K. | last=Broughan | title=Relaxations of the ABC Conjecture using integer ''k''<nowiki>'</nowiki>th roots | journal=New Zealand J. Math. | volume=35 | issue=2 | year=2006 | pages=121–136 | url=http://www.math.waikato.ac.nz/~kab/papers/abc01.pdf}}</ref> Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers<ref>{{cite book | last = Ribenboim | first = P. | author-link = Paulo Ribenboim | title = 13 Lectures on Fermat's Last Theorem
 | publisher = Springer | year = 1979 | location = New York | page = 154 | url = https://books.google.com/books?id=w2o67-lBGhIC&q=%22Let+me+add+that+Schinzel+has+conjectured%22&pg=PA154 | isbn = 978-0-387-90432-0}}</ref> as well as if there exists a real number ''ξ'' such that the set {''n'' ∈ '''N''' : λ(2<sup>''n''</sup> − 1) < 2 − ''ξ''} is of [[Natural density|density]] one, where the ''index of composition'' ''λ''(''n'') of an integer ''n'' is defined as <math>\tfrac{\log n}{\log \gamma (n)}</math> and <math style="vertical-align:-65%;">\gamma (n) = \prod_{p \mid n} p</math>, meaning <math>\gamma (n)</math> gives the product of all [[prime factor]]s of ''n''.<ref name=DeKoninckDoyon />{{rp|4}}

=== Connection with Mersenne and Fermat primes ===

It is known that the ''n''th [[Mersenne number]] {{math|1=''M''<sub>''n''</sub>&nbsp;=&nbsp;2<sup>''n''</sup>&nbsp;−&nbsp;1}} is prime only if ''n'' is prime. [[Fermat's little theorem]] implies that if {{math|1=''p''&nbsp;>&nbsp;2}} is prime, then ''M''<sub>''p''−1</sub> {{math|1=(=&nbsp;2<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1)}} is always divisible by ''p''. Since Mersenne numbers of prime indices  ''M''<sub>''p''</sub> and ''M''<sub>''q''</sub> are co-prime,
::A prime divisor ''p'' of ''M''<sub>''q''</sub>, where ''q'' is prime, is a Wieferich prime if and only if ''p''<sup>2</sup> divides ''M''<sub>''q''</sub>.<ref>{{Citation |url=http://primes.utm.edu/mersenne/index.html#unknown |title=Mersenne Primes: Conjectures and Unsolved Problems }}</ref>
Thus, a Mersenne prime cannot also be a Wieferich prime. A notable [[Unsolved problems in mathematics|open problem]] is to determine whether or not all Mersenne numbers of prime index are [[Square-free integer|square-free]]. If ''q'' is prime and the Mersenne number ''M''<sub>''q''</sub> is ''not'' square-free, that is, there exists a prime ''p'' for which ''p''<sup>2</sup> divides ''M''<sub>''q''</sub>, 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.<ref>{{cite journal | last = Rotkiewicz | first = A. | title = Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturels n, tels que {{math|1=''n''<sup>2</sup>{{!}}2<sup>n</sup>&nbsp;−&nbsp;2}} | journal = Mat. Vesnik | volume = 2 | issue = 17 | pages = 78–80 | year = 1965 | language = French | url = http://resolver.sub.uni-goettingen.de/purl?PPN311571026_0017}}</ref>

Similarly, if ''p'' is prime and ''p''<sup>2</sup> divides some [[Fermat number]] ''F''<sub>''n''</sub> {{math|1==&nbsp;2<sup>2<sup>''n''</sup></sup>&nbsp;+&nbsp;1}}, then ''p'' must be a Wieferich prime.<ref>{{Citation |author-link=Paulo Ribenboim |last=Ribenboim |first=Paulo |title=The little book of big primes |place=New York |publisher=Springer |year=1991 |page=64 |isbn=978-0-387-97508-5 |url=https://books.google.com/books?id=zUCK7FT4xgAC&pg=PA64 }}</ref>

In fact, there exists a natural number ''n'' and a prime ''p'' that ''p''<sup>2</sup> divides <math>\Phi_n(2)</math> (where <math>\Phi_n(x)</math> is the ''n''-th [[cyclotomic polynomial]]) [[if and only if]] ''p'' is a Wieferich prime. For example, 1093<sup>2</sup> divides <math>\Phi_{364}(2)</math>, 3511<sup>2</sup> divides <math>\Phi_{1755}(2)</math>. Mersenne and Fermat numbers are just special situations of <math>\Phi_n(2)</math>. Thus, if 1093 and 3511 are only two Wieferich primes, then all <math>\Phi_n(2)</math> are [[square-free number|square-free]] except <math>\Phi_{364}(2)</math> and <math>\Phi_{1755}(2)</math> (In fact, when there exists a prime ''p'' which ''p''<sup>2</sup> divides some <math>\Phi_n(2)</math>, then it is a Wieferich prime); and clearly, if <math>\Phi_n(2)</math> is a prime, then it cannot be Wieferich prime. (Any odd prime ''p'' divides only one <math>\Phi_n(2)</math> and ''n'' divides {{math|1=''p''&nbsp;−&nbsp;1}}, and if and only if the period length of 1/p in [[Binary number|binary]] is ''n'', then ''p'' divides <math>\Phi_n(2)</math>. Besides, if and only if ''p'' is a Wieferich prime, then the period length of 1/p and 1/p<sup>2</sup> 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.<ref>{{citation | first1=H. G. | last1=Bray | first2=L. J. | last2=Warren | title=On the square-freeness of Fermat and Mersenne numbers | journal=Pacific J. Math. | volume=22 | issue=3 | year=1967 | pages=563–564 | url=http://projecteuclid.org/euclid.pjm/1102992105 | mr=0220666 | zbl=0149.28204 | doi=10.2140/pjm.1967.22.563| doi-access=free }}</ref>

=== Connection with other equations ===

Scott and Styer showed that the equation ''p''<sup>x</sup> – 2<sup>y</sup> = ''d'' has at most one solution in positive integers (''x'', ''y''), unless when ''p''<sup>4</sup> | 2<sup>ord<sub>''p''</sub> 2</sup> – 1 if ''p'' ≢ 65 (mod 192) or unconditionally when ''p''<sup>2</sup> | 2<sup>ord<sub>''p''</sub> 2</sup> – 1, where ord<sub>''p''</sub> 2 denotes the [[multiplicative order]] of 2 modulo ''p''.<ref>{{cite journal | last = Scott | first = R. |author2=Styer, R.  | title = On {{math|1=''p''<sup>''x''</sup>&nbsp;−&nbsp;''q''<sup>''y''</sup>&nbsp;=&nbsp;''c''}} and related three term exponential Diophantine equations with prime bases | journal = Journal of Number Theory | volume = 105 | issue = 2 | pages = 212–234 | date = April 2004 | doi = 10.1016/j.jnt.2003.11.008| doi-access =  }}</ref>{{rp|215, 217–218}} They also showed that a solution to the equation ±''a''<sup>''x''<sub>1</sub></sup> ± 2<sup>''y''<sub>1</sub></sup> = ±''a''<sup>''x''<sub>2</sub></sup> ± 2<sup>''y''<sub>2</sub></sup> = ''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 10<sup>15</sup>.<ref>{{cite journal | last = Scott | first = R. |author2=Styer, R.  | title = On the generalized Pillai equation ±''a''<sup>''x''</sup>±''b''<sup>''y''</sup> = ''c'' | journal = Journal of Number Theory | volume = 118 | issue = 2 | pages = 236–265 | year = 2006 | doi = 10.1016/j.jnt.2005.09.001| doi-access = free }}</ref>{{rp|258}}

=== Binary periodicity of ''p'' − 1 ===

Johnson observed<ref name=John>{{Citation |author=Wells Johnson |title=On the nonvanishing of Fermat quotients {{math|1=(mod&nbsp;''p'')}} |journal=J. Reine Angew. Math. |volume=292 |year=1977 |pages=196–200 |url=http://www.digizeitschriften.de/index.php?id=resolveppn&PPN=GDZPPN002193698}}</ref> that the two known Wieferich primes are one greater than numbers with periodic [[binary expansion]]s (1092 = 010001000100<sub>2</sub>=444<sub>16</sub>; 3510 = 110110110110<sub>2</sub>=6666<sub>8</sub>). 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.<ref name=DoKu>{{Cite journal |last1=Dobeš |first1=Jan |last2=Kureš |first2=Miroslav |title=Search for Wieferich primes through the use of periodic binary strings |journal=Serdica Journal of Computing |volume=4 |year=2010 |issue=3 |pages=293–300 |doi=10.55630/sjc.2010.4.293-300 |zbl=1246.11019 |url=https://www.researchgate.net/publication/267166566 |hdl=10525/1595 |hdl-access=free }}</ref>

=== Abundancy of ''p'' − 1 ===

It has been noted {{OEIS|id=A239875}} that the known Wieferich primes are one greater than mutually [[friendly number]]s (the shared abundancy index being 112/39).

=== Connection with pseudoprimes ===

It was observed that the two known Wieferich primes are the square factors of all [[Square-free integer|non-square free]] base-2 [[Fermat pseudoprime]]s up to 25{{e|9}}.<ref>{{Cite book | author1-link=Paulo Ribenboim | last1=Ribenboim | first1=P. | title=The Little Book of Bigger Primes | location=New York | publisher=Springer-Verlag | year=2004 | isbn=978-0-387-20169-6 | page=99 |chapter-url=http://trex58.files.wordpress.com/2010/01/ribenboimbook1.pdf |chapter=Chapter 2. How to Recognize Whether a Natural Number is a Prime}}</ref> Later computations showed that the only repeated factors of the pseudoprimes up to 10<sup>12</sup> are 1093 and 3511.<ref>{{Cite book | last = Pinch | first = R. G. E. | title = The Pseudoprimes up to 10<sup>13</sup> | volume = 1838 | pages = 459–473 | year = 2000  | doi = 10.1007/10722028_30 | series = Lecture Notes in Computer Science | isbn = 978-3-540-67695-9 }}</ref> In addition, the following connection exists:
:Let ''n'' be a base 2 pseudoprime and ''p'' be a prime divisor of ''n''. If <math>\tfrac{2^{n-1}-1}{n}\not\equiv 0 \pmod{p}</math>, then also <math>\tfrac{2^{p-1}-1}{p}\not\equiv 0 \pmod{p}</math>.<ref name="Dilcher, Skula" />{{rp|378}} Furthermore, if ''p'' is a Wieferich prime, then ''p''<sup>2</sup> is a [[Catalan pseudoprime]].

=== Connection with directed graphs ===

For all primes {{math|''p''}} up to {{math|1=100000}}, {{math|1=''L''(''p''<sup>''n''+1</sup>) = ''L''(''p''<sup>''n''</sup>)}} only in two cases: {{math|1=''L''(1093<sup>2</sup>) = ''L''(1093) = 364}} and {{math|1=''L''(3511<sup>2</sup>) = ''L''(3511) = 1755}}, where {{math|1=''L''(''m'')}} is the number of vertices in the cycle of 1  in the ''doubling diagram'' modulo {{math|''m''}}. Here the doubling diagram represents the [[directed graph]] with the non-negative integers less than ''m'' as vertices and with directed edges going from each vertex ''x'' to vertex 2''x'' reduced modulo ''m''.<ref name="Ehrlich">{{Citation | first=A. |  last=Ehrlich | title=Cycles in Doubling Diagrams mod m | journal=The Fibonacci Quarterly | volume=32 | issue=1 | year=1994 | pages=74–78 | doi=10.1080/00150517.1994.12429259 | url=http://www.fq.math.ca/Scanned/32-1/ehrlich.pdf | postscript=.}}</ref>{{rp|74}} It was shown, that for all odd prime numbers either {{math|1=''L''(''p''<sup>''n''+1</sup>) = ''p'' · ''L''(''p''<sup>''n''</sup>)}} or {{math|1=''L''(''p''<sup>''n''+1</sup>) = ''L''(''p''<sup>''n''</sup>)}}.<ref name="Ehrlich" />{{rp|75}}

=== Properties related to number fields ===

It was shown that <math>\chi_{D_{0}} \big(p \big) = 1</math> and <math>\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1</math> if and only if {{math|1=2<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;≢&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)}} where ''p'' is an odd prime and <math>D_{0} < 0</math> is the [[fundamental discriminant]] of the imaginary [[quadratic field]] <math>\mathbb{Q} \big(\sqrt{1 - p^2} \big)</math>. Furthermore, the following was shown: Let ''p'' be a Wieferich prime. If {{math|''p'' ≡ 3 (mod 4)}}, let <math>D_{0} < 0</math> be the fundamental discriminant of the imaginary quadratic field <math>\mathbb{Q} \big(\sqrt{1 - p} \big)</math> and if {{math|''p'' ≡ 1 (mod 4)}}, let <math>D_{0} < 0</math> be the fundamental discriminant of the imaginary quadratic field <math>\mathbb{Q} \big(\sqrt{4 - p} \big)</math>. Then <math>\chi_{D_{0}} \big(p \big) = 1</math> and <math>\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1</math> (''χ'' and ''λ'' in this context denote Iwasawa [[Invariant (mathematics)|invariants]]).<ref>{{citation | first=D. | last=Byeon | title=Class numbers, Iwasawa invariants and modular forms | journal=Trends in Mathematics | volume=9 | issue=1 | year=2006 | pages=25–29 | url=http://basilo.kaist.ac.kr/mathnet/kms_tex/985999.pdf | access-date=2012-09-05 | archive-date=2012-04-26 | archive-url=https://web.archive.org/web/20120426012529/http://basilo.kaist.ac.kr/mathnet/kms_tex/985999.pdf | url-status=dead }}</ref>{{rp|27}}

Furthermore, the following result was obtained: Let ''q'' be an odd prime number, ''k'' and ''p'' are primes such that {{math|1=''p'' = 2''k'' + 1,}} {{math|''k'' ≡ 3 (mod 4),}} {{math|''p'' ≡ −1 (mod ''q''),}} {{math|''p'' ≢ −1 (mod ''q''<sup>3</sup>)}} and the order of ''q'' modulo ''k'' is <math>\tfrac{k-1}{2}</math>. Assume that ''q'' divides ''h''<sup>+</sup>, the [[Ideal class group|class number]] of the real [[cyclotomic field]] <math>\mathbb{Q} \big( \zeta\,\!_p + \zeta\,\!_p^{-1} \big)</math>, the cyclotomic field obtained by adjoining the sum of a ''p''-th [[root of unity]] and its [[Multiplicative inverse|reciprocal]] to the field of rational numbers. Then ''q'' is a Wieferich prime.<ref name="Jakubec">{{citation | first=S. | last=Jakubec | title=Connection between the Wieferich congruence and divisibility of h<sup>+</sup> | journal=Acta Arithmetica | volume=71 | issue=1 | year=1995 | pages=55–64 | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7114.pdf| doi=10.4064/aa-71-1-55-64 | doi-access=free }}</ref>{{rp|55}} This also holds if the conditions {{math|''p'' ≡ −1 (mod ''q'')}} and {{math|''p'' ≢ −1 (mod ''q''<sup>3</sup>)}} are replaced by {{math|''p'' ≡ −3 (mod ''q'')}} and {{math|''p'' ≢ −3 (mod ''q''<sup>3</sup>)}} as well as when the condition {{math|''p'' ≡ −1 (mod ''q'')}} is replaced by {{math|''p'' ≡ −5 (mod ''q'')}} (in which case ''q'' is a [[Wall–Sun–Sun prime]]) and the incongruence condition replaced by {{math|''p'' ≢ −5 (mod ''q''<sup>3</sup>)}}.<ref>{{citation | first=S. | last=Jakubec | title=On divisibility of the class number h<sup>+</sup> of the real cyclotomic fields of prime degree l | journal=Mathematics of Computation | volume=67 | issue=221 | year=1998 | pages=369–398 | url=http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00916-8/S0025-5718-98-00916-8.pdf | doi=10.1090/s0025-5718-98-00916-8| doi-access=free }}</ref>{{rp|376}}

== Generalizations ==

=== Near-Wieferich primes ===
A prime ''p'' satisfying the congruence 2<sup>(''p''−1)/2</sup> {{math|1=≡&nbsp;±1&nbsp;+&nbsp;''Ap''}} (mod ''p''<sup>2</sup>) with small |''A''| is commonly called a ''near-Wieferich prime'' {{OEIS|id=A195988}}.<ref name=Crandall>{{Citation | first1=Richard E. |last1=Crandall |first2=Karl |last2=Dilcher |first3=Carl |last3=Pomerance | title=A search for Wieferich and Wilson primes | journal=Mathematics of Computation | volume=66 | issue=217 | pages=433–449 | year=1997 | doi=10.1090/S0025-5718-97-00791-6 |url=http://gauss.dartmouth.edu/~carlp/PDF/paper111.pdf | postscript=. |bibcode=1997MaCom..66..433C | doi-access=free }}</ref><ref name=Knauer>{{Citation |author=Joshua Knauer |author2=Jörg Richstein |title=The continuing search for Wieferich primes |journal=Mathematics of Computation |volume=74 |year=2005 |pages=1559–1563 |doi=10.1090/S0025-5718-05-01723-0 |url=http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-05-01723-0/S0025-5718-05-01723-0.pdf |postscript=. |issue=251|bibcode=2005MaCom..74.1559K |doi-access=free }}</ref> 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.<ref name=Dorais /><ref name=elmath>{{Cite web |url=http://www.elmath.org/index.php?id=main |title=About project Wieferich@Home |access-date=2010-07-05 |archive-date=2012-03-22 |archive-url=https://web.archive.org/web/20120322140402/http://www.elmath.org/index.php?id=main |url-status=dead }}</ref> The following table lists all near-Wieferich primes with |''A''| ≤ 10 in the interval [1{{e|9}}, 3{{e|15}}].<ref>PrimeGrid, [http://www.primegrid.com/download/wieferich_list.pdf Wieferich & near Wieferich primes p < 11e15] {{Webarchive|url=https://web.archive.org/web/20121018120338/http://www.primegrid.com/download/wieferich_list.pdf |date=2012-10-18 }}</ref> This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.<ref name="Ribenboim, 2004">{{Citation | last = Ribenboim | first = Paulo | author-link = Paulo Ribenboim | title = Die Welt der Primzahlen: Geheimnisse und Rekorde | publisher = Springer | year = 2004 | location = New York | page = 237 | language = de | url = https://books.google.com/books?id=-nEM9ZVr4CsC&pg=PA237 | isbn = 978-3-540-34283-0}}</ref><ref>{{Citation | last = Ribenboim | first = Paulo | author-link = Paulo Ribenboim | title = My numbers, my friends: popular lectures on number theory | publisher = Springer | year = 2000 | location = New York | pages = 213–229 | isbn = 978-0-387-98911-2}}</ref> Bigger entries are by PrimeGrid.
{| class="wikitable" style="width:40%; border:0; text-align:right;"
|-
! p !! 1 or −1 !! A
|-
| 3520624567 || +1 || −6
|-
| 46262476201 || +1 || +5
|-
| 47004625957 || −1 || +1
|-
| 58481216789 || −1 || +5
|-
| 76843523891 || −1 || +1
|-
| 1180032105761 || +1 || −6
|-
| 12456646902457 || +1 || +2
|-
| 134257821895921 || +1 || +10
|-
| 339258218134349 || −1 || +2
|-
| 2276306935816523 || −1 || −3
|-
| 82687771042557349 || -1 || -10
|-
| 3156824277937156367 || +1 || +7
|}

The sign +1 or -1 above can be easily predicted by [[Euler's criterion]] (and the second supplement to the law of [[quadratic reciprocity]]).

Dorais and Klyve<ref name=Dorais /> used a different definition of a near-Wieferich prime, defining it as a prime ''p'' with small value of <math>\left|\tfrac{\omega(p)}{p}\right|</math> where <math>\omega(p)=\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math> is the [[Fermat quotient]] of 2 with respect to ''p'' modulo ''p'' (the [[modulo operation]] here gives the residue with the smallest absolute value). The following table lists all primes ''p'' ≤ {{math|1=6.7&nbsp;×&nbsp;10<sup>15</sup>}} with <math>\left|\tfrac{\omega(p)}p\right|\leq 10^{-14}</math>.

{| class="wikitable" style="width:40%; border:0; text-align:right;"
|-
! ''p'' !! <math>\omega(p)</math> !! <math>\left|\tfrac{\omega(p)}{p}\right|\times 10^{14}</math>
|-
| 1093 || 0 || 0
|-
| 3511 || 0 || 0
|-
| 2276306935816523 || +6 || 0.264
|-
| 3167939147662997 || −17 || 0.537
|-
| 3723113065138349 || −36 || 0.967
|-
| 5131427559624857 || −36 || 0.702
|-
| 5294488110626977 || −31 || 0.586
|-
| 6517506365514181 || +58 || 0.890
|}

The two notions of nearness are related as follows. If <math>2^{(p-1)/2}\equiv \pm 1 + Ap \pmod{p^2}</math>, then by squaring, clearly <math>2^{p-1}\equiv 1 \pm 2Ap \pmod{p^2}</math>. So if {{mvar|A}} had been chosen with <math>|A|</math> small, then clearly <math>|\!\pm 2A| = 2|A|</math> is also (quite) small, and an even number. However, when <math>\omega(p)</math> is odd above, the related {{mvar|A}} from before the last squaring was not "small". For example, with <math>p=3167939147662997</math>, we have <math>2^{(p-1)/2}\equiv -1 - 1583969573831490p \pmod{p^2}</math> which reads extremely non-near, but after squaring this is <math>2^{p-1}\equiv 1 - 17p \pmod{p^2}</math> which is a near-Wieferich by the second definition.

=== Base-''a'' Wieferich primes ===
{{Main|Fermat quotient#Generalized Wieferich primes|l1=Fermat quotient}}
A ''Wieferich prime base a'' is a prime ''p'' that satisfies
: {{math|1=''a''<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)}},<ref name="Solutions" /> with ''a'' less than ''p'' but greater than 1.
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 {{math|''a''<sup>''p''−1</sup> ≡ 1 (mod ''q'')}}, {{math|''a''<sup>''q''−1</sup> ≡ 1 (mod ''p'')}}, then {{math|''a''<sup>''pq''−1</sup> ≡ 1 (mod ''pq'')}}. Setting ''p'' = ''q'' leads to {{math|''a''<sup>''p''<sup>2</sup>−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}.<ref name="Kiss">{{cite journal | last = Kiss | first = E. |author2=Sándor, J. | title = On a congruence by János Bolyai, connected with pseudoprimes | journal = Mathematica Pannonica | volume = 15 | issue = 2 | pages = 283–288 | year = 2004 | url = http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp15-2/mp15-2-283-288.pdf}}</ref>{{rp|284}} It was shown that {{math|''a''<sup>''p''<sup>2</sup>−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} if and only if {{math|''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}.<ref name="Kiss" />{{rp|285–286}}

Known solutions of {{math|''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} for small values of ''a'' are:<ref>[http://primes.utm.edu/glossary/xpage/FermatQuotient.html Fermat Quotient] at ''The Prime Glossary''</ref> (checked up to 5 × 10<sup>13</sup>)

:{| class="wikitable"
|-
! ''a''
! primes ''p'' such that ''a''<sup>''p'' − 1</sup> = 1 (mod ''p''<sup>2</sup>)
! [[OEIS]] sequence
|-
| 1 || 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)
| {{OEIS link|id=A000040}}
|-
| 2 || 1093, 3511, ...
| {{OEIS link|id=A001220}}
|-
| 3 || 11, 1006003, ...
| {{OEIS link|id=A014127}}
|-
| 4 || 1093, 3511, ...
|
|-
| 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...
| {{OEIS link|id=A123692}}
|-
| 6 || 66161, 534851, 3152573, ...
| {{OEIS link|id=A212583}}
|-
| 7 || 5, 491531, ...
| {{OEIS link|id=A123693}}
|-
| 8 || 3, 1093, 3511, ...
|
|-
| 9 || 2, 11, 1006003, ...
|
|-
| 10 || 3, 487, 56598313, ...
| {{OEIS link|id=A045616}}
|-
| 11 || 71, ...
|
|-
| 12 || 2693, 123653, ...
| {{OEIS link|id=A111027}}
|-
| 13 || 2, 863, 1747591, ...
| {{OEIS link|id=A128667}}
|-
| 14 || 29, 353, 7596952219, ...
| {{OEIS link|id=A234810}}
|-
| 15 || 29131, 119327070011, ...
| {{OEIS link|id=A242741}}
|-
| 16 || 1093, 3511, ...
|
|-
| 17 || 2, 3, 46021, 48947, 478225523351, ...
| {{OEIS link|id=A128668}}
|-
| 18 || 5, 7, 37, 331, 33923, 1284043, ...
| {{OEIS link|id=A244260}}
|-
| 19 || 3, 7, 13, 43, 137, 63061489, ...
| {{OEIS link|id=A090968}}
|-
| 20 || 281, 46457, 9377747, 122959073, ...
| {{OEIS link|id=A242982}}
|-
| 21 || 2, ...
|
|-
| 22 || 13, 673, 1595813, 492366587, 9809862296159, ...
| {{OEIS link|id=A298951}}
|-
| 23 || 13, 2481757, 13703077, 15546404183, 2549536629329, ...
| {{OEIS link|id=A128669}}
|-
| 24 || 5, 25633, ...
|
|-
| 25 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...
|
|-
| 26 || 3, 5, 71, 486999673, 6695256707, ...
| {{OEIS link|id=A306255}}
|-
| 27 || 11, 1006003, ...
|
|-
| 28 || 3, 19, 23, ...
|
|-
| 29 || 2, ...
|
|-
| 30 || 7, 160541, 94727075783, ...
| {{OEIS link|id=A306256}}
|-
| 31 || 7, 79, 6451, 2806861, ...
| {{OEIS link|id=A331424}}
|-
| 32 || 5, 1093, 3511, ...
|
|-
| 33 || 2, 233, 47441, 9639595369, ...
|
|-
| 34 || 46145917691, ...
|
|-
| 35 || 3, 1613, 3571, ...
|
|-
| 36 || 66161, 534851, 3152573, ...
|
|-
| 37 || 2, 3, 77867, 76407520781, ...
| {{OEIS link|id=A331426}}
|-
| 38 || 17, 127, ...
|
|-
| 39 || 8039, ...
|
|-
| 40 || 11, 17, 307, 66431, 7036306088681, ...
|
|-
| 41 || 2, 29, 1025273, 138200401, ...
| {{OEIS link|id=A331427}}
|-
| 42 || 23, 719867822369, ...
|
|-
| 43 || 5, 103, 13368932516573, ...
|
|-
| 44 || 3, 229, 5851, ...
|
|-
| 45 || 2, 1283, 131759, 157635607, ...
|
|-
| 46 || 3, 829, ...
|
|-
| 47 || ...
|
|-
| 48 || 7, 257, ...
|
|-
| 49 || 2, 5, 491531, ...
|
|-
| 50 || 7, ...
|
|}

For more information, see<ref>{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort|title=Wieferich primes to base 1052|access-date=2014-07-22|archive-date=2015-09-11|archive-url=https://web.archive.org/web/20150911144520/http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort|url-status=dead}}</ref><ref>{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt|title=Wieferich primes to base 10125}}</ref><ref>{{cite web | title=Fermat quotients ''q''<sub>''p''</sub>(''a'') that are divisible by ''p''  | website=www1.uni-hamburg.de | date=2014-08-09 | url=http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html | archive-url=https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html | archive-date=2014-08-09 | url-status=dead | access-date=2019-09-18}}</ref> and.<ref>{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/FermatQ3|title=Wieferich primes with level ≥ 3}}</ref> (Note that the solutions to ''a'' = ''b<sup>k</sup>'' is the union of the prime divisors of ''k'' which does not divide ''b'' and the solutions to ''a'' = ''b'')

The smallest solutions of {{math|''n''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} are

: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 > 4.9×10<sup>13</sup>) {{OEIS|id=A039951}}

There are no known solutions of {{math|''n''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} 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 {{math|''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} for every natural number ''a''.

The bases ''b'' < ''p''<sup>2</sup> which ''p'' is a Wieferich prime are (for ''b'' > ''p''<sup>2</sup>, the solutions are just shifted by ''k''·''p''<sup>2</sup> for ''k'' > 0), and there are {{math|1=''p''&nbsp;−&nbsp;1}} solutions < ''p''<sup>2</sup> of ''p'' and the set of the solutions [[Modular arithmetic|congruent]] to ''p'' are {1, 2, 3, ..., {{math|1=''p''&nbsp;−&nbsp;1})}} {{OEIS|id=A143548}}

: {|class="wikitable"
|-
!''p''
!values of ''b'' < ''p''<sup>2</sup>
|-
|2
|1
|-
|3
|1, 8
|-
|5
|1, 7, 18, 24
|-
|7
|1, 18, 19, 30, 31, 48
|-
|11
|1, 3, 9, 27, 40, 81, 94, 112, 118, 120
|-
|13
|1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168
|-
|17
|1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288
|-
|19
|1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360
|-
|23
|1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528
|-
|29
|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
|}

The least base ''b'' > 1 which prime(''n'') is a Wieferich prime are
: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, ... {{OEIS|id=A039678}}

We can also consider the formula <math>(a+1)^{p-1}-a^{p-1} \equiv 0 \pmod{p^2}</math>, (because of the generalized Fermat little theorem, <math>(a+1)^{p-1}-a^{p-1} \equiv 0 \pmod{p^2}</math> is true for all prime ''p'' and all natural number ''a'' such that both ''a'' and {{math|1=''a''&nbsp;+&nbsp;1}} are not divisible by ''p''). It's a conjecture that for every natural number ''a'', there are infinitely many primes such that <math>(a+1)^{p-1}-a^{p-1} \equiv 0 \pmod{p^2}</math>.

Known solutions for small ''a'' are: (checked up to 4 × 10<sup>11</sup>) <ref>{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/PQuotient_Sort.txt|title=Solution of {{math|1=(''a''&nbsp;+&nbsp;1)<sup>''p''−1</sup>&nbsp;−&nbsp;''a''<sup>''p''−1</sup> ≡ 0 (mod ''p''<sup>2</sup>)}}}}</ref>

: {|class="wikitable"
!{{tmath|a}}
!primes {{tmath|p}} such that <math>(a+1)^{p-1}-a^{p-1} \equiv 0 \pmod{p^2}</math>
|-
|1
|1093, 3511, ...
|-
|2
|23, 3842760169, 41975417117, ...
|-
|3
|5, 250829, ...
|-
|4
|3, 67, ...
|-
|5
|3457, 893122907, ...
|-
|6
|72673, 1108905403, 2375385997, ...
|-
|7
|13, 819381943, ...
|-
|8
|67, 139, 499, 26325777341, ...
|-
|9
|67, 887, 9257, 83449, 111539, 31832131, ...
|-
|10
|...
|-
|11
|107, 4637, 239357, ...
|-
|12
|5, 11, 51563, 363901, 224189011, ...
|-
|13
|3, ...
|-
|14
|11, 5749, 17733170113, 140328785783, ...
|-
|15
|292381, ...
|-
|16
|4157, ...
|-
|17
|751, 46070159, ...
|-
|18
|7, 142671309349, ...
|-
|19
|17, 269, ...
|-
|20
|29, 162703, ...
|-
|21
|5, 2711, 104651, 112922981, 331325567, 13315963127, ...
|-
|22
|3, 7, 13, 94447, 1198427, 23536243, ...
|-
|23
|43, 179, 1637, 69073, ...
|-
|24
|7, 353, 402153391, ...
|-
|25
|43, 5399, 21107, 35879, ...
|-
|26
|7, 131, 653, 5237, 97003, ...
|-
|27
|2437, 1704732131, ...
|-
|28
|5, 617, 677, 2273, 16243697, ...
|-
|29
|73, 101, 6217, ...
|-
|30
|7, 11, 23, 3301, 48589, 549667, ...
|-
|31
|3, 41, 416797, ...
|-
|32
|95989, 2276682269, ...
|-
|33
|139, 1341678275933, ...
|-
|34
|83, 139, ...
|-
|35
|...
|-
|36
|107, 137, 613, 2423, 74304856177, ...
|-
|37
|5, ...
|-
|38
|167, 2039, ...
|-
|39
|659, 9413, ...
|-
|40
|3, 23, 21029249, ...
|-
|41
|31, 71, 1934399021, 474528373843, ...
|-
|42
|4639, 1672609, ...
|-
|43
|31, 4962186419, ...
|-
|44
|36677, 17786501, ...
|-
|45
|241, 26120375473, ...
|-
|46
|5, 13877, ...
|-
|47
|13, 311, 797, 906165497, ...
|-
|48
|...
|-
|49
|3, 13, 2141, 281833, 1703287, 4805298913, ...
|-
|50
|2953, 22409, 99241, 5427425917, ...
|}

=== Wieferich pairs ===
{{Main|Wieferich pair}}
A [[Wieferich pair]] is a pair of primes ''p'' and ''q'' that satisfy
: ''p''<sup>''q'' − 1</sup> ≡ 1 (mod ''q''<sup>2</sup>) and ''q''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)
so that a Wieferich prime ''p'' ≡ 1 (mod 4) will form such a pair (''p'', 2): the only known instance in this case is {{math|1=''p''&nbsp;=&nbsp;1093}}. There are only 7 known Wieferich pairs.<ref>{{MathWorld |urlname=DoubleWieferichPrimePair |title=Double Wieferich Prime Pair}}</ref>

: (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence {{OEIS2C|id=A282293}} in [[On-Line Encyclopedia of Integer Sequences|OEIS]])

=== Wieferich sequence ===

Start with a(1) any natural number (>1), a(''n'') = the smallest prime ''p'' such that (a(''n'' − 1))<sup>''p'' − 1</sup> = 1 (mod ''p''<sup>2</sup>) but ''p''<sup>2</sup> does not divide a(''n'' − 1) − 1 or a(''n'' − 1) + 1. (If ''p''<sup>2</sup> divides a(''n'' − 1) − 1 or a(''n'' − 1) + 1, then the solution is a [[Triviality (mathematics)|trivial solution]]) It is a conjecture that every natural number ''k'' = a(1) > 1 makes this sequence become periodic, for example, let a(1) = 2: 

:2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.
:{{OEIS|id=A359952}}

Let a(1) = 83: 

:83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}.

Let a(1) = 59 (a longer sequence):

:59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ..., it also gets 5.

However, there are many values of a(1) with unknown status, for example, let a(1) = 3:

:3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).

Let a(1) = 14:

:14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 2<sup>2</sup> = 4 divides 29 − 1 = 28)

Let a(1) = 39 (a longer sequence):
:39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)

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)

=== Wieferich numbers ===
A '''Wieferich number''' is an odd natural number ''n'' satisfying the congruence 2<sup>{{φ}}(''n'')</sup> ≡ 1 (mod ''n''<sup>2</sup>), where {{φ}} denotes the [[Euler's totient function]] (according to [[Euler's theorem]], 2<sup>{{φ}}(''n'')</sup> ≡ 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:
:1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... {{OEIS|A077816}}
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.<ref name="Banks, Luca" />

More generally, a natural number ''n'' is a '''Wieferich number to base''' ''a'', if ''a''<sup>{{φ}}(''n'')</sup> ≡ 1 (mod ''n''<sup>2</sup>).<ref>{{citation | first1=T. | last1=Agoh | first2=K. | last2=Dilcher | first3=L. | last3=Skula | title=Fermat Quotients for Composite Moduli | journal=Journal of Number Theory | volume=66 | issue=1 | year=1997 | pages=29–50 | doi=10.1006/jnth.1997.2162| doi-access=free }}</ref>{{rp|31}}

Another definition specifies a '''Wieferich number''' as odd natural number ''n'' such that ''n'' and <math>\tfrac{2^m-1}{n}</math> are not [[coprime]], where ''m'' is the [[multiplicative order]] of 2 modulo ''n''. The first of these numbers are:<ref>{{cite journal | last = Müller | first = H. | journal = Mitteilungen der Mathematischen Gesellschaft in Hamburg | volume = 28 | pages = 121–130 | year = 2009 | language = de | title = Über Periodenlängen und die Vermutungen von Collatz und Crandall | url = 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}}</ref>
: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, ... {{OEIS|A182297}}
As above, if Wieferich number ''q'' is prime, then it is a Wieferich prime.

=== Weak Wieferich prime ===
A weak Wieferich prime to base ''a'' is a prime ''p'' satisfies the condition
:''a''<sup>''p''</sup> ≡ ''a'' (mod ''p''<sup>2</sup>)

Every Wieferich prime to base ''a'' is also a weak Wieferich prime to base ''a''. If the base ''a'' is [[squarefree number|squarefree]], 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)
: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, ...

=== Wieferich prime with order ''n'' ===
For integer ''n'' ≥2, a Wieferich prime to base ''a'' with order ''n'' is a prime ''p'' satisfies the condition
:''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>''n''</sup>)

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.

=== Lucas–Wieferich primes ===
Let ''P'' and ''Q'' be integers. The [[Lucas sequence]] [[Lucas sequence#Recurrence relations|of the first kind]] associated with the [[Ordered pair|pair]] (''P'', ''Q'') is defined by
:<math>\begin{align}
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{align}
</math>
for all <math>n \geq 2</math>. A '''Lucas–Wieferich prime''' associated with (''P'', ''Q'') is a prime ''p'' such that ''U''<sub>''p''−''ε''</sub>(''P'', ''Q'') ≡ 0 (mod ''p''<sup>2</sup>), where ''ε'' equals the [[Legendre symbol]] <math>\left(\tfrac{P^2-4Q}p\right)</math>. All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).<ref name="McIntosh, 2007" />{{rp|2088}}

=== Wieferich places ===
Let ''K'' be a [[global field]], i.e. a [[number field]] or a [[Algebraic function field|function field]] in one variable over a [[finite field]] and let ''E'' be an [[elliptic curve]]. If ''v'' is a [[Algebraic number field#Nonarchimedian or ultrametric places|non-archimedean place]] of [[Field norm|norm]] ''q<sub>v</sub>'' of ''K'' and a ∈ K, with ''v''(''a'') = 0 then {{math|1=''v''(a<sup>''q''<sub>''v''</sub>&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1)}} ≥ 1. ''v'' is called a ''Wieferich place'' for base ''a'', if {{math|1=''v''(a<sup>''q<sub>v</sub>''&nbsp;−&nbsp;1</sup>&nbsp;−&nbsp;1)}} > 1, an ''elliptic Wieferich place'' for base ''P'' ∈ ''E'', if ''N<sub>v</sub>P'' ∈ ''E''<sub>2</sub> and a ''strong elliptic Wieferich place'' for base ''P'' ∈ ''E'' if ''n<sub>v</sub>P'' ∈ ''E''<sub>2</sub>, where ''n<sub>v</sub>'' is the order of ''P'' modulo ''v'' and ''N<sub>v</sub>'' gives the number of [[rational point]]s (over the [[residue field]] of ''v'') of the reduction of ''E'' at ''v''.<ref>{{Citation | first = J. F. | last = Voloch | title = Elliptic Wieferich Primes | journal = Journal of Number Theory | volume = 81 | issue = 2 | year = 2000 | pages = 205–209 | doi = 10.1006/jnth.1999.2471| doi-access = free }}</ref>{{rp|206}}

== See also ==
* [[Wall–Sun–Sun prime]] – another type of prime number which in the broadest sense also resulted from the study of FLT
* [[Wolstenholme prime]] – another type of prime number which in the broadest sense also resulted from the study of FLT
* [[Wilson prime]]
* [[Table of congruences]] – lists other congruences satisfied by prime numbers
* [[PrimeGrid]] – primes search project
*[[BOINC]]
*[[Distributed computing]]

== References ==
{{Reflist|colwidth=30em}}

== Further reading ==
* {{citation | first=R. | last=Haussner | title=Über die Kongruenzen {{math|1=2<sup>''p''−1</sup>&nbsp;−&nbsp;1}} ≡ 0 (mod ''p''<sup>2</sup>) für die Primzahlen ''p''=1093 und 3511 | journal=Archiv for Mathematik og Naturvidenskab | volume=39 | issue=5 | year=1926 | language=de | page=7 | id=[[German National Library|DNB]] [http://d-nb.info/363953469/about/html 363953469] | jfm=52.0141.06 | url=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}}
* {{citation | first=R. | last=Haussner | title=Über numerische Lösungen der Kongruenz {{math|1=''u''<sup>''p''−1</sup>&nbsp;−&nbsp;1}} ≡ 0 (mod ''p''<sup>2</sup>) | journal=Journal für die Reine und Angewandte Mathematik | volume=1927 | issue=156 | year=1927 | language=de |  pages=223–226 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002169924 | doi=10.1515/crll.1927.156.223| s2cid=117969297 }}
* {{citation | first=P. |last=Ribenboim | author-link=Paulo Ribenboim | title=Thirteen lectures on Fermat's Last Theorem | publisher=[[Springer-Verlag]] | year=1979 | isbn=978-0-387-90432-0 | pages=139, 151 }}
* {{citation |first=Richard K. |last=Guy |author-link=Richard K. Guy  |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |year=2004  |isbn=978-0-387-20860-2 |page=14 }}
* {{citation | first1=R. E. | last1=Crandall | first2=C. | last2=Pomerance | title=Prime numbers: a computational perspective | publisher=Springer Science+Business Media | year=2005 | pages=31–32 | isbn=978-0-387-25282-7 | url=http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf}}
* {{citation | first=P. | last=Ribenboim | title=The new book of prime number records | publisher=Springer-Verlag |location=New York | year=1996 | pages=333–346 | isbn=978-0-387-94457-9 | url=https://books.google.com/books?id=72eg8bFw40kC&pg=PA333}}

== External links ==
* {{MathWorld |urlname=WieferichPrime |title=Wieferich prime}}
* [http://go.helms-net.de/math/expdioph/fermatquotients.pdf Fermat-/Euler-quotients {{math|1=(''a''<sup>''p''−1</sup>&nbsp;−&nbsp;1)/''p''<sup>''k''</sup>}} with arbitrary ''k'']
* [http://library.uwinnipeg.ca/people/dobson/mathematics/Wieferich_primes.html A note on the two known Wieferich primes]
* PrimeGrid's [https://www.primegrid.com/forum_thread.php?id=9436 Wieferich Prime Search project] page

{{Prime number classes|state=collapsed}}

[[Category:Classes of prime numbers]]
[[Category:Unsolved problems in number theory]]
[[Category:Abc conjecture]]