Wieferich prime

The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation {{math|2^{p -1} ≡ 1 (mod p²)}}. From the definition of the 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 $\backslash tfrac\{2^\{p-1\}-1\}\{p\}$. The following are two illustrative examples using the primes 11 and 1093:

: For p = 11, we get $\backslash tfrac\{2^\{10\}-1\}\{11\}$ 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 $\backslash tfrac\{2^\{1092\}-1\}\{1093\}$ 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.

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^p−1 ≡ 1 (mod p²)}} by 2 to get {{math|1=2^p ≡ 2 (mod p²)}}. Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies {{math|1=2^p2 ≡2^p ≡ 2 (mod p²)}}, and hence {{math|1=2^pk ≡ 2 (mod p²)}} for all {{math|1=k ≥ 1}}. The converse is also true: {{math|1=2^pk ≡ 2 (mod p²)}} for some {{math|1=k ≥ 1}} implies that the multiplicative order of 2 modulo p² divides gcd{{math|1=(p^k − 1}}, φ{{math|1=(p²)) = p − 1}}, that is, {{math|1=2^p−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: {{math|1=ord_p² 2 = ord_p 2}}, (By the way, ord₁₀₉₃2 = 364, and ord₃₅₁₁2 = 1755).

H. S. Vandiver proved that {{math|1=2^p−1 ≡ 1 (mod p³)}} if and only if $1\; +\; \backslash tfrac\{1\}\{3\}\; +\; \backslash dots\; +\; \backslash tfrac\{1\}\{p-2\}\; \backslash equiv\; 0\; \backslash pmod\{p^2\}$.{{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}}{{rp|187}}

{{Unsolved|mathematics|Are there infinitely many Wieferich primes?}}In 1902, Meyer proved a theorem about solutions of the congruence a^{p − 1} ≡ 1 (mod p^r).{{Citation |author=Wilfrid Keller |author2=Jörg Richstein |title=Solutions of the congruence {{math|1=a^p−1 ≡ 1 (mod p^r)}} |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 }}{{rp|930}}{{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}} 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.{{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=.}} In other words, if there exist solutions to x^p + y^p + z^p = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2^{p − 1} ≡ 1 (mod p²). In 1913, Bachmann examined the residues of $\backslash tfrac\{2^\{p-1\}-1\}\{p\}\backslash ,\backslash bmod\backslash ,p$. He asked the question when this residue vanishes and tried to find expressions for answering this question.{{cite journal | last = Bachmann | first = P. | author-link = Paul Gustav Heinrich Bachmann | title = Über den Rest von $\backslash tfrac\{2^\{p-1\}-1\}\{p\}\backslash ,\backslash bmod\backslash ,p$ | journal = Journal für Mathematik | volume = 142 | issue = 1 | pages = 41–50 | language=de | year = 1913 | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167646}}

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 $\backslash tfrac\{2^\{t\}-1\}\{p\}\backslash ,\backslash bmod\backslash ,p$ for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a conjecture by Grave about the impossibility of the Wieferich congruence.{{citation | first=W. |last=Meissner |title=Über die Teilbarkeit von 2^p − 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}} {{ill|Emil Haentzschel|de|lt=E. Haentzschel}} later ordered verification of the correctness of Meissner's congruence via only elementary calculations.{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2¹⁰⁹² ≡ 1 (mod 1093²) | 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}}{{rp|664}} Inspired by an earlier work of Euler, he simplified Meissner's proof by showing that 1093² | (2¹⁸² + 1) and remarked that (2¹⁸² + 1) is a factor of (2³⁶⁴ − 1).{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2¹⁰⁹² ≡ 1 (mod 1093²) | 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}} It was also shown that it is possible to prove that 1093 is a Wieferich prime without using complex numbers contrary to the method used by Meissner,{{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}} although Meissner himself hinted at that he was aware of a proof without complex values.{{rp|665}}

The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922{{citation | first=N. G. W. H. |last=Beeger | author-link=N. G. W. H. Beeger | title=On a new case of the congruence 2^{p − 1} ≡ 1 (mod p²) | journal=Messenger of Mathematics | volume=51 | year=1922 | pages=149–150 |url=https://archive.org/stream/messengerofmathe5051cambuoft#page/148/mode/2up}} and another proof of it being a Wieferich prime was published in 1965 by Guy.{{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}} In 1960, Kravitz{{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^p-1 ≡ 1 (mod p²) 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 }} doubled a previous record set by {{ill|Carl-Erik Fröberg|sv|lt=Fröberg}}{{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 }} and in 1961 Riesel extended the search to 500000 with the aid of BESK.{{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^p−1 ≡ 1 (mod p²) |journal=Mathematics of Computation |volume=18 |issue=85 |year=1964 |pages=149–150 |doi=10.1090/S0025-5718-1964-0157928-6|doi-access=free }} Around 1980, Lehmer was able to reach the search limit of 6{{e|9}}.{{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 }} This limit was extended to over 2.5{{e|15}} in 2006, 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}}.{{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}}

In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home.{{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}} In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted.{{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}} 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 primes 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.{{cite web|title=Message boards : Wieferich and Wall-Sun-Sun Prime Search|url=https://www.primegrid.com/forum_forum.php?id=127|website=PrimeGrid}} The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 2⁶⁴ (about 18{{e|18}}).{{cite web|title=WW Statistics|url=https://www.primegrid.com/stats_ww.php|website=PrimeGrid}}

It has been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log(log(x)), which is a heuristic result that follows from the plausible assumption that for a prime p, the {{math|1=(p − 1)-th}} degree roots of unity modulo p² are uniformly distributed in the multiplicative group of integers modulo p².

The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909:

:Let p be prime, and let x, y, z be integers such that {{math|1=x^p + y^p + z^p = 0}}. Furthermore, assume that p does not divide the product 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){{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 }}{{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 }} 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.{{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 }}

In 1910, Mirimanoff expanded{{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=. }} the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must also divide {{math|1=3^p − 1 − 1}}. Granville and Monagan further proved that p² must actually divide {{math|1=m^p − 1 − 1}} for every prime m ≤ 89.{{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 }} Suzuki extended the proof to all primes m ≤ 113.{{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 }}

Let H_p 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)^p−1 ≡ 1 (mod p²), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers (such an extension is known as a number field or in this particular case, where ξ is a root of unity, a cyclotomic number field).{{rp|332}}

From 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_p.{{rp|333}}

Granville and Monagan showed that (1, 1) ∈ H_p if and only if p is a Wieferich prime.{{rp|333}}

A non-Wieferich prime is a prime p satisfying {{math|1=2^p − 1 ≢ 1 (mod p²)}}. J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.{{cite web | author=Charles, D. X. | url=http://pages.cs.wisc.edu/~cdx/Criterion.pdf | title=On Wieferich primes |website=wisc.edu}} 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.{{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 }}{{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 W₂ and W₂^c respectively,{{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}} are 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.{{citation | first=K. | last=Broughan | title=Relaxations of the ABC Conjecture using integer k'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}} Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers{{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}} as well as if there exists a real number ξ such that the set {n ∈ N : λ(2ⁿ − 1) < 2 − ξ} is of density one, where the index of composition λ(n) of an integer n is defined as $\backslash tfrac\{\backslash log\; n\}\{\backslash log\; \backslash gamma\; (n)\}$ and $\backslash gamma\; (n)\; =\; \backslash prod\_\{p\; \backslash mid\; n\}\; p$, meaning $\backslash gamma\; (n)$ gives the product of all prime factors of n.{{rp|4}}

It is known that the nth Mersenne number {{math|1=M_n = 2ⁿ − 1}} is prime only if n is prime. Fermat's little theorem implies that if {{math|1=p > 2}} is prime, then M_p−1 {{math|1=(= 2^p − 1 − 1)}} is always divisible by p. Since Mersenne numbers of prime indices M_p and M_q are co-prime,

::A prime divisor p of M_q, where q is prime, is a Wieferich prime if and only if p² divides M_q.{{Citation |url=http://primes.utm.edu/mersenne/index.html#unknown |title=Mersenne Primes: Conjectures and Unsolved Problems }}

Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free. If q is prime and the Mersenne number M_q is not square-free, that is, there exists a prime p for which p² divides M_q, 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.{{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=n²{{!}}2ⁿ − 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}}

Similarly, if p is prime and p² divides some Fermat number F_n {{math|1== 2²ⁿ + 1}}, then p must be a Wieferich prime.{{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 }}

In fact, there exists a natural number n and a prime p that p² divides $\backslash Phi\_n(2)$ (where $\backslash Phi\_n(x)$ is the n-th cyclotomic polynomial) if and only if p is a Wieferich prime. For example, 1093² divides $\backslash Phi\_\{364\}(2)$, 3511² divides $\backslash Phi\_\{1755\}(2)$. Mersenne and Fermat numbers are just special situations of $\backslash Phi\_n(2)$. Thus, if 1093 and 3511 are only two Wieferich primes, then all $\backslash Phi\_n(2)$ are square-free except $\backslash Phi\_\{364\}(2)$ and $\backslash Phi\_\{1755\}(2)$ (In fact, when there exists a prime p which p² divides some $\backslash Phi\_n(2)$, then it is a Wieferich prime); and clearly, if $\backslash Phi\_n(2)$ is a prime, then it cannot be Wieferich prime. (Any odd prime p divides only one $\backslash Phi\_n(2)$ and n divides {{math|1=p − 1}}, and if and only if the period length of 1/p in binary is n, then p divides $\backslash Phi\_n(2)$. 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.{{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 }}

Scott and Styer showed that the equation p^x – 2^y = d has at most one solution in positive integers (x, y), unless when p⁴ | 2^{ord_p 2} – 1 if p ≢ 65 (mod 192) or unconditionally when p² | 2^{ord_p 2} – 1, where ord_p 2 denotes the multiplicative order of 2 modulo p.{{cite journal | last = Scott | first = R. |author2=Styer, R. | title = On {{math|1=p^x − q^y = 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 = }}{{rp|215, 217–218}} They also showed that a solution to the equation ±a^x₁ ± 2^y₁ = ±a^x₂ ± 2^y₂ = 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.{{cite journal | last = Scott | first = R. |author2=Styer, R. | title = On the generalized Pillai equation ±a^x±b^y = 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 }}{{rp|258}}

Johnson observed{{Citation |author=Wells Johnson |title=On the nonvanishing of Fermat quotients {{math|1=(mod p)}} |journal=J. Reine Angew. Math. |volume=292 |year=1977 |pages=196–200 |url=http://www.digizeitschriften.de/index.php?id=resolveppn&PPN=GDZPPN002193698}} that the two known Wieferich primes are one greater than numbers with periodic binary expansions (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.{{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 }}

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

It was observed that the two known Wieferich primes are the square factors of all non-square free base-2 Fermat pseudoprimes up to 25{{e|9}}.{{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}} Later computations showed that the only repeated factors of the pseudoprimes up to 10¹² are 1093 and 3511.{{Cite book | last = Pinch | first = R. G. E. | title = The Pseudoprimes up to 10¹³ | volume = 1838 | pages = 459–473 | year = 2000 | doi = 10.1007/10722028_30 | series = Lecture Notes in Computer Science | isbn = 978-3-540-67695-9 }} In addition, the following connection exists:

:Let n be a base 2 pseudoprime and p be a prime divisor of n. If $\backslash tfrac\{2^\{n-1\}-1\}\{n\}\backslash not\backslash equiv\; 0\; \backslash pmod\{p\}$, then also $\backslash tfrac\{2^\{p-1\}-1\}\{p\}\backslash not\backslash equiv\; 0\; \backslash pmod\{p\}$.{{rp|378}} Furthermore, if p is a Wieferich prime, then p² is a Catalan pseudoprime.

For all primes {{math|p}} up to {{math|1=100000}}, {{math|1=L(pⁿ⁺¹) = L(pⁿ)}} only in two cases: {{math|1=L(1093²) = L(1093) = 364}} and {{math|1=L(3511²) = 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 2x reduced modulo m.{{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=.}}{{rp|74}} It was shown, that for all odd prime numbers either {{math|1=L(pⁿ⁺¹) = p · L(pⁿ)}} or {{math|1=L(pⁿ⁺¹) = L(pⁿ)}}.{{rp|75}}

It was shown that $\backslash chi\_\{D\_\{0\}\}\; \backslash big(p\; \backslash big)\; =\; 1$ and $\backslash lambda\backslash ,\backslash !\_p\; \backslash big(\; \backslash mathbb\{Q\}\; \backslash big(\backslash sqrt\{D\_\{0\}\}\; \backslash big)\; \backslash big)\; =\; 1$ if and only if {{math|1=2^p − 1 ≢ 1 (mod p²)}} where p is an odd prime and is the fundamental discriminant of the imaginary quadratic field $\backslash mathbb\{Q\}\; \backslash big(\backslash sqrt\{1\; -\; p^2\}\; \backslash big)$. Furthermore, the following was shown: Let p be a Wieferich prime. If {{math|p ≡ 3 (mod 4)}}, let be the fundamental discriminant of the imaginary quadratic field $\backslash mathbb\{Q\}\; \backslash big(\backslash sqrt\{1\; -\; p\}\; \backslash big)$ and if {{math|p ≡ 1 (mod 4)}}, let be the fundamental discriminant of the imaginary quadratic field $\backslash mathbb\{Q\}\; \backslash big(\backslash sqrt\{4\; -\; p\}\; \backslash big)$. Then $\backslash chi\_\{D\_\{0\}\}\; \backslash big(p\; \backslash big)\; =\; 1$ and $\backslash lambda\backslash ,\backslash !\_p\; \backslash big(\; \backslash mathbb\{Q\}\; \backslash big(\backslash sqrt\{D\_\{0\}\}\; \backslash big)\; \backslash big)\; =\; 1$ (χ and λ in this context denote Iwasawa invariants).{{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 }}{{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 = 2k + 1,}} {{math|k ≡ 3 (mod 4),}} {{math|p ≡ −1 (mod q),}} {{math|p ≢ −1 (mod q³)}} and the order of q modulo k is $\backslash tfrac\{k-1\}\{2\}$. Assume that q divides h⁺, the class number of the real cyclotomic field $\backslash mathbb\{Q\}\; \backslash big(\; \backslash zeta\backslash ,\backslash !\_p\; +\; \backslash zeta\backslash ,\backslash !\_p^\{-1\}\; \backslash big)$, the cyclotomic field obtained by adjoining the sum of a p-th root of unity and its reciprocal to the field of rational numbers. Then q is a Wieferich prime.{{citation | first=S. | last=Jakubec | title=Connection between the Wieferich congruence and divisibility of h⁺ | 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 }}{{rp|55}} This also holds if the conditions {{math|p ≡ −1 (mod q)}} and {{math|p ≢ −1 (mod q³)}} are replaced by {{math|p ≡ −3 (mod q)}} and {{math|p ≢ −3 (mod q³)}} 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 q³)}}.{{citation | first=S. | last=Jakubec | title=On divisibility of the class number h⁺ 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 }}{{rp|376}}

A prime p satisfying the congruence 2^(p−1)/2 {{math|1=≡ ±1 + Ap}} (mod p²) with small |A| is commonly called a near-Wieferich prime {{OEIS|id=A195988}}.{{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 }}{{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 }} 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.{{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 }} The following table lists all near-Wieferich primes with |A| ≤ 10 in the interval [1{{e|9}}, 3{{e|15}}].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 }} This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.{{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}}{{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}} Bigger entries are by PrimeGrid.

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 used a different definition of a near-Wieferich prime, defining it as a prime p with small value of $\backslash left|\backslash tfrac\{\backslash omega(p)\}\{p\}\backslash right|$ where $\backslash omega(p)=\backslash tfrac\{2^\{p-1\}-1\}\{p\}\backslash ,\backslash bmod\backslash ,p$ 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 × 10¹⁵}} with $\backslash left|\backslash tfrac\{\backslash omega(p)\}p\backslash right|\backslash leq\; 10^\{-14\}$.

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

{{Main|Fermat quotient#Generalized Wieferich primes|l1=Fermat quotient}}

A Wieferich prime base a is a prime p that satisfies

: {{math|1=a^p − 1 ≡ 1 (mod p²)}}, 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^p−1 ≡ 1 (mod q)}}, {{math|a^q−1 ≡ 1 (mod p)}}, then {{math|a^pq−1 ≡ 1 (mod pq)}}. Setting p = q leads to {{math|a^p2−1 ≡ 1 (mod p²)}}.{{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}}{{rp|284}} It was shown that {{math|a^p2−1 ≡ 1 (mod p²)}} if and only if {{math|a^p−1 ≡ 1 (mod p²)}}.{{rp|285–286}}

Known solutions of {{math|a^p−1 ≡ 1 (mod p²)}} for small values of a are:[http://primes.utm.edu/glossary/xpage/FermatQuotient.html Fermat Quotient] at The Prime Glossary (checked up to 5 × 10¹³)

:

For more information, see{{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}}{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt|title=Wieferich primes to base 10125}}{{cite web | title=Fermat quotients q_p(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}} and.{{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/FermatQ3|title=Wieferich primes with level ≥ 3}} (Note that the solutions to a = b^k 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^p−1 ≡ 1 (mod p²)}} 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 > 4.9×10¹³) {{OEIS|id=A039951}}

There are no known solutions of {{math|n^p−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 {{math|a^p−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 {{math|1=p − 1}} solutions < p² of p and the set of the solutions congruent to p are {1, 2, 3, ..., {{math|1=p − 1})}} {{OEIS|id=A143548}}

:

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

Known solutions for small a are: (checked up to 4 × 10¹¹) {{Cite web|url=http://www.fermatquotient.com/FermatQuotienten/PQuotient_Sort.txt|title=Solution of {{math|1=(a + 1)^p−1 − a^p−1 ≡ 0 (mod p²)}}}}

:

{{Main|Wieferich pair}}

A Wieferich pair is a pair of primes p and q that satisfy

: p^{q − 1} ≡ 1 (mod q²) and q^{p − 1} ≡ 1 (mod p²)

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 = 1093}}. There are only 7 known Wieferich pairs.{{MathWorld |urlname=DoubleWieferichPrimePair |title=Double Wieferich Prime Pair}}

: (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence {{OEIS2C|id=A282293}} in OEIS)

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 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 2² = 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)

A Wieferich number is an odd natural number n satisfying the congruence 2^{{φ}}(n) ≡ 1 (mod n²), where {{φ}} denotes the Euler's totient function (according to Euler's theorem, 2^{{φ}}(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:

: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.

More generally, a natural number n is a Wieferich number to base a, if a^{{φ}}(n) ≡ 1 (mod n²).{{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 }}{{rp|31}}

Another definition specifies a Wieferich number as odd natural number n such that n and $\backslash tfrac\{2^m-1\}\{n\}$ are not coprime, where m is the multiplicative order of 2 modulo n. The first of these numbers are:{{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}}

: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.

A weak Wieferich prime to base a is a prime p satisfies the condition

:a^p ≡ a (mod p²)

Every Wieferich prime to base a is also a weak Wieferich prime to base a. If the base a is 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, ...

For integer n ≥2, a Wieferich prime to base a with order n is a prime p satisfies the condition

:a^p−1 ≡ 1 (mod pⁿ)

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.

Let P and Q be integers. The Lucas sequence of the first kind associated with the pair (P, Q) is defined by

:$\backslash begin\{align\}$

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}

for all $n\; \backslash geq\; 2$. A Lucas–Wieferich prime associated with (P, Q) is a prime p such that U_p−ε(P, Q) ≡ 0 (mod p²), where ε equals the Legendre symbol $\backslash left(\backslash tfrac\{P^2-4Q\}p\backslash right)$. All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).{{rp|2088}}

Let K be a global field, i.e. a number field or a function field in one variable over a finite field and let E be an elliptic curve. If v is a non-archimedean place of norm q_v of K and a ∈ K, with v(a) = 0 then {{math|1=v(a^q_v − 1 − 1)}} ≥ 1. v is called a Wieferich place for base a, if {{math|1=v(a^q_v − 1 − 1)}} > 1, an elliptic Wieferich place for base P ∈ E, if N_vP ∈ E₂ and a strong elliptic Wieferich place for base P ∈ E if n_vP ∈ E₂, where n_v is the order of P modulo v and N_v gives the number of rational points (over the residue field of v) of the reduction of E at v.{{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 }}{{rp|206}}

• 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

{{Reflist|colwidth=30em}}

• {{citation | first=R. | last=Haussner | title=Über die Kongruenzen {{math|1=2^p−1 − 1}} ≡ 0 (mod p²) für die Primzahlen p=1093 und 3511 | journal=Archiv for Mathematik og Naturvidenskab | volume=39 | issue=5 | year=1926 | language=de | page=7 | id=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^p−1 − 1}} ≡ 0 (mod p²) | 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}}

• {{MathWorld |urlname=WieferichPrime |title=Wieferich prime}}
• [http://go.helms-net.de/math/expdioph/fermatquotients.pdf Fermat-/Euler-quotients {{math|1=(a^p−1 − 1)/p^k}} 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