Wieferich prime

Wieferich prime
Named after	Arthur Wieferich
Publication year	1909
Author of publication	Wieferich, A.
Number of known terms	2
Subsequence of	$Crandall numbers [1]$ $Wieferich numbers [2]$ $Lucas-Wieferich primes [3]$ $near-Wieferich primes$
First terms	1093, 3511
Largest known term	3511
OEIS index	A001220

In number theory, a Wieferich prime is a prime number p such that p² divides $2 p - 1 - 1$ ,^[4] therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides $2 p - 1 - 1$ . 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.^[5]^[6]

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 and Fermat numbers, specific types of pseudoprimes 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 fields and the abc conjecture.

As of October 2014, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in OEIS).

Equivalent definitions

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

For p = 11, we get

\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

\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 $2 p -1 \equiv 1 (mod p 2)$ by 2 to get $2 p \equiv 2 (mod p 2)$ . Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies $2 p 2 \equiv2 p \equiv 2 (mod p 2)$ , and hence $2 p k \equiv 2 (mod p 2)$ for all $k \geq 1$ . The converse is also true: $2 p k \equiv 2 (mod p 2)$ for some $k \geq 1$ implies that the multiplicative order of 2 modulo p² divides gcd $(p k - 1$ ,φ $(p 2)) = p - 1$ , that is, $2 p -1 \equiv 1 (mod p 2)$ 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: $ord p 2 2 = ord p 2$ , (By the way, ord₁₀₉₃2 = 364, and ord₃₅₁₁2 = 1755).

H. S. Vandiver proved that $2 p -1 \equiv 1 (mod p 3)$ if and only if $1 + \tfrac{1}{3} + \dots + \tfrac{1}{p-2} \equiv 0 \pmod{p^2}$ .^[7]^:187

History and search status

In 1902, W. F. Meyer proved a theorem about solutions of the congruence a^{p − 1} ≡ 1 (mod p^r).^[8]^:930 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. 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 $\tfrac{2^{p-1}-1}{p}\,\bmod\,p$ . He asked the question when this residue vanishes and tried to find expressions for answering this question.^[9]

The prime 1093 was found to be a Wieferich prime by Waldemar Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of $\tfrac{2^{t}-1}{p}\,\bmod\,p$ for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a conjecture by Grawe about the impossibility of the Wieferich congruence.^[10] E. Haentzschel later ordered verification of the correctness of Meissners congruence via only elementary calculations.^[11]^:664 Inspired by an earlier work of Euler, he simplified Meissners proof by showing that 1093² | (2¹⁸² + 1) and remarked that (2¹⁸² + 1) is a factor of (2³⁶⁴ − 1).^[12] 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,^[13] although Meissner himself hinted at that he was aware of a proof without complex values.^[10]^:665

The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922^[14] and another proof of it being a Wieferich prime was published in 1965 by Guy.^[15] In 1960, Kravitz^[16] doubled a previous record set by Fröberg^[17] and in 1961 Riesel extended the search to 500000 with the aid of BESK.^[18] Around 1980, Lehmer was able to reach the search limit of 6×10⁹.^[19] This limit was extended to over 2.5×10¹⁵ in 2006,^[20] finally reaching 3×10¹⁵. It is now known, that if any other Wieferich primes exist, they must be greater than 6.7×10¹⁵.^[21] The search for new Wieferich primes is currently performed by the distributed computing project Wieferich@Home. In December 2011, another search was started by the PrimeGrid project.^[22] As of November 2015, PrimeGrid has extended the search limit to over 4.9×10¹⁷ and continues.^[23]

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 $(p - 1)-th$ degree roots of unity modulo p² are uniformly distributed in the multiplicative group of integers modulo p².^[24]

Properties

Connection with Fermat's last theorem

The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:^[25]

Let p be prime, and let x, y, z be integers such that

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)^[26]^[27] 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.^[28] In 1910, Mirimanoff expanded^[29] the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must also divide $3 p - 1 - 1$ . Granville and Monagan further proved that p² must actually divide $m p - 1 - 1$ for every prime m ≤ 89.^[30] Suzuki extended the proof to all primes m ≤ 113.^[31]

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).^[30]^: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.^[30]^:333 Granville and Monagan showed that (1, 1) ∈ H_p if and only if p is a Wieferich prime.^[30]^:333

Connection with the abc conjecture and non-Wieferich primes

A non-Wieferich prime is a prime p satisfying $2 p - 1 ≢ 1 (mod p 2)$ . J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.^[32] 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.^[33]^: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,^[34] are complementary sets, so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers. 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.^[35] Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers^[36] 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 $\tfrac{\log n}{\log \gamma (n)}$ and $\gamma (n) = \prod_{p \mid n} p$ , meaning $\gamma (n)$ gives the product of all prime factors of n.^[34]^:4

Connection with Mersenne and Fermat primes

It is known that the nth Mersenne number $M n = 2 n - 1$ is prime only if n is prime. Fermat's little theorem implies that if $p > 2$ is prime, then M_p−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.^[37]

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.^[38]

Similarly, if p is prime and p² divides some Fermat number F_n $= 2 2 n + 1$ , then p must be a Wieferich prime.^[39]

In fact, if and only if there exists a natural number n and a prime p that p² divides $\Phi_n(2)$ (where $\Phi$ is Cyclotomic polynomial), then p is a Wieferich prime. For example, 1093² divides $\Phi_{364}(2)$ , 3511² divides $\Phi_{1755}(2)$ . Mersenne and Fermat numbers are just special situations of $\Phi_n(2)$ . Thus, if 1093 and 3511 are only two Wieferich primes, then all $\Phi_n(2)$ are square-free except $\Phi_{364}(2)$ and $\Phi_{1755}(2)$ (In fact, when there exists a prime p which p² divides some $\Phi_n(2)$ , then it is a Wieferich prime); and clearly, if $\Phi_n(2)$ is a prime, then it cannot be Wieferich prime. (Notice any odd prime p divides only one $\Phi_n(2)$ and n divides $p - 1$ , and if and only if the period length of 1/p in binary is n, then p divides $\Phi_n(2)$ . 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.^[40]

Connection with other equations

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.^[41]^{: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.^[42]^:258

Binary periodicity of p − 1

Johnson observed^[43] 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 searches 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.^[44]

Abundancy of p − 1

It has been noted (sequence A239875 in OEIS) that the known Wieferich primes are one greater than mutually friendly numbers (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 non-square free base-2 Fermat pseudoprimes up to 25×10⁹.^[45] Later computations showed that the only repeated factors of the pseudoprimes up to 10¹² are 1093 and 3511.^[46] In addition, the following connection exists: Let n be a base 2 pseudoprime and p be a prime divisor of n. If $\tfrac{2^{n-1}-1}{n}\not\equiv 0 \pmod{p}$ , then also $\tfrac{2^{p-1}-1}{p}\not\equiv 0 \pmod{p}$ .^[28]^:378 Furthermore if p is a Wieferich prime, then p² is a Catalan pseudoprime.^[47]

Connection with directed graphs

For all primes up to $100000 L (p n +1) = L (p n)$ only for two cases: $L (1093 2) = L (1093) = 364$ and $L (3511 2) = L (3511) = 1755$ , where m is the modulus of the doubling diagram and $L (m)$ gives the number of vertices in the cycle of 1. The term doubling diagram refers to the directed graph with 0 and the natural numbers less than m as vertices with arrows pointing from each vertex x to vertex 2x reduced modulo m.^[48]^:74 It was shown, that for all odd prime numbers either $L (p n +1) = p \cdot L (p n)$ or $L (p n +1) = L (p n)$ .^[48]^:75

Properties related to number fields

It was shown that $\chi_{D_{0}} \big(p \big) = 1$ and $\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1$ if and only if $2 p - 1 ≢ 1 (mod p 2)$ where p is an odd prime and $D_{0} < 0$ is the fundamental discriminant of the imaginary quadratic field $\mathbb{Q} \big(\sqrt{1 - p^2} \big)$ . Furthermore the following was shown: Let p be a Wieferich prime. If $p \equiv 3 (mod 4)$ , let $D_{0} < 0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb{Q} \big(\sqrt{1 - p} \big)$ and if $p \equiv 1 (mod 4)$ , let $D_{0} < 0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb{Q} \big(\sqrt{4 - p} \big)$ . Then $\chi_{D_{0}} \big(p \big) = 1$ and $\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1$ (χ and λ in this context denote Iwasawa invariants).^[49]^:27

Furthermore the following result was obtained: Let q be an odd prime number, k and p are primes such that $p = 2 k + 1,$ $k \equiv 3 (mod 4),$ $p \equiv -1 (mod q),$ $p ≢ -1 (mod q 3)$ and the order of q modulo k is $\tfrac{k-1}{2}$ . Assume that q divides h⁺, the class number of the real cyclotomic field $\mathbb{Q} \big( \zeta\,\!_p + \zeta\,\!_p^{-1} \big)$ , 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.^[50]^:55 This also holds if the conditions $p \equiv -1 (mod q)$ and $p ≢ -1 (mod q 3)$ are replaced by $p \equiv -3 (mod q)$ and $p ≢ -3 (mod q 3)$ as well as when the condition $p \equiv -1 (mod q)$ is replaced by $p \equiv -5 (mod q)$ (in which case q is a Wall−Sun−Sun prime) and the incongruence condition replaced by $p ≢ -5 (mod q 3)$ .^[51]^:376

Generalizations

Near-Wieferich primes

A prime p satisfying the congruence 2^(p−1)/2 $\equiv \pm1 + Ap$ (mod p²) with small |A| is commonly called a near-Wieferich prime (sequence A195988 in OEIS).^[24]^[52] 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.^[21]^[53] The following table lists all near-Wieferich primes with |A| ≤ 10 in the interval [1×10⁹, 3×10¹⁵].^[54] This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.^[20]^[55]

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

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^[21] used a different definition of a near-Wieferich prime, defining it as a prime p with small value of $\left|\tfrac{\omega(p)}{p}\right|$ where $\omega(p)=\tfrac{2^{p-1}-1}{p}\,\bmod\,p$ 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 ≤ $6.7 \times 1015$ with $\left|\tfrac{\omega(p)}p\right|\leq 10^{-14}$ .

p	$\omega(p)$	$\left\|\tfrac{\omega(p)}{p}\right\|\times 10^{14}$
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 $2^{(p-1)/2}\equiv \pm 1 + Ap \pmod{p^2}$ , then by squaring, clearly $2^{p-1}\equiv 1 \pm 2Ap \pmod{p^2}$ . So if $A$ had been chosen with $|A|$ small, then clearly $|\pm 2A|=2|A|$ is also (quite) small, and an even number. However, when $\omega(p)$ is odd above, the related $A$ from before the last squaring was not "small". For example with $p=3167939147662997$ , we have $2^{(p-1)/2}\equiv -1 - 1583969573831490p \pmod{p^2}$ which reads extremely non-near, but after squaring this is $2^{p-1}\equiv 1 - 17p \pmod{p^2}$ which is a near-Wieferich by the second definition.

Base-a Wieferich primes

Main article: Fermat quotient

A Wieferich prime base a is a prime p that satisfies

a p - 1 \equiv 1 (mod p 2)

.^[8]

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 $a p -1 \equiv 1 (mod q)$ , $a q -1 \equiv 1 (mod p)$ , then $a pq -1 \equiv 1 (mod pq)$ . Setting p = q leads to $a p 2 -1 \equiv 1 (mod p 2)$ .^[56]^:284 It was shown that $a p 2 -1 \equiv 1 (mod p 2)$ if and only if $a p -1 \equiv 1 (mod p 2)$ .^[56]^:285–286

Known solutions of $a p -1 \equiv 1 (mod p 2)$ for small values of a are:^[57] (checked up to 5 × 10¹³)

a	primes p such that a^{p − 1} = 1 (mod p²)	OEIS sequence
1	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)	A000040
2	1093, 3511, ...	A001220
3	11, 1006003, ...	A014127
4	1093, 3511, ...
5	2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...	A123692
6	66161, 534851, 3152573, ...	A212583
7	5, 491531, ...	A123693
8	3, 1093, 3511, ...
9	2, 11, 1006003, ...
10	3, 487, 56598313, ...	A045616
11	71, ...
12	2693, 123653, ...	A111027
13	2, 863, 1747591, ...	A128667
14	29, 353, 7596952219, ...	A234810
15	29131, 119327070011, ...	A242741
16	1093, 3511, ...
17	2, 3, 46021, 48947, 478225523351, ...	A128668
18	5, 7, 37, 331, 33923, 1284043, ...	A244260
19	3, 7, 13, 43, 137, 63061489, ...	A090968
20	281, 46457, 9377747, 122959073, ...	A242982
21	2, ...
22	13, 673, 1595813, 492366587, 9809862296159, ...
23	13, 2481757, 13703077, 15546404183, 2549536629329, ...	A128669
24	5, 25633, ...
25	2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...
26	3, 5, 71, 486999673, 6695256707, ...
27	11, 1006003, ...
28	3, 19, 23, ...
29	2, ...
30	7, 160541, 94727075783, ...

For more information, see^[58]^[59]^[60] and.^[61] (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 $n p -1 \equiv 1 (mod p 2)$ 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) (sequence A039951 in OEIS)

There are no known solutions of $n p -1 \equiv 1 (mod p 2)$ for n = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 347, 355, 435, 454, 542, 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 infinity many solutions of $a p -1 \equiv 1 (mod p 2)$ for every natural number a.

The bases b < p² which p is a Wieferich prime are (for b > p², the solutions are just shifted by k·p² for k > 0), and there are $p - 1$ solutions < p² of p and the set of the solutions congruent to p are {1, 2, 3, ..., $p - 1})$ (sequence A143548 in OEIS)

p	values of b < p²
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, ... (sequence A039678 in OEIS)

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

Known solutions for small a are: (checked up to 4 × 10¹¹) ^[62]

$a$	primes $p$ such that $(a+1)^{p-1}-a^{p-1} \equiv 0 \pmod{p^2}$
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, ...

Wieferich pairs

Main article: 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 $p = 1093$ . There are only 7 known Wieferich pairs.^[63]

(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequences A124121, A124122 and A126432 in OEIS)

Wieferich sequence

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}.

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)

Wieferich numbers

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, … (sequence A077816 in OEIS)

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 105 Wieferich numbers, which matches the number of Wieferich numbers currently known.^[2]

More generally, a natural number n is a Wieferich number to base a, if a^{$φ$ (n)} ≡ 1 (mod n²).^[64]^:31

Another definition specifies a Wieferich number as odd natural number n such that n and $\tfrac{2^m-1}{n}$ are not coprime, where m is the multiplicative order of 2 modulo n. The first of these numbers are:^[65]

21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, … (sequence A182297 in OEIS)

As above, if Wieferich number q is prime, then it is a Wieferich prime.

Lucas–Wieferich primes

A Lucas–Wieferich prime associated with the pair of integers (P, Q) is a prime p such that U_p−ε(P, Q) ≡ 0 (mod p²), where U_n(P, Q) denotes the Lucas sequence of the first kind and ε equals the Legendre symbol $\left(\tfrac{P^2-4Q}p\right)$ . All Wieferich primes are Lucas-Wieferich primes associated with the pair (3, 2).^[3]^:2088

Fibonacci–Wieferich primes

Let Q = −1, P be any natural number, these primes are called P-Fibonacci–Wieferich primes or P-Wall–Sun–Sun primes, and if P = 1, they are called Fibonacci-Wieferich primes, and if P = 2, they are called Pell–Wieferich primes. For example, 241 is a Wieferich prime when P = 3, so it is a 3-Fibonacci-Wieferich prime or 3-Wall–Sun–Sun prime. In fact, 3 is an n-Fibonacci-Wieferich prime if and only if n congruent to 0, 4, or 5 (mod 9), like the traditional Wieferich primes, 3 is a base n Wieferich prime if and only if n congruent to 1 or 8 (mod 9).

Wieferich places

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 $v (a q v - 1 - 1)$ ≥ 1. v is called a Wieferich place for base a, if $v (a q v - 1 - 1)$ > 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.^[66]^:206

References

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↑ PrimeGrid Announcement of Wieferich and Wall-Sun-Sun searches
↑ PrimeGrid Wieferich prime search server statistics
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↑ About project Wieferich@Home
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↑ Wieferich primes to base 10125
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External links

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 100 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

List of prime numbers

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