[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.
The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation 2p -1 ≡ 1 (mod p2).
[7]: 187 In 1902, Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr).
[8]: 930 [9] 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.
[10] In other words, if there exist solutions to xp + yp + zp = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2p − 1 ≡ 1 (mod p2).
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.
[12] E. Haentzschel [de] later ordered verification of the correctness of Meissner's congruence via only elementary calculations.
[13]: 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).
[14] 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,[15] although Meissner himself hinted at that he was aware of a proof without complex values.
[17] In 1960, Kravitz[18] doubled a previous record set by Fröberg [sv][19] and in 1961 Riesel extended the search to 500000 with the aid of BESK.
[25] While these projects reached search bounds above 1×1017, neither of them reported any sustainable results.
In 2020, PrimeGrid started another project that searched for Wieferich and 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.
[26] The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 264 (about 18×1018).
[34] Let Hp 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.
[33]: 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 Hp.
J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.
[35] 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.
[38] Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers[39] 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
A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free.
[43] Scott and Styer showed that the equation px – 2y = d has at most one solution in positive integers (x, y), unless when p4 | 2ordp 2 – 1 if p ≢ 65 (mod 192) or unconditionally when p2 | 2ordp 2 – 1, where ordp 2 denotes the multiplicative order of 2 modulo p.[44]: 215, 217–218 They also showed that a solution to the equation ±ax1 ± 2y1 = ±ax2 ± 2y2 = c must be from a specific set of equations but that this does not hold, if a is a Wieferich prime greater than 1.25 x 1015.
[45]: 258 Johnson observed[46] 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.
[47] It has been noted (sequence A239875 in the OEIS) 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 25×109.
[49] In addition, the following connection exists: For all primes p up to 100000, L(pn+1) = L(pn) only in two cases: L(10932) = L(1093) = 364 and L(35112) = L(3511) = 1755, where L(m) is the number of vertices in the cycle of 1 in the doubling diagram modulo m. Here the doubling diagram represents the 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.[50]: 74 It was shown, that for all odd prime numbers either L(pn+1) = p · L(pn) or L(pn+1) = L(pn).
Assume that q divides h+, the class number of the real cyclotomic field
The sign +1 or -1 above can be easily predicted by Euler's criterion (and the second supplement to the law of quadratic reciprocity).
[58]: 285–286 Known solutions of ap−1 ≡ 1 (mod p2) for small values of a are:[59] (checked up to 5 × 1013) For more information, see[60][61][62] and.
[65] 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.