In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 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. 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 2026, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).
History and search status
In 1902, Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr). 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 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). In 1913, Bachmann examined the residues of 2 p − 1 − 1 p mod p {\displaystyle {\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. The prime 1093 was found to be a Wieferich prime by W. Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of 2 t − 1 p mod p {\displaystyle {\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p} for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a conjecture by Grave about the impossibility of the Wieferich congruence. E. Haentzschel later ordered verification of the correctness of Meissner's congruence via only elementary calculations. 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). 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, although Meissner himself hinted at that he was aware of a proof without complex values. The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922 and another proof of it being a Wieferich prime was published in 1965 by Guy. In 1960, Kravitz doubled a previous record set by Fröberg and in 1961 Riesel extended the search to 500000 with the aid of the computer BESK. Around 1980, Lehmer was able to reach the search limit of 6×109. This limit was extended to over 2.5×1015 in 2006, finally reaching 3×1015. Eventually, it was shown that if any other Wieferich primes exist, they must be greater than 6.7×1015. In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home. In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted. 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. The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 264 (about 18×1018). 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.
Properties
Connection with Fermat's Last Theorem The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909:
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