History and Search Status
In 1902 W. F. 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 . 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 Waldemar Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of 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. E. Haentzschel later ordered verification of the correctness of Meissners congruence via only elementary calculations. 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). 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 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. It is now known, that if any other Wieferich primes exist, they must be greater than 6.7×1015. 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. As of May 2012, PrimeGrid has extended the search limit to 17×1015 and continues.
Chris Caldwell conjectured that only a finite number of Wieferich primes exist. It has also 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.
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