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- R. Crandall, K. Dilcher and C. Pomerance, "A search for Wieferich and Wilson primes," Math. Comp., 66:217 (1997) 433--449. MR 97c:11004
An odd prime p is called a Wieferich prime if
2p-1 ≡ 1 (mod p2) ; alternatively, a Wilson prime if
(p-1)!≡ -1 (mod p2) . To date the only known Wieferich primes are p=1093 and 3511, while the only known Wilson primes are p=5, 13, and 563. We report that there exist no new Wieferich primes p< 4× 1012, and no new Wilson primes p< 5× 108. It is elementary that both defining congruences above hold merely (mod p) , and it is sometimes estimated on heuristic grounds that the "probability" that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p) .