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CDP97

R. Crandall, K. Dilcher and C. Pomerance, "A search for Wieferich and Wilson primes," Math. Comp., 66:217 (1997) 433--449.  MR 97c:11004

Abstract: An odd prime p is called a Wieferich prime if

2^p-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) .