1997/01/01 by Richard E. Crandall, Karl Dilcher, Carl Pomerance · 2 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #History and Theory of Mathematics #Mathematics #Congruence relation #Prime (order theory) #Fermat's Last Theorem #Quotient #Combinatorics
paper · pdf · doi:10.1090/s0025-5718-97-00791-6
openalex publication_date 1997/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
An odd prime p is called a Wieferich prime if 2p-1 ≡ 1 \pmod p2; alternatively, a Wilson prime if (p-1)! ≡ -1 \pmod 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 âprobability" 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).