A search for Wieferich and Wilson primes

Computer verification of the Ankeny--Artin--Chowla Conjecture for all primes less than $100000000000$

“…and it has only recently been possible to compute the values of the Wilson quotients w p = (p − 1)! + 1 /p up to 5 × 10 8 (see Crandall, Dilcher and Pomerance [8]). In fact, in all previous attempts to verify the AAC conjecture for all primes < L, the value of u was computed modulo p. We summarize this work in Table 1.1. Ankeny, Artin and Chowla did not provide any algebraic justification that would suggest a negative response to their question.…”

Congruences involving the Fermat quotient

“…Notice that the congruences s(0, For an odd prime p not dividing xyz, A. Wieferich [43] showed that x p +y p +z p = 0 implies q p (2) ≡ 0 (mod p). The only known such primes (the so called Wieferich primes) 1093 and 3511 have long been known, and it was reported in [5] The connection of Fermat quotients with the first case of the Fermat Last Theorem retains its historical interest despite the complete proof of this theorem by A. Wiles in 1995, and Skula's demonstration in 1992 [30] that the failure of the first case of the Fermat Last Theorem would imply the vanishing of many similar sums but with much smaller ranges (sums of Lerch's type which cannot be evaluated in terms of Fermat quotients). Some criteria concerning the first case of the Fermat Last Theorem on Lerch's type sums were established in Ribenboim's book [27], in 1995 by Dilcher and Skula [6] (cf.…”

Around Pelikán’s conjecture on very odd sequences

“…Although it seems very likely there are indeed infinitely many such primes, proving this is quite another matter. A prime being in P 2 is closely related to the Artin primitive root conjecture and a prime p satisfying 2 [5]. Note that 3511 ∈ P 2 \P ′ 2 and that 1093 ∈ ∪ ∞ m=1 (P m \P ′ m ).…”