Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials

Abstract

By recent work of the author, Wilson's theorem as well as the Wilson quotient can be described by supercongruences of power sums of Fermat quotients modulo every higher prime power. We translate these congruences into congruences of power sums and Bernoulli numbers. This together provides relatively short proofs of the congruences compared to former approaches. As an application, we compute, e.g., the Wilson quotient up to modulo $p^4$ and equivalently the factorial $(p-1)!$ up to modulo $p^5$, which can be extended to any higher prime power with some effort. As a by-product, we determine some power sums of the Fermat quotients up to modulo $p^4$.