Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials
Bernd C. Kellner
TL;DR
This work addresses the problem of evaluating the Wilson quotient $\mathcal{W}_p$ and the factorial $(p-1)!$ modulo high powers of a prime $p$ by extending Wilson's theorem through supercongruences of Fermat-quotient sums. The authors translate these power-sum congruences into a Bernoulli-number framework using the polynomials $\psi_\nu$ and divided Bernoulli numbers $\overline{\mathcal{B}}_n$, enabling compact $p$-adic expansions. Key contributions include explicit expansions for $\mathcal{W}_p$ modulo $p^r$ (up to $r=4$) in terms of $\overline{\mathcal{B}}_n$ and $\overline{\mathcal{B}}_{n,2}$, and by-product results for the power sums $Q_p(n)$. The approach unifies and extends classical congruences (Glaisher, Sun, Levaillant) and provides a practical route to compute Wilson quotients and factorial congruences for arbitrary prime powers with manageable effort.
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 $p^4$.