Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients

Abstract

We show that Wilson's theorem as well as the Wilson quotient can be described by supercongruences modulo any higher prime power involving terms of power sums of Fermat quotients. The new approach uses Bell polynomials and Newton's identities relating elementary symmetric polynomials to power sums. This enables us to compute certain multivariate polynomials recursively that are needed to establish the supercongruences. Subsequently, we give a recurrence formula for these polynomials and show further properties.