The universal Kummer congruences

Abstract

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number $\frac{{\hat{B}}_n}{n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers $\frac{{\hat{B}}_n}{n}$ when $n$ is divisible by $p-1$.

20 pages. To appear in Journal of the Australian Mathematical Society

References in corpus (1)

• Bernoulli Numbers, Wolstenholme's Theorem, and p^5 Variations of Lucas' Theorem