The universal Kummer congruences

TL;DR

This paper provides a detailed $p$-adic analysis of factorials and Bernoulli numbers, establishing universal Kummer congruences modulo prime powers for certain divided Bernoulli numbers.

Contribution

It introduces new bounds for $p$-adic sizes of coefficients and proves universal Kummer congruences for divided universal Bernoulli numbers when $n$ is divisible by $p-1$.

Findings

Established bounds for $p$-adic coefficients.

Proved universal Kummer congruences modulo prime powers.

Extended classical results to a universal setting.

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 ${{\hat B_n}\over n}$ 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 ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.