The universal Kummer congruences

Details The universal Kummer congruences The universal Kummer congruences

2008-08-26

arxiv.org/abs/0808.3544v1

Mathematics

Number Theory

25 pages

Scientific paper

Let $p$ be a prime. In this paper, we present 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 $p-1$. This strengthens the modulo primes theorems obtained by Clark and recently by Adelberg, Hong and Ren. It also complements Adelberg's modulo prime powers result.

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