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THE UNIVERSAL KUMMER CONGRUENCES
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AbstractLet $p$ be a prime. In this paper, we present a detailed $p$-adic analysis on 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 ${B}_{n} / n$ 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 ${B}_{n} / n$ when $n$ is divisible by $p- 1$.
Cambridge University Press (CUP)
Title: THE UNIVERSAL KUMMER CONGRUENCES
Description:
AbstractLet $p$ be a prime.
In this paper, we present a detailed $p$-adic analysis on 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 ${B}_{n} / n$ 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 ${B}_{n} / n$ when $n$ is divisible by $p- 1$.
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