The Universal Kummer Congruences
نویسندگان
چکیده
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 B̂n 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 B̂n 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.
منابع مشابه
Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences
Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p− 1. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the case (p − 1)|n, which is a new result.
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