On the Kummer-Mirimanoff congruences

The Universal Kummer Congruences

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...

Universal Kummer Congruences Mod Prime Powers

Kummer Type Congruences and Stickelberger Subideals

Searching for Kummer congruences in an infinite slope family

We consider powers of a grossencharacter, the corresponding Lfunctions twisted with quadratic Dirichlet characters, and their central critical values. We state several conjectures concerning Kummer-type congruences between these numbers for a ramified prime and describe specific numerical data in support of these conjectures.

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.