The Universal Bernoulli Numbers And The Universal Kummer Congruence

Let p be a prime. In this paper, we give generalizations of Wilson's theoremand Adelberg's congruence. We obtain good bounds for the p-adic valuations of thecoefficients of the universal divided Bernoulli number ((B|^)_n)/n when n is divisible by p?1. Asan application, we give a simple proof of Clarke's 1989 universal von Staudt theorem.We establish the universal Kummer congruence modulo p for the universal dividedBernoulli numbers for the case (p - 1)|n which is not currently known.