 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 136, Number 1, January 2008, Pages 6171
S 00029939(07)090259
Article electronically published on August 14, 2007
BOUNDS OF DIVIDED UNIVERSAL BERNOULLI NUMBERS
AND UNIVERSAL KUMMER CONGRUENCES
ARNOLD ADELBERG, SHAOFANG HONG, AND WENLI REN
(Communicated by WenChing Winnie Li)
Abstract. Let p be a prime. We obtain good bounds for the padic sizes of the
coefficients of the divided universal Bernoulli number
^Bn
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.
1. Introduction
There are many beautiful and useful congruences in number theory. Some ex
