 
Summary: UNIVERSAL KUMMER CONGRUENCES MOD PRIME POWERS
ARNOLD ADELBERG
Abstract. We have previously proved Kummer congruences mod primes p
such that p 16 j n for the universal divided Bernoulli numbers ^
Bn=n. In this
paper we strengthen these congruences to hold mod powers of p.
1. Introduction
The strongest form of the classical Kummer congruences says that if p is prime
and p 16 j m and b m = (1 p m 1 )Bm=m, where Bm is a Bernoulli number, then
if is the Euler function and n m mod (p N+1 ), then
(1.1) b n b m mod p N+1 :
This periodic behavior of the divided Bernoulli numbers Bm=m is closely related
to the existence of a padic zeta function (cf. [17]). The factor 1 p m 1 is called an
Euler factor. The congruence is now usually proved by means of padic measures
and padic integration (cf. [22]). As a corollary, we have a congruence without
Euler factors, namely if p 16 j m and n m mod (p N+1 ) and n; m N + 2, then
(1.2) Bn=n Bm=m mod p N+1 :
This congruence is the one that we generalize in this paper to the divided uni
versal Bernoulli numbers ^
Bn=n. As our concluding examples in Section 4 show, the
