UNIVERSAL KUMMER CONGRUENCES MOD PRIME POWERS ARNOLD ADELBERG 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 p-adic zeta function (cf. [17]). The factor 1 p m 1 is called an Euler factor. The congruence is now usually proved by means of p-adic measures and p-adic 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 Collections: Mathematics