Some Congruences Involving Euler Numbers

Abstract

In this paper, we obtain some explicit congruences for Euler numbers modulo an odd prime power in an elementary way. The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually deflned by the exponential generating functions: te xt e t i 1 = 1 X n=0 Bn(x) t n n! and 2e xt e t + 1 = 1 X n=0 En(x) t n n! :