Congruences involving Bernoulli and Euler numbers

Abstract

Let [ x ] be the integral part of x. Let p > 5 be a prime. In the paper we mainly determine ∑ x = 1 [ p / 4 ] 1 x k ( mod p 2 ) , ( p − 1 ) ( mod p 3 ) , ∑ k = 1 p − 1 2 k k ( mod p 3 ) and ∑ k = 1 p − 1 2 k k 2 ( mod p 2 ) in terms of Euler and Bernoulli numbers. For example, we have ∑ x = 1 [ p / 4 ] 1 x 2 ≡ ( − 1 ) p − 1 2 ( 8 E p − 3 − 4 E 2 p − 4 ) + 14 3 p B p − 3 ( mod p 2 ) , where E n is the nth Euler number and B n is the nth Bernoulli number.