Congruences concerning Bernoulli numbers and Bernoulli polynomials Original Research Article

Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummerʹs congruences by determining Bk(p−1)+b(x)/(k(p−1)+b) (mod pn), where p is an odd prime, x is a p-integral rational number and p−1∤b. As applications we obtain explicit formulae for ∑x=1p−1(1/xk) (mod p3), ∑x=1(p−1)/2(1/xk) (mod p3), (p−1)! (mod p3) and Ar(m,p) (mod p), where k∈{1,2,…,p−1} and Ar(m,p) is the least positive solution of the congruence px≡r (mod m). We also establish similar congruences for generalized Bernoulli numbers {Bn,χ}.