Congruences involving Bernoulli polynomials

Let {Bn(x)}{Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for Bj(x)(modpn), where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p  -regular functions, the congruences for h(-sp)(modp)(s=3,5,8,12) and the sum ∑k≡r(modm)(pk), where h(d)h(d) is the class number of the quadratic field Q(d) of discriminant d and p-regular functions are those functions f   such that f(k)(k=0,1,…) are rational p  -integers and ∑k=0n(nk)(-1)kf(k)≡0(modpn) for n=1,2,3,…n=1,2,3,… . We also establish many congruences for Euler numbers.