Von Staudt congruences for Bernoulli numbers
by
I. Sh. Slavutskii
Akko, Israel

Let h = j(p^l), k, l, n in \bold N and p a prime number. We consider the von Staudt type congruences for Bernoulli numbers B_n with arbitrary indices n (not excepting the case n \equiv 0 mod p-1). The proved theorems generalize the well-known results due to H. S. Vandiver, L. Carlitz and others. As an application we obtain the congruence

pBhk \equiv p-1 + kpl wp mod pl+1,        p >= 5, wp = ((p-1)! + 1)/p,

from which it is possible to obtain several classical results due to N. W. G. H. Beeger, M. Lerch and E. Lehmer (e.g., the survey [1]), and the generalized Sun's congruence [2]

pBhk \equiv pB2hk(k-1)/2 - pBh k(k-2) + (p-1)(k-1)(k-2)/2 mod p2l+1,        p >= l+3.

References

1. Agoh, T.: On Fermat and Wilson quotients. Expos. Math. 14 (1996), 145-170.
2. Sun, Z.-H.: Congruences for Bernoulli numbers and Bernoulli polynomials. Discrete Math. 163 (1997), 153-163.

Date received: February 19, 1999