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About von Staudt congruences for Bernoulli numbers. (English) Zbl 1024.11011

From the text: The author considers the von Staudt type congruences for Bernoulli numbers B n with arbitrary indices n (the case n0 pmod-1 being no exception). The theorems proved generalize well-known results due to H. S. Vandiver, L. Carlitz and others.

Theorem 1. Let p be an odd prime, n=k(p-1)p l-1 , k and l. Then pB n p-1(modp l ) or more exactly

pB n p-1+kp l a=1 p-1 (a p-1 -1)/p(modp l+1 ),p>3·

Theorem 2. Let k,l,t,u and z{0}. Then

δ t+k-1 B hz+u 0(modp lt ),

where h=ϕ(p l )=(p-1)p l-1 , p k (2k+lt)(p-1)+1 and ul(t+k-1). In particular, for k=2 we have

δ t+1 B hz+u 0(modp lt ),

or (in the usual symbolic form)

B hz+u (B h -1) t+1 0(modp lt ),

where plt+3 and ul(t+1). B n denotes the Bernoulli number.

11B68Bernoulli and Euler numbers and polynomials
11A07Congruences; primitive roots; residue systems