Voronoï Type Congruences for Bernoulli Numbers and Its Applications
by
Takashi Agoh
Department of Mathematics, Tokyo University of Science

Let B_m (m >= 0) be the mth Bernoulli number in the even suffix notation defined by the recurrence relation (B+1)^m+1=B_m+1 (m >= 1) with B₀=1. Letting m >= 2 be even and n >= 1, the Voronoï congruence can be stated as

(am - 1)Nm \equiv mDm n-1 å j=1  (aj)m-1 [\fracajn] mod n,

where a >= 1 is an integer with (a, n)=1 and B_m=N_m/D_m (N_m, D_m in Z, D_m > 0) in lowest terms.

In this talk, we shall introduce several Voronoï type congruences and extend some important arithmetic properties that are playing central parts in the theory of Bernoulli numbers to a more general situation making use of these congruences. Further, we shall discuss some open questions related to numerators of B_m/m for even m >= 2.

Date received: April 30, 2004