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Voronoi’s congruence via Bernoulli distributions. (English) Zbl 0543.10012

The author proves two congruences modulo an arbitrary positive integer \(N\) for Bernoulli numbers \(B_{2k}\), \(k\geq 1\). Let \(c\) be a rational number with numerator and denominator prime to \(N\). The first congruence is essentially Voronoi’s congruence, where the integral parameter is replaced by \(c\). The second congruence reads \[ (c-1) B_{2k} N/2\equiv \sum^{N-1}_{s=1}s^{2k} [sc/N]\pmod N. \] Cf. the author [J. Number Theory 16, 87–94 (1983; Zbl 0507.10008)].

MSC:

11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 0507.10008
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References:

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