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

Zbl 0507.10008
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References:

 [1] W. Johnson: p-adic proofs of congruences for the Bernoulli numbers. J. Number Th. 7 (1975), 251-265. · Zbl 0308.10006 [2] O. Grün: Eine Kongruenz für Bernoullische Zahlen. Jahresber. d. Deutschen Math. Verein. 50 (1940), 111-112. · Zbl 0023.20302 [3] S. Lang: Cyclotomic Fields. Springer-Verlag, New York 1978. · Zbl 0395.12005 [4] J. Uspenski, M. Heaslet: Elementary Number Theory. McGraw-Hill, New York 1939. · Zbl 0022.30602 [5] J. Slavutskij: Generalized Voronoi’s congurence and the number of classes of ideals of an imaginary quadratic field II. (Russian), Izv. Vyšš. Učebn. Zavedenij, Math. 4 (53) (1966), 118-126. [6] H. S. Vandiver: Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat’s quotient and Bernoulli numbers. Ann. Math. 18 (1917), 105-114. · JFM 46.1444.03 [7] H. S. Vandiver: On Bernoulli numbers and Fermat’s last theorem. Duke Math. J. 3 (1937), 569-584. · Zbl 0018.00505 [8] G. F. Voronoi: On Bernoulli numbers. (Russian), Commen. Charkov Math. Soc. 2 (1890), 129-148; or in Collected Papers, Vol. I, Publ. House Of the Ukrainian Acad. Sci., Kiev 1952.
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