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Voronoï type congruences for Bernoulli numbers. (English) Zbl 0948.11012
Engel, P. (ed.) et al., Voronoï’s impact on modern science. Book I. Transl. from the Ukrainian. Kyiv: Institute of Mathematics. Proc. Inst. Math. Natl. Acad. Sci. Ukr., Math. Appl. 21(1), 71-98 (1998).
Bernoulli numbers $B_m (m\ge 0)$ are defined by the formal power series expansion $x/(e^x-1)=\sum^{\infty}_{m=0} (B_m/m!)x^m$. These numbers may be also defined by the familiar symbolic notation $(B+1)^m=B_m \ (m\ge 2), \ B_0=1$. It is clear that $B_{m}=0$ if $m\ge 3$ is odd and $(-1)^{m/2-1}B_{m}>0$ if $m\ge 2$ is even. In 1890, while he was still a student, G. F. Voronoï proved that if we express $B_m$ ($m\ge 2$, even) as $B_m=P_m/Q_m$ ($P_m, Q_m\in {\Bbb Z}$, $Q_m>0$) in lowest terms, then $$(b^m - 1)P_m\equiv mQ_m\sum_{j=1}^{N-1} (bj)^{m-1} \left[\frac{bj}{N}\right]\pmod N,$$ where $N, b$ are any positive integers with $(b, N)=1$ and $[x]$ means the greatest integer $\le x$ for a real number $x$. It is needless to say that this is one of the most significant congruences in the theory of Bernoulli numbers. In this survey article, the author looks over a surrounding landscape of Bernoulli numbers and argues wide-ranging subjects (e.g., Fermat’s Last Theorem, Fermat quotients, regular and irregular primes, class numbers of quadratic and cyclotomic fields, $p$-adic $L$-functions and others) in number theory which are deeply connected with the above Voronoï congruence, interweaving historical details. Further, many kinds of generalizations of Voronoï’s congruence devised up to the present by various mathematicians are also introduced in this article. The reviewer believes that this is a well-written survey on Voronoï’s congruence and its applications, and it will be very useful for many readers to understand how Voronoï’s and other congruences of his type have made an important contribution to number theory. For the entire collection see [Zbl 0928.00008].
Reviewer: Takashi Agoh (Noda)
##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11F85 $p$-adic theory, local fields 11-02 Research monographs (number theory)