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Zbl 0948.11012
Porubský, Štefan
Vorono\" i type congruences for Bernoulli numbers. (English)
[A] Engel, P. (ed.) et al., Vorono\" i'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). ISBN 966-02-0643-7

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. \par In 1890, while he was still a student, G.~F.~Vorono\" i 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. \par 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 Vorono\" i congruence, interweaving historical details. Further, many kinds of generalizations of Vorono\" i's congruence devised up to the present by various mathematicians are also introduced in this article. \par The reviewer believes that this is a well-written survey on Vorono\" i's congruence and its applications, and it will be very useful for many readers to understand how Vorono\" i's and other congruences of his type have made an important contribution to number theory.
[Takashi Agoh (Noda)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11F85 p-adic theory, local fields
11-02 Research monographs (number theory)

Keywords: Bernoulli numbers; generalized Bernoulli numbers; Vorono\" i's congruence; Kummer's congruence; Raabe's formula; Vandiver's congruence; $p$-adic $L$-functions; class numbers of quadratic and cyclotomic fields

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