zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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)
11B68Bernoulli and Euler numbers and polynomials
11F85$p$-adic theory, local fields
11-02Research monographs (number theory)