Congruences for Bernoulli numbers and Bernoulli polynomials
type: Item
instance of: scholarly article
title: Congruences for Bernoulli numbers and Bernoulli polynomials
zbMATH Open document ID: 0872.11012
DOI: 10.1016/S0012-365X(97)81050-3
published in: Discrete Mathematics
publication date: 1997-10-15T00:00:00Z
review text: The Bernoulli numbers and polynomials are defined by \(B_0=1\), \(\sum^{n-1}_{k=0}{n\choose k} B_k= 0\) \((n=2,3,\dots)\) and \(B_n(x)= \sum^n_{k=0}{n\choose k} B_{n-k} x^k\), respectively. Two basic congruences for Bernoulli numbers are the Kummer congruences (used in the theory of Fermat's last theorem) and the von Staudt-Clausen theorem. There exist many extensions and generalizations for these two theorems. The author obtains a new generalization of the von Staudt-Clausen theorem, by proving that \[ \begin{multlined} pB_{k(p-1)+b}(x)- p^{k(p-1)+b} B_{k(p-1)+b}\Biggl({x+\langle-x\rangle\over p}\Biggr)\equiv\\ a_nk^n+ a_{n-1}k^{n-1}+\cdots+ a_1k+a_0\pmod{p^n}\quad (k=0,1,\dots),\end{multlined} \] where \(p\) is a prime; \(b\in\mathbb{Z}^+\cup\{0\}\); \(x\in\mathbb{Z}_p\); \(\langle-x\rangle\) is the least nonnegative residue of \(-x \text{mod }p\); and \(a_0,a_1,\dots,a_n\) are all (explicit) integers. For the reader's convenience we quote the excellent work by \textit{K. Dilcher}, \textit{L. Skula} and \textit{I. Sh. Slavutskij} [Bernoulli numbers. Bibliography (1713-1990). (1991; Zbl 0741.11001)].
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 1018625
zbMATH Keywords: Bernoulli polynomials
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: congruences
zbMATH Keywords: von Staudt-Clausen theorem
author: Zhihong Sun
reviewed by: József Sándor
MaRDI profile type: Publication
cites work: A Divisibility Property of the Bernoulli Polynomials
cites work: Some Congruences for the Bernoulli Numbers
cites work: Q3272544
cites work: On representations of the braid group
cites work: Q4001342
cites work: Bernoulli Numbers Modulo 27000
cites work: Q4022800
cites work: Q3939864
cites work: Q3715182
cites work: Certain congruences involving the Bernoulli numbers
cites work: Q3282037
cites work: Q3943047
full work available at URL: s0012-365x(97)81050-3
OpenAlex ID: W2095478631
Recommended article: A general family of congruences for Bernoulli numbers
Recommended article: Q3715182
Recommended article: Q4696673
Recommended article: Q4951548
Recommended article: General congruences for Bernoulli polynomials.
Recommended article: Q4951551
Recommended article: Q3788079
Recommended article: Congruences concerning Bernoulli numbers and Bernoulli polynomials
Recommended article: On the Kummer system of congruences and the Fermat quotients
Recommended article: The distribution of Bernoulli numbers modulo primes
description: scientific article; zbMATH DE number 1018625