Congruences involving Bernoulli numbers and Fermat-Euler quotients.
type: Item
instance of: scholarly article
title: Congruences involving Bernoulli numbers and Fermat-Euler quotients.
zbMATH Open document ID: 1034.11014
author: Takashi Agoh
published in: Journal of Number Theory
publication date: 2002-08-15T00:00:00Z
review text: The classical Voronoï congruence for Bernoulli numbers \(B_m\) (in the even suffix notation) expresses the residue mod \(N\) of \((b^m-1)B_m/m\) as a sum of multiples of \((m-1)\)st powers of \(bj\,(j=1,\dots,N-1)\), provided \(b\) and \(N\) are coprime. As a nice supplement to this result the author finds an analogous expression for the residue mod \(p^e\) of \((p^{m-1}-1)B_m/m\), where \(e\geq 1\) and \(p\) is a prime \(\geq 5\) with \(p-1\) not dividing \(m\). This expression contains a similar sum and moreover an additional term depending on the Fermat-Euler quotient \((r^{\varphi(p^e)}-1)/p^e\), where \(r\) is a primitive root mod \(p^e\). As applications the author proves several congruences involving Bernoulli numbers and, occasionally, the Fermat quotient \((r^{p-1}-1)/p\). Some of these are known, but the proofs are different. A particular theme is the connection of the Bernoulli numbers \(B_{(p+1)/2}\) and \(B_{(p-1)/2}\) to the class numbers of quadratic fields.
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1781932
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Fermat quotients
zbMATH Keywords: Fermat-Euler quotients
zbMATH Keywords: class numbers of quadratic fields
MaRDI profile type: Publication
full work available at URL: jnth.2001.2728
OpenAlex ID: W2013658735
cites work: Q3682545
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DOI: 10.1006/JNTH.2001.2728
description: scientific article; zbMATH DE number 1781932