Note on Wieferich's congruence for primes \(p\equiv 1\pmod 4\)
type: Item
instance of: scholarly article
title: Note on Wieferich's congruence for primes \(p\equiv 1\pmod 4\)
zbMATH Open document ID: 0954.11009
DOI: 10.1007/BF02942562
author: Stanislav Jakubec
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: For primes \(p \equiv 1 \pmod 4\) it is known that the Euler number \(E_{p-1}\) is divisible by \(p\). In this note the author computes \(E_{p-1}/p\) modulo \(p\). Writing \(p\) in the form \(a^2 + 4b^2\) he in fact shows that \[ E_{p-1}/p + 2Q_2 + 2Q_a + 1/(2a^2) \equiv 0 \pmod p, \] where \(Q_n = (n^{p-1}-1)/p\), the Fermat quotient. The rather involved computation is based on the author's previous work, particularly on a result about the Gauss sum attached to a character mod \(p\) of order 4 [Acta Arith. 85, No. 4, 377-388 (1998; Zbl 0912.11041)]. Of special interest is the case that \(Q_2 \equiv 0 \pmod p\), in other words, \(p\) is a Wieferich prime.
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 1416680
zbMATH Keywords: Euler numbers
zbMATH Keywords: power residues
zbMATH Keywords: Fermat quotients
zbMATH Keywords: cyclotomic fields
MaRDI profile type: Publication
cites work: Q5797106
cites work: Note on the congruence of Ankeny-Artin-Chowla type modulo p²
cites work: The congruence for Gauss period
cites work: On Vandiver's conjecture
full work available at URL: https://doi.org/10.1007/bf02942562
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description: scientific article; zbMATH DE number 1416680