The abc conjecture and non-Wieferich primes in arithmetic progressions
type: Item
instance of: scholarly article
title: The abc conjecture and non-Wieferich primes in arithmetic progressions
zbMATH Open document ID: 1272.11014
author: Hester Graves
author: M. Ram Murty
published in: Journal of Number Theory
publication date: 2013-04-15T00:00:00Z
review text: If a prime \(p\) satisfies the congruence \(2^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime and if it satisfies the congruence \(a^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime for base \(a\). As was noted by the authors it can be shown by elementary tools that under the \textbf{abc conjecture} there are infinitely many non-Wieferich primes. In this paper the authors show that under the abc~conjecture the number of non-Wieferich primes is \(\ll \frac{\log x}{\log\log x}\).
reviewed by: Mohamed El Bachraoui
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 6154092
zbMATH Keywords: Wieferich primes
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: abc conjecture
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.10.012
OpenAlex ID: W2015320374
Wikidata QID: Q122891782
DOI: 10.1016/J.JNT.2012.10.012
description: scientific article; zbMATH DE number 6154092