On the divisibility of Fermat quotients
type: Item
instance of: scholarly article
title: On the divisibility of Fermat quotients
zbMATH Open document ID: 1223.11116
author: Kevin Ford
author: Igor E. Shparlinski
author: Jean Bourgain
author: Sergei Konyagin
published in: Michigan Mathematical Journal
publication date: 2010-09-28T00:00:00Z
review text: For a prime \(p\) and an integer \(a\) the Fermat quotient is defined as \[ q_p(a)=\frac{a^{p-1}-1}{p}. \] Let \(l_p\) be the smallest value of \(a\) for which \(q_p(a)\not\equiv 0\pmod p\). The authors prove that \[ l_p\leq (\log p)^{\frac{463}{252}+o(1)} \] as \(p\to\infty\). As a corollary, for every \(\varepsilon>0\) and a sufficiently large integer \(n\), if \(a^{n-1}\equiv 1\pmod n\) for every positive integer \(a\leq (\log n)^{\frac{463}{252}+\varepsilon}\) then \(n\) is squarefree. Furthermore, for every \(\varepsilon>0\), there is \(\delta>0\) such that for all but one prime \(Q^{1-\delta} <p\leq Q\), we have \(l_p\leq (\log p)^{\frac{59}{35}+\varepsilon}\). Finally, for every \(\varepsilon>0\), there is \(\delta>0\) such that for all but \(O(Q^{1-\delta})\) primes \(p\leq Q\), we have \(l_p\leq (\log p)^{\frac{5}{3}+\varepsilon}\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 5792255
zbMATH Keywords: Fermat quotient
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1281531459
OpenAlex ID: W2093265813
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DOI: 10.1307/MMJ/1281531459
description: scientific article; zbMATH DE number 5792255