An upper bound for the g.c.d. of \(a^n-1\) and \(b^n -1\)
type: Item
instance of: scholarly article
title: An upper bound for the g.c.d. of \(a^n-1\) and \(b^n -1\)
zbMATH Open document ID: 1021.11001
author: Yann Bugeaud
author: Pietro Corvaja
author: Umberto Zannier
published in: Mathematische Zeitschrift
publication date: 2003-06-02T00:00:00Z
review text: Recently, \textit{P. Corvaja} and \textit{U. Zannier} [Invent. Math. 149, No. 2, 431--451 (2002; Zbl 1026.11021)] proved a remarkable result inter alia entailing that if \(a^n-1\) divides \(b^n-1\) for infinitely many positive integers \(n\) then \(a\) and \(b\) are multiplicatively dependent. Here the authors show that the techniques of the general argument may be applied quantitatively, here to prove that if \(a\) and \(b\) are multiplicatively independent then, for every \(\varepsilon>0\) and \(n\) sufficiently large, \(\gcd(a^n-1,b^n-1)\) is no greater than \(\exp(\varepsilon n)\).
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11D75
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 1922486
MaRDI profile type: Publication
full work available at URL: s00209-002-0449-z
OpenAlex ID: W1983904105
DOI: 10.1007/S00209-002-0449-Z
description: scientific article; zbMATH DE number 1922486