Bugeaud, Yann; Corvaja, Pietro; Zannier, Umberto An upper bound for the g.c.d. of \(a^n-1\) and \(b^n -1\). (English) Zbl 1021.11001 Math. Z. 243, No. 1, 79-84 (2003). Recently, P. Corvaja and 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 \(\exp(\varepsilon n)\). Reviewer: A. J. van der Poorten (North Ryde) Cited in 14 ReviewsCited in 48 Documents MSC: 11J25 Diophantine inequalities 11D75 Diophantine inequalities 11B37 Recurrences Citations:Zbl 1026.11021 PDFBibTeX XMLCite \textit{Y. Bugeaud} et al., Math. Z. 243, No. 1, 79--84 (2003; Zbl 1021.11001) Full Text: DOI