An upper bound for the g.c.d. of \(a^n-1\) and \(b^n -1\) (Q1566412)

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 \(\exp(\varepsilon n)\).