An upper bound for the g.c.d. of \(a^n-1\) and \(b^n -1\). (English) Zbl 1021.11001

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


11J25 Diophantine inequalities
11D75 Diophantine inequalities
11B37 Recurrences


Zbl 1026.11021
Full Text: DOI