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

##### MSC:
 11J25 Diophantine inequalities 11D75 Diophantine inequalities 11B37 Recurrences
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