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There has been interest during the last decade in properties of the sequence{gcd(a^n-1,b^n-1)}, n=1,2,3,..., where a,b are fixed (multiplicativelyindependent) elements in either the rational integers, the polynomials in onevariable over the complex numbers, or the polynomials in one variable over afinite field. In the case of the rational integers, Bugeaud, Corvaja andZannier have obtained an upper bound exp(\epsilon n) for any given \epsilon >0and all large n, and demonstrate its approximate sharpness by extracting from apaper of Adleman, Pomerance, and Rumely a lower bound \exp(\exp(c\frac{logn}{loglog n})) for infinitely many n, where c is an absolute constant. Theupper bound generalizes immediately to gcd(\Phi_N(a^n), \Phi_N(b^n)) for anypositive integer N, where \Phi_N(x)$ is the Nth cyclotomic polynomial, thepreceding being the case N=1. The lower bound has been generalized in the firstauthor's Ph.D. thesis to N=2. In this paper we generalize the lower bound forarbitrary N but under GRH (the generalized Riemann Hypothesis). The analogue ofthe lower bound result for gcd(a^n-1,b^n-1) over F_q[T] was proved bySilverman; we prove a corresponding generalization (without GRH).

On GCD(\Phi_N(a^n),\Phi_N(b^n))