Jack Sonn (IIT)
On the sequence {gcd(a^n-1,b^n-1)}, n=1,2,3,... 
and a cyclotomic polynomial generalization.

Abstract:  
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 (multiplicatively independent) 
elements in either the rational integers, 
the polynomials in one variable over the complex numbers, 
or the polynomials in one variable over a finite field. 
In the case of the rational integers, 
Bugeaud, Corvaja and Zannier have obtained an upper bound 
exp(\epsilon n) for any given \epsilon >0 and all large n, 
and demonstrated its sharpness by extracting 
from a paper of Adleman, Pomerance, and Rumely a lower bound
\exp(\exp(c\frac{log n}{loglog n}))
for infinitely many n, where c is an absolute constant. 
The upper bound generalizes immediately to 
gcd(\Phi_N(a^n), \Phi_N(b^n)) for any positive integer N, 
where \Phi_N(x)$ is the Nth cyclotomic polynomial, 
the preceding being the case N=1. 
The lower bound has been generalized in Yossi Cohen's Ph.D. thesis to N=2. 
In this paper we generalize the lower bound for arbitrary N  under GRH 
(the generalized Riemann Hypothesis), 
using an effective version of the Chebotarev density theorem due to 
Lagarias and Odlyzko. 
The analogue of the lower bound result for gcd(a^n-1,b^n-1) over F_q[T] 
was proved by Silverman; 
we prove a corresponding generalization (without GRH).  
(Joint work with Yossi Cohen)