  We study the behavior of the greatest common divisor of a^k-1 and b^k-1,
where a,b are fixed integers or polynomials, and k varies. In the integer case,
we conjecture that when a and b are multiplicatively independent and in
addition a-1 and b-1 are coprime, then a^k-1 and b^k-1 are coprime infinitely
often. In the polynomial case, we prove a strong version of this conjecture. To
do this we use a result of Lang's on the finiteness of torsion points on
algebraic curves. We also give a matrix analogue of these results, where for a
unimodular matrix A, we look at the greatest common divisor of the elements of
the matrix A^k-I.
