Torsion Points And

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 on the finiteness of torsion points on algebraic curves. We also give a matrix analogue of these results, where for a nonsingular integer matrix A, we look at the greatest common divisor of the elements of the matrix A k − I.