Torsion points on curves and common divisors of a^k-1 and b^k-1

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63314We 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.Conjecture B and Theorem 3 extended to the case of matrices with arbitrary determinant. Replaced the proof of Theorem 2 with a simpler oneNumber TheoryAlgebraic Geometry11D61Torsion points on curves and common divisors of a^k-1 and b^k-1text