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

TL;DR

This paper investigates the gcd of a^k-1 and b^k-1 for fixed a, b, proposing conjectures in the integer case and proving strong results in the polynomial case, using algebraic geometry tools.

Contribution

It introduces a conjecture about gcd behavior for integers and proves a stronger version for polynomials, applying Lang's theorem on torsion points.

Findings

Conjecture that a^k-1 and b^k-1 are coprime infinitely often for multiplicatively independent a, b.

Proved a strong polynomial analogue of the conjecture.

Extended results to matrix analogues involving gcd of matrix elements.

Abstract

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.