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

Nir Ailon·Zeev Rudnick
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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.

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Nir Ailon, Zeev Rudnick. 2002-12-17. Torsion points on curves and common divisors of a^k-1 and b^k-1. https://arxiv.org/abs/math/0202102

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