On a conjecture of Levesque and Waldschmidt II

TL;DR

This paper confirms a conjecture that twisted parametrized Diophantine equations related to simplest cubic fields only have non-trivial solutions when the twisting exponents are small, extending previous results to a doubly-twisted case.

Contribution

It proves that the conjecture by Levesque and Waldschmidt holds for the doubly-twisted family of equations when the exponents are not too large.

Findings

Non-trivial solutions only occur for small exponents s, t.

The conjecture is confirmed for the doubly-twisted case.

Results extend previous work on singly-twisted equations.

Abstract

Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations, \[ x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - \lambda_0 y\right) \left(x-\lambda_1 y\right) \left(x - \lambda_2 y\right) = \pm 1, \] was studied and solved effectively by Thomas and later solved completely by Mignotte. An open conjecture of Levesque and Waldschmidt states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at \[ f_{n,s,t}(x,y) = \left( x - \lambda_0^s \lambda_1^t y \right) \left( x - \lambda_1^s\lambda_2^t y \right) \left( x - \lambda_2^s\lambda_0^t y \right) = \pm 1, \] retains a result similar to what Thomas obtained in the original or Levesque and Waldschidt in the once-twisted ($t = 0$) case; namely, that non-trivial solutions can only appear in equations where the parameters are small. We confirm this conjecture, given that the absolute values of the exponents $s, t$ are not too large compared to the base parameter $n$.