On a conjecture of Levesque and Waldschmidt II

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 $n$.