Related to Shank’s notion of simplest cubic fields, the family of parametrised Diophantine equations, \(\begin{aligned} x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - \lambda _0 y\right) \left( x-\lambda _1 y\right) \left( x - \lambda _2 y\right) = \pm 1, \end{aligned}\) was studied and solved effectively by Thomas and later solved completely by Mignotte. An open conjecture of Levesque and Waldschmidt [4] states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at \(\begin{aligned} f_{n,s,t}(x,y) = \left( x - \lambda _0^s \lambda _1^t y \right) \left( x - \lambda _1^s\lambda _2^t y \right) \left( x - \lambda _2^s\lambda _0^t y \right) = \pm 1, \end{aligned}\) 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.
