  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 $n$.
