A family of Thue equations involving powers of units of the simplest cubic fields

Abstract: E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $$F_n(X, Y )= X³ -(n-1)X^2Y -(n+2)XY² -Y<sup>3$$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(\lambda)$ of D. Shanks, $\lambda$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1)$ of $\lambda$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters $n$ and $a$.