Complete Resolution Of A Family Of Twisted Thue Equations

Abstract

One of the first infinite families of Thue equations, $$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(λ)$ of Shanks, where $λ$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(λ)/\mathbb{Q}}(X-λ^t Y)=\pm 1,$$ where $t\in \mathbb{Z}$ with $t\neq 0$. In this paper, we find all solutions $(X,Y,n,t)\in \mathbb{Z}^4$ with $n,t\in\mathbb{Z}$ and $t\neq 0$ to this family of twisted Thue equations, thereby answering a question of Levesque and Waldschmidt.