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arXiv · 2608.20995

Complete Resolution Of A Family Of Twisted Thue Equations

Tobias Hilgart·Carina Premstaller·Volker Ziegler
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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 $\mathbb{Q}(\lambda)$ of Shanks, where $\lambda$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(\lambda)/\mathbb{Q}}(X-\lambda^t Y)=\pm 1,$$ 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.

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Tobias Hilgart, Carina Premstaller, Volker Ziegler. 2026-08-21. Complete Resolution Of A Family Of Twisted Thue Equations. https://arxiv.org/abs/2608.20995

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