AI 中文总结
本文求解了Lévesque与Waldschmidt提出的一类扭曲Thue方程，找出其所有整数解，解答了相关公开问题。
AI 中文摘要
Thue方程的首个无限族之一为$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$（$n\in \mathbb{Z}$），由Thomas于1990年求解，该族与Shanks的最简三次域$\mathbb{Q}(\lambda)$相关，其中$\lambda$是$F_n(X,1)$的根。Lévesque与Waldschmidt将Thue方程按指数$t$扭曲，研究方程$N_{\mathbb{Q}(\lambda)/\mathbb{Q}}(X-\lambda^t Y)=\pm 1$（$t\in \mathbb{Z}$且$t\neq 0$）。本文找出该扭曲Thue方程族的所有整数解$(X,Y,n,t)$（$n,t\in\mathbb{Z}$，$t\neq 0$），从而解答了Lévesque与Waldschmidt的问题。
英文摘要
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.
Comments28 pages
