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arXiv · 1505.06708

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

Claude Levesque·Michel Waldschmidt
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Abstract

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ 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 $a$.

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Claude Levesque, Michel Waldschmidt. 2015-05-25. A family of Thue equations involving powers of units of the simplest cubic fields. https://arxiv.org/abs/1505.06708

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