A family of Thue equations involving powers of units of the simplest cubic fields
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, p. 537-563

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms ${F}_{n}\left(X,Y\right)={X}^{3}-\left(n-1\right){X}^{2}Y-\left(n+2\right)X{Y}^{2}-{Y}^{3}$ and the family of equations ${F}_{n}\left(X,Y\right)=±1$, $n\in ℕ$. This family is associated to the family of the simplest cubic fields $ℚ\left(\lambda \right)$ of D. Shanks, $\lambda$ being a root of ${F}_{n}\left(X,1\right)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial ${F}_{n}\left(X,1\right)$ 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 $a$.

E. Thomas fut l’un des premiers à résoudre une famille infinie d’équations de Thue, lorsqu’il a considéré les formes ${F}_{n}\left(X,Y\right)={X}^{3}-\left(n-1\right){X}^{2}Y-\left(n+2\right)X{Y}^{2}-{Y}^{3}$ et la famille d’équations ${F}_{n}\left(X,Y\right)=±1$, $n\in ℕ$. Cette famille est associée à la famille des corps cubiques les plus simples $ℚ\left(\lambda \right)$ de D. Shanks, $\lambda$ étant une racine de ${F}_{n}\left(X,1\right)$. Nous introduisons dans cette famille un second paramètre en remplaçant les racines du polynôme minimal ${F}_{n}\left(X,1\right)$ de $\lambda$ par les puissances $a$-ièmes des racines et nous résolvons de façon effective la famille d’équations de Thue que nous obtenons et qui dépend maintenant des deux paramètres $n$ et $a$.

DOI : https://doi.org/10.5802/jtnb.913
Classification:  11D61,  11D41,  11D59
Keywords: Simplest cubic fields, family of Thue equations, diophantine equations, linear forms of logarithms.
@article{JTNB_2015__27_2_537_0,
author = {Levesque, Claude and Waldschmidt, Michel},
title = {A family of Thue equations involving powers of units of the simplest cubic fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {27},
number = {2},
year = {2015},
pages = {537-563},
doi = {10.5802/jtnb.913},
mrnumber = {3393166},
language = {en},
url = {http://www.numdam.org/item/JTNB_2015__27_2_537_0}
}

Levesque, Claude; Waldschmidt, Michel. A family of Thue equations involving powers of units of the simplest cubic fields. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 537-563. doi : 10.5802/jtnb.913. http://www.numdam.org/item/JTNB_2015__27_2_537_0/

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