Complete solutions of a family of cubic Thue equations

Alain Togbé

doi:10.5802/jtnb.544

Alain Togbé

Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 285-298.

Résumé
Abstract

Dans cet article, nous utilisons la méthode de Baker, basée sur les formes linéaires en logarithmes, pour résoudre une famille d’équations de Thue liée à une famille de corps de nombres de degré 3. Nous obtenons toutes les solutions de l’équation de Thue

\begin{matrix} Φ_{n} (x, y) & = x^{3} + (n^{8} + 2 n^{6} - 3 n^{5} + 3 n^{4} - 4 n^{3} + 5 n^{2} - 3 n + 3) x^{2} y \\ - (n^{3} - 2) n^{2} x y^{2} - y^{3} = \pm 1, \end{matrix}

pour $n \geq 0$ .

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation

\begin{matrix} Φ_{n} (x, y) & = x^{3} + (n^{8} + 2 n^{6} - 3 n^{5} + 3 n^{4} - 4 n^{3} + 5 n^{2} - 3 n + 3) x^{2} y \\ - (n^{3} - 2) n^{2} x y^{2} - y^{3} = \pm 1, \end{matrix}

for $n \geq 0$ .

Reçu le : 2004-05-18
Publié le : 2008-12-02

MR 2245886 | Zbl 05070458

DOI : https://doi.org/10.5802/jtnb.544

@article{JTNB_2006__18_1_285_0,
     author = {Alain Togb\'e},
     title = {Complete solutions of a family of cubic {Thue} equations},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {285--298},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.544},
     zbl = {05070458},
     mrnumber = {2245886},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.544/}
}

Alain Togbé. Complete solutions of a family of cubic Thue equations. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 285-298. doi : 10.5802/jtnb.544. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.544/

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