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 (X,Y)=X 3 -(n-1)X 2 Y-(n+2)XY 2 -Y 3 and the family of equations F n (X,Y)=±1, n. This family is associated to the family of the simplest cubic fields (λ) of D. Shanks, λ 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 λ 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 (X,Y)=X 3 -(n-1)X 2 Y-(n+2)XY 2 -Y 3 et la famille d’équations F n (X,Y)=±1, n. Cette famille est associée à la famille des corps cubiques les plus simples (λ) de D. Shanks, λ étant une racine de F n (X,1). Nous introduisons dans cette famille un second paramètre en remplaçant les racines du polynôme minimal F n (X,1) de λ 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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