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

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.

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.

DOI : 10.5802/jtnb.913
Classification : 11D61, 11D41, 11D59
Mots clés : Simplest cubic fields, family of Thue equations, diophantine equations, linear forms of logarithms.
Claude Levesque 1 ; Michel Waldschmidt 2

1 Dép. de mathématiques et de statistique Université Laval, Québec Québec CANADA G1V 0A6
2 UPMC Univ Paris 06 UMR 7586-IMJ 75005 Paris France
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Claude Levesque; Michel Waldschmidt. A family of Thue equations involving powers of units of the simplest cubic fields. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 537-563. doi : 10.5802/jtnb.913. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/

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