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

作者：Levesque Claude^*; Waldschmidt Michel

来源：Journal de Theorie des Nombres de Bordeaux, 2015, 27(2): 537-563.

DOI：10.5802/jtnb.913

• Simplest cubic fields
• family of Thue equations
• diophantine equations
• linear forms of logarithms

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)(XY)-Y-2-(n+2)XY2-Y-3 and the family of equations F-n(X, Y) = +/- 1, n is an element of N. This family is associated to the family of the simplest cubic fields 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.

• 出版日期2015