A family of Thue equations involving powers of units of the simplest cubic fields. (English. French summary) Zbl 1395.11059
Summary: E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms \(F_n(X, Y) = X^3 -(n - 1) X^2 Y -(n + 2) X Y^2 - Y^3\) and the family of equations \(F_n(X, Y) = \pm 1\), \(n \in \mathbb{N}\). This family is associated to the family of the simplest cubic fields \(\mathbb{Q}(\lambda)\) 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 \(a\).
MSC:
| 11D59 | Thue-Mahler equations |
| 11D25 | Cubic and quartic Diophantine equations |
| 11J86 | Linear forms in logarithms; Baker’s method |
Keywords:
simplest cubic fields; family of Thue equations; Diophantine equations; linear forms of logarithmsReferences:
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