To appear in J. Symbolic Comput. ALL SOLUTIONS TO THOMAS’ FAMILY OF THUE EQUATIONS OVER IMAGINARY QUADRATIC NUMBER FIELDS

To appear in J. Symbolic Comput. THOMAS’ FAMILY OF THUE EQUATIONS OVER IMAGINARY QUADRATIC FIELDS

We consider the family of relative Thue equations x − (t− 1)xy − (t+ 2)xy − y = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. We prove that there are only trivial solutions (with |x|, |y| ≤ 1), if |t| is large enough or if the discriminant of the quadratic number field is large enough or if Re t = −1/2 (there a...

All solutions to Thomas' family of Thue equations over imaginary quadratic number fields

We completely solve the family of relative Thue equations x − (t − 1)xy − (t + 2)xy − y = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for |t| ≥ 53 and Baker’s method combined with a computer search using continued fractions for the remaining values of t.

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

On a Parametric Family of Thue Inequalities over Function Fields

is called a Thue equation, due to Thue [22] who proved, in the case R = Z, that such an equation has finitely many solutions. In the last decade, starting with the result of Thomas in [21], several families (at the moment up to degree 8; see [9] and the references mentioned therein) of Thue equations have been considered, where the coefficients of the form Fc(X,Y ) depend on an integral paramet...

Calculating Power Integral Bases by Solving Relative Thue Equations

In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic...