Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

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.

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...

Power Integral Bases in the Family of Simplest Quartic Fields

To appear in J. Symbolic Comput. 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.