To appear in J. Symbolic Comput. THOMAS’ FAMILY OF THUE EQUATIONS OVER IMAGINARY QUADRATIC FIELDS
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چکیده
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 are a few more solutions in this case which are explicitly listed). In the case Re t = −1/2, an algebraic method is used, in the general case, Baker’s method yields the result.
منابع مشابه
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.
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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.
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