Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the family of relative Thue equations x3 - (t - 1)x2y - (t + 2)xy2 - y3 = 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. 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 Ret = -1/2 (there are a few more solutions in this case which are explicitly listed). In the case Ret = -1/2, an algebraic method is used, in the general case, Baker's method yields the result.
['Clemens Heuberger', 'Attila Pethö', 'Robert F. Tichy']
I am writing a research paper on a idea Thomas' family of thue equations over imaginary quadratic fields
[{"from": "human", "value": "I am writing a research paper on a idea Thomas' family of thue equations over imaginary quadratic fields"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the family of relative Thue equations x3 - (t - 1)x2y - (t + 2)xy2 - y3 = \u00b5, where the parameter t, the root of unity \u00b5 and the solutions x and y are integers in the same imaginary quadratic number field. prove that there are only trivial solutions (with |x|, |y| \u2264 1), if |t| is large enough or if the discriminant of the quadratic number field is large enough or if Ret = -1/2 (there are a few more solutions in this case which are explicitly listed). In the case Ret = -1/2, an algebraic method is used, in the general case, Baker's method yields the result."}]