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Thomas’ Family of Thue Equations Over Imaginary Quadratic Fields

✍ Scribed by Clemens Heuberger; Attila Pethő; Robert F Tichy

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
310 KB
Volume
34
Category
Article
ISSN
0747-7171

No coin nor oath required. For personal study only.

✦ Synopsis

We consider the family of relative Thue equations

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.

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