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A parametric family of quintic Thue equations

Authors: István Gaál and Günter Lettl
Journal: Math. Comp. 69 (2000), 851-859
MSC (1991): Primary 11D57; Secondary 11Y50
Published electronically: May 24, 1999
MathSciNet review: 1659855
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Abstract: For an integral parameter $t \in \mathbb{Z}$ we investigate the family of Thue equations

\begin{multline*}F(x,y) = x^{5} + (t-1)^{2}x^{4}y - (2t^{3}+4t+4)x^{3}y^{2}\\ + (t^{4}+t^{3}+2t^{2}+4t-3)x^{2}y^{3} + (t^{3}+t^{2}+5t+3)xy^{4} + y^{5} = \pm 1\,, \end{multline*}

originating from Emma Lehmer's family of quintic fields, and show that for $|t| \ge 3.28 \cdot 10^{15}$ the only solutions are the trivial ones with $x=0$ or $y=0$. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

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Additional Information

István Gaál
Affiliation: Kossuth Lajos University, Mathematical Institute, H–4010 Debrecen Pf.12., Hungary

Günter Lettl
Affiliation: Karl-Franzens-Universität Graz, Institut für Mathematik, A–8010 Graz, Heinrichstraße 36, Austria

Keywords: Parametric Thue equation, Baker's method
Received by editor(s): December 12, 1997
Received by editor(s) in revised form: July 14, 1998
Published electronically: May 24, 1999
Additional Notes: The first author’s research was supported in part by Grants 16791 and 16975 from the Hungarian National Foundation for Scientific Research
Article copyright: © Copyright 2000 American Mathematical Society