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A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

Published online by Cambridge University Press:  20 November 2018

Yann Bugeaud ,
Maurice Mignotte  and
Samir Siksek
Yann Bugeaud
Affiliation:
Université Louis Pasteur, U.F.R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg Cedex, France e-mail:, [email protected]:, [email protected]
Maurice Mignotte
Affiliation:
Institute of Mathematics, University of Warwick, Coventry, CV4 7AL, U.K. e-mail:, [email protected]
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Abstract

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We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation

$${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$

in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

Keywords

11F8011D6111D5911J8611Y50Diophantine equationsFrey curveslevel-loweringlinear forms in logarithmsThue equations
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 3 , 01 June 2008 , pp. 491 - 519
Copyright
Copyright © Canadian Mathematical Society 2008

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