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Quelques résultats sur les équations $a{{x}^{p}}\,+\,b{{y}^{p}}\,=\,c{{z}^{2}}$

Published online by Cambridge University Press:  20 November 2018

W. Ivorra  and
A. Kraus
W. Ivorra
Affiliation:
Institut de Mathématiques, Université Paris VI, Équipe de Théorie des Nombres, UMR 7586 du CNRS, 175 Rue du Chevaleret, Paris 75013, France e-mail: ivorra@math.jussieu.fr, e-mail: kraus@math.jussieu.fr
A. Kraus
Affiliation:
Institut de Mathématiques, Université Paris VI, Équipe de Théorie des Nombres, UMR 7586 du CNRS, 175 Rue du Chevaleret, Paris 75013, France e-mail: ivorra@math.jussieu.fr, e-mail: kraus@math.jussieu.fr
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Abstract

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Let $p$ be a prime number ≥ 5 and $a,\,b,\,c$ be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation $a{{x}^{p}}\,+\,b{{y}^{p}}\,=\,c{{z}^{2}}$, in case the product of the prime divisors of $abc$ divides $2\ell $, with $\ell $ an odd prime number. For instance, under some conditions on $a,\,b,\,c$, we provide a constant $f(a,\,b,\,c)$ such that there are no such solutions if $p\,>\,f(a,\,b,\,c)$. In application, we obtain information concerning the $\mathbb{Q}$-rational points of hyperelliptic curves given by the equation ${{y}^{2}}\,=\,{{x}^{p}}\,+\,d$ with $d\,\in \,\mathbb{Z}$.

Keywords

11G
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 1 , 01 February 2006 , pp. 115 - 153
Copyright
Copyright © Canadian Mathematical Society 2006

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