Canadian Mathematical Society
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Quelques résultats sur les équations $ax^p+by^p=cz^2$

Open Access article
 Printed: Feb 2006
  • W. Ivorra
  • A. Kraus
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Let $p$ be a prime number $\geq 5$ 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 $ax^p+by^p=cz^2$, 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 $\Q$-rational points of hyperelliptic curves given by the equation $y^2=x^p+d$ with $d\in \Z$.
MSC Classifications: 11G show english descriptions unknown classification 11G 11G - unknown classification 11G

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