Abstract view
Quelques résultats sur les équations $ax^p+by^p=cz^2$


Published:20060201
Printed: Feb 2006
Abstract
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 $d\in \Z$.