On the Diophantine equation $x^2+q^{2m}=2y^p$

Details On the Diophantine equation $x^2+q^{2m}=2y^p$ On the Diophantine equation $x^2+q^{2m}=2y^p$

2006-05-09

arxiv.org/abs/math/0605227v1

Mathematics

Number Theory

Scientific paper

In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where
$m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and
$\gcd(x,y)=1.$ We prove that there are only finitely many solutions
$(m,p,q,x,y)$ for which $y$ is not a sum of two consecutive squares. We also
study the above equation with fixed $y$ and with fixed $q.$

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