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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 alsostudy the above equation with fixed $y$ and with fixed $q.

On the Diophantine equation x2+q2m=2yp

On the Diophantine equation x²+q^2m=2y^p