An exponential Diophantine equation (Q2748270)

The author gives all the positive integer solutions \((x,y,m,n)\) of the equation NEWLINE\[NEWLINE x^2+p^{2m}=y^n, \quad \text{\(x\), \(y\), \(m\), \(n\in {\mathbb N}\), \(\gcd(x,y)=1\), \(n>2\)} NEWLINE\]NEWLINE satisfying \(2 |n\) or \(2\nmid n\) and \(p\not\equiv (-1)^{(p-1)/2} \pmod{4n}\), where \(p\) is a prime number \({}>3\). The proof combines many results on several Diophantine equations. In particular, it uses a new deep result of Bilu-Hanrot-Voutier on primitive divisors of Lucas and Lehmer numbers.