I consider the following equation with conditions of obtaining solutions
$$a^m+nx^2 = y^n$$
This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = 1$ and $n>1$.
If we fix $a$ as an odd prime and without restriction on $x, y, m$ and $n$ (may be odd or even) with $n > 1$ and $(nx, y) =1$, can we have solutions?
If yes, how to determine such solutions?
If there is empty solution, how to conclude?