What is known about $x^m + y^m = z^n$ over $\mathbb{N}$ when $m,n \geq 2$ and $m \neq n$? So Fermat's Last Theorem resolves the question of positive integer solutions to $x^m + y^m = z^n$ when $m = n \geq 3$. But what about if $m \neq n$ and $m,n \geq 2$? Is anything general known about when there are positive integer solutions? Clearly if $gcd(m,n) \geq 3$ then there are no solutions by Fermat's Last Theorem. So let's assume $gcd(m,n) \leq 2$.

Suppose for example that $m$ and $n$ are relatively prime. Let $x=y=2^s$ and $z=2^t$. Then our equation becomes $2^{1+sm}=2^{tn}$. 
There are infinitely many pairs of positive integers $(s,t)$ such that $1+sm=tn$, so the equation $x^m+y^m=z^n$ has infinitely many positive (but uninteresting!) solutions. 
