In what follows, suppose that $u, v \in \mathbb{N}$. Furthermore, assume that $u \neq v$.
Under what conditions is $$u^x + v^y$$ factorable for odd $x$ and $y$?
(I do know that it is factorable when $x=y$. I would of course be interested in other conditions.)
For example if $d:=\gcd(x,y)\ne 1$. Then $x=dx'$ and $y=dy'$ and we have:
$$ (u^{x'})^d+(v^{y'})^d = m^d+n^d = (m+n)(m^{d-1}-m^{d-2}n+...+n^{d-1})$$
wher $m= u^{x'}$ and $n=v^{y'}$