Are there coprime integer solutions to: $$ \frac{x^n \pm y^n}{x \pm y}=z^m $$ with $n>5 , m>1$ and excluding $z=0$?
I suppose the abc conjecture implies finitely many solutions.
Are there coprime integer solutions to: $$ \frac{x^n \pm y^n}{x \pm y}=z^m $$ with $n>5 , m>1$ and excluding $z=0$? I suppose the abc conjecture implies finitely many solutions. 


I suspect this question is very difficult to answer without additional hypotheses. The case $y=1$ is the classic NagellLjunggren equation, where, unlike the (on the surface) very similar equation of Catalan, we do not even know whether there exist finitely many solutions in the variables $(x,z,n,m)$. If we fix $m=2$ and $n > 5$ prime, say, then it is still a substantial problem to solve the corresponding equation; Ivorra [Dissertationes Math. 444 (2007)] treats the cases $n \in \{ 7, 11, 13, 17 \}$ via elliptic Chabauty. 

