Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$

2011-07-07T11:25:56Z

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.

Answer by Mike Bennett for Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$

2011-07-08T09:52:45Z

I suspect this question is very difficult to answer without additional hypotheses. The case $y=1$ is the classic Nagell-Ljunggren 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.

Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$ - MathOverflow

Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$

Answer by Mike Bennett for Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$