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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.

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I suppose you also want to rule out $x=y=z=1$. – Gerry Myerson Jul 7 '11 at 12:26
Indeed......... – joro Jul 7 '11 at 13:17

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.

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@Mike Thank you. Is the case $n=5, m=2$ clear? It appears to be a quartic model of elliptic curve for fixed y. – joro Jul 8 '11 at 10:05
This curve has positive rank and (I think) conductor $200$, if memory serves, so there are infinitely many solutions to the corresponding equation.... – Mike Bennett Jul 8 '11 at 12:29
For $n=5,m=2$, $\langle x,y,z \rangle =\langle 3k, k, 11k^2\rangle$ is a family of solutions (with both $\pm$ being $-$), but I suppose we are to assume that $\gcd(x,y)=1$. – Kevin O'Bryant Jul 8 '11 at 23:06

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