Coprime integer solutions to $\frac{x^n \pm y^n}{x \pm y}=z^m$ with $n>5 , m>1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:57:31Z http://mathoverflow.net/feeds/question/69702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69702/coprime-integer-solutions-to-fracxn-pm-ynx-pm-yzm-with-n5-m1 Coprime integer solutions to $\frac{x^n \pm y^n}{x \pm y}=z^m$ with $n>5 , m>1$ joro 2011-07-07T11:25:56Z 2011-07-08T09:52:45Z <p>Are there coprime integer solutions to: $$\frac{x^n \pm y^n}{x \pm y}=z^m$$ with $n>5 , m>1$ and excluding $z=0$?</p> <p>I suppose the abc conjecture implies finitely many solutions.</p> http://mathoverflow.net/questions/69702/coprime-integer-solutions-to-fracxn-pm-ynx-pm-yzm-with-n5-m1/69782#69782 Answer by Mike Bennett for Coprime integer solutions to $\frac{x^n \pm y^n}{x \pm y}=z^m$ with $n>5 , m>1$ Mike Bennett 2011-07-08T09:52:45Z 2011-07-08T09:52:45Z <p>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.</p>