Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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
add comment

1 Answer

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.

share|improve this answer
    
@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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.