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Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$

The question is: Does DMT apply to the equation $x^n - y^n = z^2$?

Yes
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  • I am so sorry for my consecutive mistakes. They were horrible. My apology. Now I trust the question is in its definitive form. – Yes Aug 17 '14 at 12:23
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    No solutions for $x,y\le500$, and $n\le20$. – Lucian Aug 17 '14 at 13:14
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    The Darmon Merel reference https://www.imj-prg.fr/~loic.merel/winding.pdf proves this by appealing to Fermat for $n=4$, and by proving there are no coprime (non-trivial) integer solutions for $n$ an odd prime. In particular, they don't require $x,y,z$ all positive, which is crucial to creating the appropriate Frey curves. For your equation, if $p$ an odd prime divides $n$, then take $y=-y$ and apply DMT for that $p$. Then you just have to do $n=4$, which factors nicely. For $n=3$, the paper gives an infinite family of solutions which can be adapted to your equation. – Zack Wolske Aug 18 '14 at 14:24
  • @Zack Wolske: I appreciate that. Thank you for taking time to write, I see. – Yes Aug 18 '14 at 14:30

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