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# On the equation $a^n + b^n = c^2$

I am interested in the possible natural solutions of the equation $a^n + b^n = c^2$ where $n \geq 4$ is fixed. I am not sure if it is well-known or not, so any suggestion would be helpful.

$2^{2n+1}+2^{2n+1}=( 2^{n+1})^2$.

To complement Boris Novikov's answer, Darmon and Merel have shown that the equation has no solution with $\gcd(a,b,c)=1$. See J. reine angew. Math. 490 (1997), 81-100.

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# On the equation $a^n + b^n = c^2$

I am interested in the possible natural solutions of the equation $a^n + b^n = c^2$ where $n \geq 4$ is fixed. I am not sure if it is well-known or not, so any suggestion would be helpful.

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$2^{2n+1}+2^{2n+1}=( 2^{n+1})^2$.

Sorry, I don't understand your comment. - Boris Novikov Dec 11 '13 at 20:18

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