TITLE: On the equation $a^n + b^n = c^2$
QUESTION [7 upvotes]: 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.

REPLY [12 votes]: 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.

REPLY [9 votes]: $2^{2n+1}+2^{2n+1}=( 2^{n+1})^2$.

REPLY [5 votes]: Assuming $gcd(a,b,c)=1$ there are no solutions (see GH's answer), and this fits well for the hyperbolic case of the generalised Fermat equation 
 $x^p+y^q=z^r$ with
$1/p+1/q+1/r<1$, and $gcd(x,y,z)=1$, because for $n>4$ above we are in this case. 
By Darmon-Granville theorem we know that there are only finitely many solutions, and conjecturally at most $10$ solutions. None of these is for $(p,q,r)=(n,n,2)$ with $n>4$.