For which values of $n$ can $x^n+y^n$ be a perfect square? Let $x, y$ and $n$ be positive integers. Using Fermat's Last Theorem we can show that $x^n+y^n$ can't be a perfect square if $n$ is divisible by $4$, but when $n=3$ we have some simple solutions like $$1^3+2^3=3^2 \\ 2^3+2^3=4^2$$
I have conjectured that

There are nontrivial integer solutions for $x,y$ and $z$ in $$x^n+y^n=z^2,\,\,\,\,\forall n\in\mathbb{N}\,\,\,\,\text{which is not a multiple of 4.}$$

How can I attempt to prove this? A hint or a reference would be appreciated.
Thank you.

We say that a solution $(x,y,z)$ is trivial if $xyz = 0$ or $xyz = \pm 1$, and is primitive if $\gcd(x,y,z) = 1$.
Then (theorem): if $n \geq 4$, there are no nontrivial primitive solutions to $x^n + y^n = z^2$. 
This is proved in Darmon-Merel, "Winding quotients and some variants of
Fermat’s Last Theorem", which can be found here on Merel's webpage. Simpler cases (for instance, $n \equiv -1 \mod 4$, or $n \leq 7$) had been solved before their paper. They give a brief discussion of the history of the problem in the paper.
