Let $n$ be an odd prime. I know that the equation $x^n+y^n = z^2$ has no nonzero coprime solution in integers whenever $n \geq 5$, and that there are infinitely many solutions as soon as one drops the coprimality condition. As a curiosity, my question is: is there a way to parametrize all the noncoprime nonzero integer solutions to this equation (using different parametrization families, if necessary)? If it helps, take $n$ small, say $n = 5$ or $n = 7$.

The answer to the question depends on the parity of $n$. Suppose that $n=2k$ is even. Let $(x,y,z)$ be a nonzero solution to your equation, with $y\neq 0$. Then $(x/y,1,z/y^k)$ is also a solution of the equation. Hence to construct all the integral solutions you need to find all the rational points on the hyperelliptic curve $z^2=1+x^n$. For $n\geq 5$ there are only finitely many such points and for each point you find a onedimensional family of solutions of the form $(tx,ty,t^k z)$ with $\gcd(x,y,z)=1$. If $n=2k1$ is odd then you can find all the integral solutions as follows. Let $x,y$ be coprime integers and write $x^n+y^n=ts^2$. Then $(tx,ty,t^k s)$ is a solution. So $((x^n+y^n)x,(x^n+y^n)y,(x^n+y^n)^k)$, $x,y\in \mathbb{Z}$ parametrizes a subset of all the integral solutions. For fixed $x/y$ you will miss at most finitely many solutions. However, I doubt that you can give an algebraic description of all the integral solutions. 

