Show how to determine that there are infinite solutions to x^2 + y^2 =z^4 where x, y, and z > 0 and (x,y,z)=1
$\displaystyle x=a^4-6a^2b^2+b^4, \ y=4ab(a^2-b^2), \ z=a^2+b^2.$ you don't need to worry about positivity of $\displaystyle x,y,z$ because if $\displaystyle (x,y,z)$ is a solution, then $\displaystyle (\pm x, \pm y, \pm z)$ will be a solution too.
in order to make sure that $\displaystyle \gcd(x,y,z)=1,$ you should choose $\displaystyle a,b$ so that $\displaystyle \gcd(a,b)=1$ and not both $\displaystyle a,b$ are odd.