$\mod 3$ gives that $3|x,y$, giving that $3|z$, thus infinite descent gives the only integral solution $x=y=z=0$. This works for any number $k$ that is not the sum of two squares for the equation $x^2+y^2=kz^2$.

$\mod 4$ and infinite descendant. @ZetaX. Sorry for posting the same at the same time