The trivial solutions of the diophantine equation $x^2+y^2=z^2$

Question

The trivial solutions of the diophantine equation $x^2+y^2=z^2$ are the following: $$x=0, y=n, z=\pm n, n \in \mathbb{Z}$$ $$x= \xi , y=0, z= \pm \xi, \xi \in \mathbb{Z}$$ $$$$ My question is, why is the $\pm$ only at $z$?? Why isn't it for example $x=0, y= \pm n, z=\pm n, n \in \mathbb{Z}$ ??

Answer 1

Because if you choose $x=0$ and $y=n\in\mathbb{Z}$, then there are only two solutions for $z$: $z=+n$ and $z=-n$. (and of course it works similar for $y=0$ and $x=\xi$)

In other words, every $y=n$ gives you two solutions for $z$.