Proof that $x^2+y^2+z^2=7$ does not have a rational solution

Question

Proof that $x^2+y^2+z^2=7$ does not have a rational solution.

I figured out that I have to prove it by reductio ad absurdum, but I try and can't still found how to prove it. Or using proof by infinite descent but I have no idea

Answer 1

Any such rational solution yields a primitive integral solution to $$X^2+Y^2+Z^2=7W^2.$$ Now reducing mod $8$ shows that $W$ is even, but then also $X$, $Y$ and $Z$ are even, a contradiction.