a proof of contradiction

Question

I am wondering whether the following is a valid proof?

Answer 1

Very well-written, good job! I have a minor thing to point out.

When it comes to divisibility proofs, I would avoid fractions as much as possible and just stick to the integers. Notice that after you show that $$ 4y = -2 + x^2 $$ for some $y \in \mathbb Z$, you can immediately say that $4 \mid (x^2 - 2)$. There is no need to divide each side of the equation by $4$.

Answer 2

Looks fine to me. Although, "if $x$ is odd then $x^2-2$ is odd and therefore not divisible by 4" is rather quicker.