Exercise 1.2.15 (Divisibility of $x^2+y^2$ by $4$)

Proof. If $x,y$ are odd, then $x^2$ and $y^2$ are also odd, thus $x^2+y^2$ is even.

By Exercise 15, knowing that $x$ and $y$ are odd, $8\mid x^2-1$ and $8\mid y^2-1$. Therefore $8\mid x^2+y^2-2$, a fortiori $4\mid x^2+y^2-2$.

Reasoning by contradiction, if $4\mid x^2+y^2$, then $4\mid 2$: this is false. This proves that $x^2+y^2$ is not divisible by $4$. □