Exercise 1.3.11 (If $x$ and $y$ are odd, $x^2 + y^2$ cannot be a perfect square)

Proof. Assume for contradiction that $x^2+y^2=z^2$ is the square of $z$, where $x,y$ are odd.

Then $x=2k+1$ for some integer $k$, thus $x^2=4k^2+4k+1\equiv 1(\mathit{mod}4)$. Similarly, $y^2\equiv 1(\mathit{mod}4)$. Therefore

Thus $2\mid z^2$, therefore $2\mid z$, and $4\mid z^2$, so $z^2\equiv 0(\mathit{mod}4)$. We obtain $z^2\equiv 0$ and $z^2\equiv 2(\mathit{mod}4)$: this is absurd. Therefore, if $x$ and $y$ are odd, $x^2+y^2$ cannot be a perfect square. □