Exercise 5.3.5 (Consequences of $2uv$ is a perfect square (where $u \wedge v = 1$))

Proof. We suppose that $2\mathit{uv}=w^2>0$ is a perfect square. Then $w^2$ is even, thus $w$ is even: $w=2t$ for some integer $t$. Then $\mathit{uv}=2t^2$. This shows that $u$ or $v$ is even (but not both since $u$ and $v$ are relatively prime positive integers).

• If $2\mid u$, then $u=2u^{\prime }$ for some positive integer $u^{\prime }$, and $u^{\prime }v=t^2$, where $u^{\prime }$ and $v$ are relatively prime (since $u^{\prime }\mid u$). By Lemma 5.4, $u^{\prime }$ and $v$ are both perfect squares, so $u^{\prime }=r^2,v=s^2$, and

  $$u=2r^2,v=s^2,u,v\in {\mathbb{N}}^{\text{∗}}.$$

• If $2\mid v$, we obtain similarly by exchanging the roles of $u$ and $v$

  $$u=r^2,v=2s^2,u,v\in {\mathbb{N}}^{\text{∗}}.$$