(Find all integers $n$ such that $2^{n+1} n$ is a perfect square.

)
CN : We make a case distinction based on the parity of $n$.

If $n$ is even, $n=2 p$ and $2^{n+1} n=2^{2 p+1} 2 p=2^{p+1} p$. Therefore, $p$ must be a square.

If $n$ is odd, we have $n=2 p+1$ and $2^{n+1} n=2^{2 p+2}(2 p+1)=2^{p+1}(2 p+1)$. Therefore, $n$ must be a square.

CS : Conversely, we verify that $\left\{2 p^{2}, p \in \mathbb{N}\right\} \cup\left\{(2 p+1)^{2}, p \in \mathbb{N}\right\}$ is a solution.