# Prove that 2022/n+4n is a perfect square iff 2022/n−8n is a perfect square

Charlie Conner 2022-10-06 Answered
Prove that $\frac{2022}{n}+4n$ is a perfect square iff $\frac{2022}{n}-8n$ is a perfect square
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omeopata25
You can greatly reduce the number of cases to check by working modulo $4$. We have
$2022=2\cdot 3\cdot 337\equiv 2\cdot 3\cdot 1\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}4\right).$
Since a square must be $0$ or $1$ mod $4$, we can immediately reduce to the two cases $n\in \left\{6,2022\right\}$. And of course $n=2022$ is impossible since $\frac{2022}{n}-8n$ will be negative.