Can the product of three positive consecutive integers be a perfect square?
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Assume the three integers are $n-1, n, n+1$ and that $(n-1)n(n+1) = n(n^2-1)$ is a perfect square.
Since $n$ is relatively prime to both $n-1$ and $n+1$ (and hence their product $n^2-1$), we must have that $n^2-1$ is a perfect square too.
