Problem

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Tags: arithmetic sequence, number theory unsolved, number theory



Call a triple $(a, b, c)$ of positive integers a nice triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.