Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Prove it Belarus

Does there exist a positive integer $n$ which can be represented both as $n = a^2 - b$ and $n = b^2 - c$ , where $a$ , $b$ , $c$ are three distinct divisors of $n$ ?

Solution

Answer: no, such $n$ doesn't exist.

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