Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Prove it Croatia

Do there exist positive integers $m$ and $n$ such that $m^2 + n$ and $n^2 + m$ are squares of positive integers?

Solution

Without loss of generality we can assume that $m \ge n$ . Obviously, $m^2 < m^2 + n$ . On the other hand, $(m+1)^2 = m^2 + 2m + 1 > m^2 + n$ because $2m > n$ .
Hence we have
$m^2 < m^2 + n < (m + 1)^2,$
which means that $m^2+n$ is between two consecutive squares, so it can not be a square of an integer. Such positive integers $m$ and $n$ do not exist.

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.