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Austrian Mathematical Olympiad

Number theory Difficulty 5.3 AIME, harder Prove it Austria

Determine all integers $n \ge 2$ that have a representation
$n = a^2 + b^2,$
where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$ .

Solutions — 2

Solution 1

If $n$ is odd, then both $a$ and $b$ are odd and therefore $n = a^2 + b^2$ is even, contradiction. Therefore, $n$ is even and $a = 2$ . This also shows that $b$ is even. Furthermore, $b \mid (n - b^2) = a^2 = 4$ . Thus $b \in \{2, 4\}$ , which results in $n = 8$ and $n = 20$ , respectively.

Solution 2

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