Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum
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Theorem
Let $m, n \in \Z_{>0}$ be a positive integer.
Let the divisor sum of $m$ and $n$ both be square.
Let $m$ and $n$ be coprime.
Then the divisor sum of $m n$ is square.
Proof
Let $\map {\sigma_1} m = k^2$.
Let $\map {\sigma_1} n = l^2$.
Thus:
\(\ds \map {\sigma_1} {m n}\) | \(=\) | \(\ds \map {\sigma_1} m \map {\sigma_1} n\) | Divisor Sum Function is Multiplicative | |||||||||||
\(\ds \) | \(=\) | \(\ds k^2 l^2\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k l}^2\) | from above |
$\blacksquare$