# Product of Coprime Numbers whose Sigma is Square has Square Sigma

## Theorem

Let $m, n \in \Z_{>0}$ be a positive integer.

Let the $\sigma$ value of $m$ and $n$ both be square.

Let $m$ and $n$ be coprime.

Then the $\sigma$ value of $m n$ is square.

## Proof

Let $\sigma \left({m}\right) = k^2$.

Let $\sigma \left({n}\right) = l^2$.

Thus:

 $\displaystyle \sigma \left({m n}\right)$ $=$ $\displaystyle \sigma \left({m}\right) \sigma \left({n}\right)$ Sigma Function is Multiplicative $\displaystyle$ $=$ $\displaystyle k^2 l^2$ from above $\displaystyle$ $=$ $\displaystyle \left({k l}\right)^2$ from above

$\blacksquare$