12 times Sigma of 12 equals 14 times Sigma of 14

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Theorem

$x = 12$ and $y = 14$ are solutions to the indeterminate equation:

$x \ \sigma \left({x}\right) = y \ \sigma \left({y}\right)$

where $\sigma$ denotes the $\sigma$ function.


Proof

\(\displaystyle 12 \ \sigma \left({12}\right)\) \(=\) \(\displaystyle 12 \times 28\) $\sigma$ of $12$
\(\displaystyle \) \(=\) \(\displaystyle 12 \times 2 \times 14\)
\(\displaystyle \) \(=\) \(\displaystyle 14 \times 24\)
\(\displaystyle \) \(=\) \(\displaystyle 14 \ \sigma \left({14}\right)\) $\sigma$ of $14$

$\blacksquare$


Sources