12 times Sigma of 12 equals 14 times Sigma of 14

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

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$