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
\(\ds 12 \ \sigma \left({12}\right)\) | \(=\) | \(\ds 12 \times 28\) | $\sigma$ of $12$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 12 \times 2 \times 14\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14 \times 24\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14 \ \sigma \left({14}\right)\) | $\sigma$ of $14$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$