12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14
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Theorem
$x = 12$ and $y = 14$ are solutions to the indeterminate equation:
- $x \, \map {\sigma_1} x = y \, \map {\sigma_1} y$
where $\sigma_1$ denotes the divisor sum function.
Proof
| \(\ds 12 \, \map {\sigma_1} {12}\) | \(=\) | \(\ds 12 \times 28\) | $\sigma_1$ of $12$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 12 \times 2 \times 14\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 14 \times 24\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 14 \, \map {\sigma_1} {14}\) | $\sigma_1$ of $14$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$