Perfect Number/Examples/496
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Example of Perfect Number
$496$ is a perfect number:
- $1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$
Proof
\(\ds 496\) | \(=\) | \(\ds 16 \times 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{5 - 1} \paren {2^5 - 1}\) |
Thus $496$ is in the form $2^{p - 1} \paren {2^p - 1}$.
$2^5 - 1 = 31$ is prime.
So $496$ is perfect by the Theorem of Even Perfect Numbers.
The aliquot parts of $496$ are enumerated at $\sigma_0$ of $496$:
- $1, 2, 4, 8, 16, 31, 62, 124, 248$
$\blacksquare$
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): perfect number
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): perfect number