Perfect Number/Examples/496

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