Perfect Number/Examples/8128

Example of Perfect Number

$8128$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128$

Proof

\(\ds 8128\)

\(=\)

\(\ds 64 \times 127\)

\(\ds \)

\(=\)

\(\ds 2^{7 - 1} \paren {2^7 - 1}\)

Thus $8128$ is in the form $2^{p - 1} \paren {2^p - 1}$.

$2^7 - 1 = 127$ is prime.

So $8128$ is perfect by the Theorem of Even Perfect Numbers.

The aliquot parts of $8128$ are enumerated at $\sigma_0$ of $8128$:

$1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064$

$\blacksquare$