Perfect Number/Examples/6

Example of Perfect Number

$6$ is a perfect number:

$1 + 2 + 3 = 6$


Proof

\(\ds 6\) \(=\) \(\ds 2 \times 3\)
\(\ds \) \(=\) \(\ds 2^{2 - 1} \paren {2^2 - 1}\)

Thus $6$ is in the form $2^{n - 1} \paren {2^n - 1}$.

$2^2 - 1 = 3$ is prime.

So $6$ is perfect by the Theorem of Even Perfect Numbers.


The aliquot parts of $6$ are enumerated at $\sigma_0$ of $6$:

$1, 2, 3$

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): perfect number
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): perfect number
  • 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
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