Perfect number

A positive integer $n$ is called a perfect number if it is the sum of its proper divisors.

The first four perfect numbers are:

  • $6 = 3 + 2 + 1$
  • $28 = 14 + 7 + 4 + 2 + 1$
  • $496 = 248 + 124 + 62 + 31 + 16 + 8 + 4 + 2 + 1$
  • $8128 = 4064 + 2032 + 1016 + 508 + 254 + 127 + 64 + 32 + 16 + 8 + 4 + 2 + 1$

These were the only perfect numbers known to ancient mathematicians. Some other perfect numbers include:

  • $33550336$
  • $8589869056$
  • $137438691328$
  • $2305843008139952128$


Even Perfect Numbers

$n$ is an even perfect number if and only if $n=\frac{p(p+1)}{2}$, where $p$ is a prime number of the form $2^k-1$ for some $k$. Primes of the form $2^k-1$ are called Mersenne primes.

Proof

It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No odd perfect numbers are known, and any that do exist must be greater than $10^{500}$. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.


Numbers which are not perfect may be either deficient numbers or abundant numbers.

Other Resources

http://mathforum.org/dr.math/faq/faq.perfect.html

Invalid username
Login to AoPS