Archive for February, 2001

Perfect numbers

Sunday, February 25th, 2001

If you take any number and find all of its factors (including 1 but excluding the original number), then take the sum of all these factors, you will end up with another number. If this number is the same as the one you started with, it is a perfect number.

Perfect numbers are very rare. Only about 31 are known. The first few are:

  • 6
  • 28
  • 496
  • 8128
  • 33550336
  • 8589869056

Here are a few facts about perfect numbers:

  • All perfect numbers end in either 6 or 8.
  • If 2^n-1 is prime, then 2^(n-1)(2^n-1) is perfect. All perfect numbers are of this form.
  • All perfect numbers are triangular.
  • The digital root, or remainder when divided by 9, of any perfect number except 6 is 1.
  • Every perfect number except 6 is the sum of consecutive odd cubes.
  • The sum of the reciprocals of all the divisors of any perfect number is equal to 2.