perfect number

Many ancient cultures endowed certain integers with special religious and magical significance. One example is the perfect numbers, those integers which are the sum of their positive proper divisors. The first three perfect numbers are

6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14, and
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.

The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the moon's period of 28 days.

Whatever significance ascribed to them, these three perfect numbers above, and 8128, were known to be "perfect" by the ancient Greeks, and the search for perfect numbers was behind some of the greatest discoveries in number theory. For example, in Book IX of Euclid's elements we find the first part of the following theorem (completed by Euler some 2000 years later).

Theorem:: If 2^k-1 is prime, then 2^k-1 (2^k-1) is perfect and every even perfect number has this form.

It turns out that for 2^k-1 to be prime, k must also be prime--so the search for Perfect numbers is the same as the search for Mersenne primes. Armed with this information it does not take too long, even by hand, to find the next two perfect numbers: 33550336 and 8589869056. See the first page on Mersennes below for a list of all known perfect numbers.

While seeking perfect and amicable numbers, Pierre de Fermat discovered Fermat’s Little Theorem, and communicated a simplified version of it to Mersenne in 1640.

It is unknown if there are any odd perfect numbers. If there are some, then they are quite large (over 300 digits) and have numerous prime factors. But this will no doubt remain an open problem for quite some time.

Related pages (outside of this work)

Perfect Numbers and Mersennes (definitions and theorems)
Great Internet Mersenne Prime Search (which is the search for even perfect numbers)