perfect numberMany 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.
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).
- If 2k-1 is prime, then 2k-1 (2k-1) is perfect and every even perfect number has this form.
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.
See Also: AmicableNumber, AbundantNumber, DeficientNumber, SigmaFunction
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)