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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
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.
- 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).
It turns out that for 2k-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.
- 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
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)