Today I’d like to talk about perfect numbers, which touch on
some clever algebra and some neat topics in number theory. At
least, I’ll start talking about them, since it will probably
take me three or four posts to say everything I want to say. =)
Without further ado, the definition of perfect numbers is very
simple: a number n is perfect if its proper divisors add up
exactly to n. A divisor of n, of course, is a number which
evenly divides n; talking about the “proper” divisors of n
simply excludes n itself. For example, the proper divisors of 6
are 1, 2, and 3, which have the sum 1 + 2 + 3 = 6 — so 6 is a
perfect number. An equivalent way of expressing the definition
is that n is perfect if all its divisors (including n itself)
add up to twice n. The sum of all the divisors of n is often
written , so formally, we can write that n is perfect if . The
next perfect number after 6 is 28 (28 = 1 + 2 + 4 + 7 + 14), and
the next after that is 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124
+ 248. Are there any more? Well