A number n is called perfect if the sum of its positive divisors (excluding n) is equal to n. For example, 6= 1+ 2+3 and 28 = 1 + 2 + 4 + 7 + 14 are perfect. Perfect numbers are rare; after 6 and 28, the next three perfect numbers are 496, 8128 and 33550336. Perfect numbers have been studied since ancient times. Biblical writers attributed perfection to these numbers because God created the world in 6 days and the lunar cycle has 28 days.

Euclid derived a formula for even perfect numbers. The formula in Euclid's words is the following.

If as many numbers as we please, beginning from unity, be set out continuously in double proportion until the sum of all becomes prime, and if the sum is multiplied by the last, the product will be perfect.

Even perfect numbers were completely characterized by Euler in terms of Mersenne primes. A number n is perfect if and only if it is of the form tex2html_wrap_inline182 where tex2html_wrap_inline184 and p are primes. If tex2html_wrap_inline188 is prime, then it is called a Mersenne prime. There are 35 known Mersenne primes, hence 34 even perfect numbers. No odd perfect numbers are known and it is conjectured that none exist.

See Student's Mersenne Prime Page and Yahoo's Mersenne Page for additional information.