Perfect number-a number that is the sum of its proper divisors (a proper divisor is a divisor smaller than the number itself). For example, 6 is perfect, since its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Twenty-eight is a perfect number, since its proper divisors are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. Perfect numbers are very rare; the first four, 6, 28, 496, and 8128 appear to have been known since ancient times, and since then, 33 more perfect numbers have been found. The largest known perfect number, discovered in 1998, has 1819050 digits.

Perfect numbers have been studied since the time of the Greeks (perhaps even earlier). To Pythagoras, the perfect numbers had mystical significance. The first recorded mathematical theorem concerning perfect numbers is from 300 B.C., in Euclid's Elements. Euclid proves the following statement: If the sums of the powers of 2 are added until their sum is a prime number, then that sum multiplied by the largest power...