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 of 2 in the sum is a perfect number. Thus, for example, 1 + 2 + 4 = 7, which is prime; thus by Euclid's theorem, 7 x 4 = 28 is a perfect number. And 1 + 2 + 4 + 8 + 16 = 31 which is prime, so 31x16=496 is a perfect number. An equivalent way to formulate Euclid's theorem is that whenever 2n-1 is a prime number, 2n-1x(2n-1) is a perfect number.
Although Euclid treated perfect numbers from a purely logical perspective, many of the mathematicians and philosophers that followed him associated a moral significance to the idea. The mathematician Nicomachus of Gerasa published a treatise around 100 AD in which he discussed superabundant numbers, whose divisors add up to more that the number itself, and deficient numbers, whose divisors add up to less than the number. Nicomachus likened superabundant numbers to excessive behavior, and deficient numbers to insufficiencies; perfect numbers were like perfect virtue in humans. Early Christian thinkers attached a theological importance to perfect numbers, observing that God created the world in 6 days, and made the moon go around the earth in 28 days. Saint Augustine wrote, "Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect."
Euclid's theorem gives a way to generate even perfect numbers. For many years after the time of Euclid, mathematicians asked whether the converse of Euclid's theorem is true: if an even number is perfect, is it necessarily of the form 2n-1x(2n-1), where 2n-1 is prime? Many mathematicians assumed that the answer to that question is 'yes', but a formal proof was not given until the 18th century, by Leonhard Euler. The question whether there are any odd perfect numbers is still open; it is one of the oldest open questions in mathematics.
This is the complete article, containing 499 words
(approx. 2 pages at 300 words per page).
