Perfect Number

A perfect number is a positive integer that equals the sum of its own positive divisors, not counting itself: 6 is perfect because its proper divisors 1, 2 and 3 add up to 6, and the next one is 28. The idea appears as early as Euclid's Elements, and Euclid himself proved that whenever 2 to the p minus 1 is prime, the number 2 to the p-1 times that quantity is a perfect number. Ancient Greek mathematicians knew only the first four perfect numbers, 6, 28, 496 and 8128, the last of which the mathematician Nicomachus recorded around AD 100. Leonhard Euler later proved the converse, that every even perfect number must take this exact form, a result now called the Euclid-Euler theorem; whether any odd perfect number exists, or whether the list of perfect numbers is infinite, both remain unresolved.