Perfect number

A perfect number is a positive integer equal to the sum of its positive proper divisors. The first example is 6, since 1 + 2 + 3 = 6.

Proper divisor: A positive divisor of an integer that is smaller than the integer itself. Perfectness is defined by adding every positive divisor except the number itself.
Euclid–Euler theorem: The theorem that an even positive integer is perfect exactly when it has the form 2^(p−1)(2^p−1), with 2^p−1 prime. It gives the complete characterization of even perfect numbers.
Euclid's Elements: A mathematical treatise by Euclid that organized geometry and number theory into a deductive system. Book IX contains the construction later recognized as producing even perfect numbers.
Odd perfect number: A perfect number that is odd; none has been found, and whether any exist is unknown. The central unresolved question is whether this type exists at all.
Divisor function: A function that assigns to each positive integer the sum of its positive divisors. A number is perfect exactly when its divisor sum is twice the number.
Euclid: An ancient Greek mathematician whose Elements systematized geometry and number theory. Euclid proved that 2^(p−1)(2^p−1) is perfect whenever 2^p−1 is prime.
Nicomachus: A Greek mathematician of the first century CE known for the Introduction to Arithmetic. He described perfect numbers and their place in numerical classifications.
Mersenne prime: A prime number of the form 2^p − 1, where p is a positive integer. Finding further such primes yields further even perfect numbers, but no odd ones.
Prime number: A positive integer greater than 1 whose only positive divisors are 1 and itself. Prime factorization makes the divisor sums of perfect numbers calculable.
Euler: An eighteenth-century mathematician whose work shaped analysis, number theory, and mathematical notation. Euler proved that every even perfect number has Euclid’s form.

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