In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, σ1(n) = 2n where σ1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Elements, Prop. IX.36).
For example, the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number, as follows:
Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare—of the 2,610,944 prime numbers p up to 43,112,609, 2p − 1 is prime for only 47 of them.
An exhaustive search by the GIMPS distributed computing project has shown that the first 47 even perfect numbers are 2p−1(2p − 1) for
Four higher perfect numbers have also been discovered, namely those for which p = 57885161, 74207281, 77232917, and 82589933, though there may be others within this range. As of December 2018, 51 Mersenne primes are known, and therefore 51 even perfect numbers (the largest of which is 282589932 × (282589933 − 1) with 49,724,095 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.
As well as having the form 2p−1(2p − 1), each even perfect number is the (2p − 1)th triangular number (and hence equal to the sum of the integers from 1 to 2p − 1) and the 2p−1th hexagonal number. Furthermore, each even perfect number except for 6 is the ((2p + 1)/3)th centered nonagonal number and is equal to the sum of the first 2(p−1)/2 odd cubes:
with each resulting triangular number T7 = 28, T31 = 496, T127 = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T2 = 3, T10 = 55, T42 = 903, T2730 = 3727815, ... This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2p−1(2p − 1) with odd prime p and, in fact, with all numbers of the form 2m−1(2m − 1) for odd integer (not necessarily prime) m.
Owing to their form, 2p−1(2p − 1), every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example,
It is unknown whether there is any odd perfect number, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists. Euler stated: "Whether (...) there are any odd perfect numbers is a most difficult question".
More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.
...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.
Furthermore, several minor results are known concerning to the exponents e1, ..., ek in
All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.