Perfect number

[1] For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number).

Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.

In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 1) is perfect.

[4] In modern language, Nicomachus states without proof that every perfect number is of the form

Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.

Philo is followed by Origen,[7] and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000.

[8] St Augustine defines perfect numbers in City of God (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number.

The Egyptian mathematician Ismail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect.

[9] The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.

[10] In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.

[14] While Nicomachus had stated (without proof) that all perfect numbers were of the form

is prime (though he stated this somewhat differently), Ibn al-Haytham (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number.

[15] It was not until the 18th century that Leonhard Euler proved that the formula

Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa.

An exhaustive search by the GIMPS distributed computing project has shown that the first 48 even perfect numbers are

Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for p below 109332539.

As of October 2024[update], 52 Mersenne primes are known,[16] and therefore 52 even perfect numbers (the largest of which is 2136279840 × (2136279841 − 1) with 82,048,640 digits).

-th triangular number (and hence equal to the sum of the integers from 1 to

-th centered nonagonal number and is equal to the sum of the first

with odd prime p and, in fact, with all numbers of the form

for odd integer (not necessarily prime) m. Owing to their form,

every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example:

It is unknown whether any odd perfect numbers exist, though various results have been obtained.

In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers,[18] thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question".

[19] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.

[21] Any odd perfect number N must satisfy the following conditions: Furthermore, several minor results are known about the exponents e1, ..., ek.

In 1888, Sylvester stated:[49] ... 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.All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare.

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.

By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.