Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
For the 2012 film, see Perfect Number (film).
The sum of proper divisors of a number 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; in symbols, where is the sum-of-divisors function.
This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby is an even perfect number whenever is a prime of the form for positive integer —what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.[1] This is known as the Euclid–Euler theorem.
It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128.[2]
History[edit]
In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.[3] In modern language, Nicomachus states without proof that every perfect number is of the form where is prime.[4][5] He seems to be unaware that n itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) 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,[6] and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19).[7] 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.[8] The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.[9] 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.[10][11][12]
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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,[17] thus implying that no odd perfect number exists. Euler stated: "Whether ... there are any odd perfect numbers is a most difficult question".[18] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.[19] All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.[20]
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:[48]
The even perfect numbers are not ; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form formed as the product of a Fermat prime with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.[52]
trapezoidal numbers
The number of perfect numbers less than n is less than , where c > 0 is a constant. In fact it is , using little-o notation.[54]
[53]
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: