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Perfect number

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For the 2012 film, see Perfect Number (film).

Illustration of the perfect number status of the number 6

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ₁(n) = 2n.
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 ${\displaystyle q(q+1)/2}$ is an even perfect number whenever ${\displaystyle q}$ is a prime of the form ${\displaystyle 2^{p}-1}$ for prime ${\displaystyle p}$—what is now called a Mersenne prime. Much later, 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.

Contents

 [hide] 

• 1 Examples
• 2 History
• 3 Even perfect numbers
• 4 Odd perfect numbers
• 5 Minor results
• 6 Related concepts
• 7 See also
• 8 Notes
• 9 References
• 10 Further reading
• 11 External links

Examples[edit]

The first perfect number is 6. Its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in the OEIS).

History[edit]

In about 300 BC Euclid showed that if 2^p−1 is prime then (2^p−1)2^p−1 is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus had noted 8128 as early as 100 AD.^[2] 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,^[3] 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).^[4] St Augustine defines perfect numbers in City of God (Part 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.^[5] In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest European reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time.^[6][7] In 1588, the Italian mathematician Pietro Cataldi also 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.^[8][9][10]