Perfect number
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In mathematics, a perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n.
The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS).
These first four perfect numbers were the only ones known to early Greek mathematics.
Even perfect numbers
Euclid discovered that the first four perfect numbers are generated by the formula 2^{n−1}(2^{n} − 1):
 for n = 2: 2^{1}(2^{2} − 1) = 6
 for n = 3: 2^{2}(2^{3} − 1) = 28
 for n = 5: 2^{4}(2^{5} − 1) = 496
 for n = 7: 2^{6}(2^{7} − 1) = 8128.
Noticing that 2^{n} − 1 is a prime number in each instance, Euclid proved that the formula 2^{n−1}(2^{n} − 1) gives an even perfect number whenever 2^{n} − 1 is prime (Euclid, Prop. IX.36).
Ancient mathematicians made many assumptions about perfect numbers based on the four they knew, but most of those assumptions would later prove to be incorrect. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2^{11} − 1 = 2047 = 23 × 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:
 The fifth perfect number would have five digits in base 10 since the first four had 1, 2, 3, and 4 digits respectively.
 The perfect numbers would alternately end in 6 or 8.
The fifth perfect number () has 8 digits, thus refuting the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show that the last digit of any even perfect number must be 6 or 8.
In order for to be prime, it is necessary but not sufficient that should be prime. Prime numbers of the form 2^{n} − 1 are known as Mersenne primes, after the seventeenthcentury monk Marin Mersenne, who studied number theory and perfect numbers.
Over a millennium after Euclid, Ibn alHaytham (Alhazen) circa 1000 AD realized that every even perfect number is of the form 2^{n−1}(2^{n} − 1) where 2^{n} − 1 is prime, but he was not able to prove this result. It was not until the 18th century that Leonhard Euler proved that the formula 2^{n−1}(2^{n} − 1) will yield all the even perfect numbers. Thus, there is a concrete onetoone association between even perfect numbers and Mersenne primes. This result is often referred to as the "EuclidEuler Theorem". As of September 2007, only 44 Mersenne primes are known, which means there are 44 perfect numbers known, the largest being 2^{32,582,656} × (2^{32,582,657} − 1) with 19,616,714 digits.
The first 39 even perfect numbers are 2^{n−1}(2^{n} − 1) for
 n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 (sequence A000043 in OEIS)
The other 5 known are for n = 20996011, 24036583, 25964951, 30402457, 32582657. It is not known whether there are others between them.
It is still uncertain whether there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS distributed computing project.
Since any even perfect number has the form 2^{n−1}(2^{n} − 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2^{n} − 1. Furthermore, any even perfect number except the first one is the sum of the first 2^{(n−1)/2} odd cubes:
Even perfect numbers (except 6) give remainder 1 when divided by 9. 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 is obtained — the resuting number is called the digital root — produces the number 1. For example the digital root of 8128 = 1, since 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1.
Odd perfect numbers
Are there any odd perfect numbers? 
It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would imply that there are no odd perfect numbers.
Any odd perfect number N must satisfy the following conditions:
 N > 10^{300}. A search is on to prove that N > 10^{500} is also required.
 N is of the form

 where:
 The largest prime factor of N is greater than 10^{8} (Takeshi Goto and Yasuo Ohno, 2006).
 The second largest prime factor is greater than 10^{4}, and the third largest prime factor is greater than 100 (Iannucci 1999, 2000).
 N has at least 75 prime factors; and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors (Nielsen 2006; Kevin Hare 2005).
 When ≤ 2 for every i
 The smallest prime factor of N is at least 739 (Cohen 1987).
 α ≡ 1 (mod 12) or α ≡ 9 (mod 12) (McDaniel 1970).
Quote
In 1888, Sylvester stated:
“  …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. 
” 
Minor results
Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist. 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:
 An odd perfect number is not divisible by 105 (Kühnel 1949).
 Every odd perfect number is of the form 12m + 1 or 36m + 9 (Touchard 1953; Holdener 2002).
 The only even perfect number of the form is 28 (Makowski 1962).
 A Fermat number cannot be a perfect number (Luca 2000).
 By dividing the definition through by the perfect number N, the reciprocals of the factors of a perfect number N must add up to 2:
 For 6, we have ;
 For 28, we have , etc.
 The number of divisors of a perfect number (whether even or odd) must be even, since N cannot be a perfect square.
 From these two results it follows that every perfect number is an Ore's harmonic number.
Related concepts
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.
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.