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 is an even perfect number whenever is what is now called a Mersenne prime. Much later, Euler proved that all even perfect numbers are of this form. 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 perfect number is 6, because 1, 2, and 3 are its proper positive divisors, 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 OEIS).
These 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. 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. St Augustine defines perfect numbers in The 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. In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time. In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers.
Even perfect numbers
Are there infinitely many perfect numbers?
Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Euclid, 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:
- for p = 2: 21(22 − 1) = 6
- for p = 3: 22(23 − 1) = 28
- for p = 5: 24(25 − 1) = 496
- for p = 7: 26(27 − 1) = 8128.
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 9,592 prime numbers p less than 100,000, 2p − 1 is prime for only 28 of them.
Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of the form 2p−1(2p − 1) where 2p − 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 2p−1(2p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem. As of April 2015[update], 48 Mersenne primes are known, and therefore 48 even perfect numbers (the largest of which is 257885160 × (257885161 − 1) with 34,850,340 digits).
An exhaustive search by the GIMPS distributed computing project has shown that the first 44 even perfect numbers are 2p−1(2p − 1) for
- p = 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, 20996011, 24036583, 25964951, 30402457, and 32582657 (sequence A000043 in OEIS).
Four higher perfect numbers have also been discovered, namely p = 37156667, 42643801, 43112609, and 57885161, though there may be others within this range. 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:
Even perfect numbers (except 6) are of the form
with each resulting triangular number (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with 3, 55, 903, 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 as p ones followed by p − 1 zeros:
- 610 = 1102
- 2810 = 111002
- 49610 = 1111100002
- 812810 = 11111110000002
- 3355033610 = 11111111111110000000000002.
Thus every even perfect number is a pernicious number.
Odd perfect numbers
Are there any odd perfect numbers?
It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument that suggests that no odd perfect numbers 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.
Any odd perfect number N must satisfy the following conditions:
- N > 101500.
- N is not divisible by 105.
- N is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).
- N is of the form
- The largest prime factor of N is greater than 108.
- The second largest prime factor is greater than 104, and the third largest prime factor is greater than 100.
- N has at least 101 prime factors and at least 10 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.
...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.
Euler stated: "Whether (...) there are any odd perfect numbers is a most difficult question".
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 only even perfect number of the form x3 + 1 is 28 (Makowski 1962).
- 28 is also the only even perfect number that is a sum of two positive integral cubes (Gallardo 2010).
- The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, , and divide both sides by n):
- For 6, we have ;
- For 28, we have , etc.
- The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
- From these two results it follows that every perfect number is an Ore's harmonic number.
- The even perfect numbers are not trapezoidal numbers; 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.
- 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.
- Every even perfect number ends in 6 or 28, base ten.
- The only square-free perfect number is 6.
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. All perfect numbers are also -perfect numbers, or Granville numbers.
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.
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