Perfect number


In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number.
The sum of divisors of a number, excluding the number itself, 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 including itself; in symbols, σ1 = 2n where σ1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
This definition is ancient, appearing as early as Euclid's Elements where it is called τέλειος ἀριθμός. Euclid also proved a formation rule whereby is an even perfect number whenever is a prime of the form for prime —what is now called a Mersenne prime. Two millennia 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 few perfect numbers are 6, 28, 496 and 8128.

History

In about 300 BC Euclid showed that if 2p − 1 is prime then 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. In modern language, Nicomachus states without proof that every perfect number is of the form where is prime. He seems to be unaware that itself has to be prime. He also says that the perfect numbers end in 6 or 8 alternately. 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, and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000.. St Augustine defines perfect numbers in City of God 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 mentioned the next three perfect numbers and listed a few more which are now known to be incorrect. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. In 1588, the Italian mathematician Pietro Cataldi identified the sixth and the seventh perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.

Even perfect numbers

proved that 2p−1 is an even perfect number whenever 2p − 1 is prime.
For example, the first four perfect numbers are generated by the formula 2p−1, with p a prime number, as follows:
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 2,610,944 prime numbers p up to 43,112,609,
2p − 1 is prime for only 47 of them.
Although Nicomachus had stated that all perfect numbers were of the form where is prime, Ibn al-Haytham circa AD 1000 conjectured only that every even perfect number is of that form. It was not until the 18th century that Leonhard Euler proved that the formula 2p−1 will yield all the even perfect numbers. 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. This result is often referred to as the Euclid–Euler theorem.
An exhaustive search by the GIMPS distributed computing project has shown that the first 47 even perfect numbers are 2p−1 for
Four higher perfect numbers have also been discovered, namely those for which p = 57885161, 74207281, 77232917, and 82589933, though there may be others within this range., 51 Mersenne primes are known, and therefore 51 even perfect numbers. 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, each even perfect number is the triangular number and the hexagonal number. Furthermore, each even perfect number except for 6 is the centered nonagonal number and is equal to the sum of the first odd cubes:
Even perfect numbers are of the form
with each resulting triangular number T7 = 28, T31 = 496, T127 = 8128 ending in 3 or 5, the sequence starting with T2 = 3, T10 = 55, T42 = 903, T2730 = 3727815, ... This can be reformulated as follows: adding the digits of any even perfect number, then adding the digits of the resulting number, and repeating this process until a single digit 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 with odd prime p and, in fact, with all numbers of the form 2m−1 for odd integer m.
Owing to their form, 2p−1, every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example,
and
Thus every even perfect number is a pernicious number.
Every even perfect number is also a practical number.

Odd perfect numbers

It is unknown whether there is any odd perfect number, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists. Euler stated: "Whether there are any odd perfect numbers is a most difficult question". More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should 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:
Furthermore, several minor results are known concerning to the exponents
e1, ..., ek in
In 1888, Sylvester stated:

Minor results

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 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, 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.