Perfect number

# Perfect number

Overview
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, a perfect number is a positive integer that is equal to the sum of its proper positive divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s, 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
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...

(n) = 2n.
Discussion

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Encyclopedia
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, a perfect number is a positive integer that is equal to the sum of its proper positive divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s, 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
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...

(n) = 2n.

## Examples

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
28 (number)
28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....

= 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496
496 (number)
Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.-In mathematics:...

and 8128
8128 (number)
8128 is the natural number following 8127 and preceding 8129.It is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 127, 27 − 1, with 26 · yielding 8128...

.

## Discovery

These first four perfect numbers were the only ones known to early Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, and the mathematician Nicomachus
Nicomachus
Nicomachus was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria , and was strongly influenced by Aristotle...

Then, in 1456, 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
Pietro Cataldi
Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to...

identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers.

## Even perfect numbers

Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

discovered that the first four perfect numbers are generated by the formula 2p−1(2p−1), with p a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

:
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.

Noticing that in each of these cases 2p−1 is a prime number, Euclid proved that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime (Euclid, Prop. IX.36).

For 2p−1 to be prime, it is necessary that p itself be prime. Prime numbers of the form 2p−1 are known as Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

s, after the seventeenth-century monk Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

, who studied number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and perfect numbers. 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. (All factors of 2p−1 will be congruent to 1 mod 2p. For example, 211−1 = 2047 = 23 × 89 and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever p is a Sophie Germain
Sophie Germain
Marie-Sophie Germain was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a gender-biased society, she gained education from books in her father's library and from correspondence with famous mathematicians such as...

prime—that is, 2p+1 is also prime—and 2p+1 is congruent to 1 or 7 mod 8, then 2p + 1 will be a factor of 2p−1.) In fact, Mersenne primes are very rare—of the 78,498 prime numbers p below 1,000,000, 2p−1 is prime for only 33 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
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

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. , 47 Mersenne primes and therefore 47 even perfect numbers are known. The largest of these is 243,112,608 × (243,112,609−1) with 25,956,377 digits.

The first 41 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 .

The other 6 known are for p = 25964951, 30402457, 32582657, 37156667, 42643801, 43112609.
It is not known whether there are others between them.

No proof is known 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.

Because any even perfect number has the form 2p−1(2p−1), it is the (2p−1)th triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

and the 2p−1th hexagonal number
Hexagonal number
A hexagonal number is a figurate number. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.The formula for the nth hexagonal number...

. Like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2p−1. Furthermore, any even perfect number except the first one is the ((2p+1)/3)th centered nonagonal number
Centered nonagonal number
A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers...

as well as the sum of the first 2(p−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 resulting number is called the digital root
Digital root
The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...

— produces the number 1. For example, the digital root of 8128 = 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. The reason that this digital root doesn't work with the perfect number 6 is because it only works with perfect numbers 2p−1(2p−1), with odd prime p; the perfect number 6 being associated with the even prime 2.

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.

## Odd perfect numbers

It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance
Carl Pomerance
Carl Bernard Pomerance is a well-known number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least 7 distinct prime factors. He immediately joined the faculty at the...

has presented a heuristic argument which 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 > 10300. A search has tentatively shown that N > 101500, but this result is as yet unpublished.

## 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
Strong Law of Small Numbers
"The Strong Law of Small Numbers" is a humorous paper by mathematician Richard K. Guy and also the so-called law that it proclaims: "There aren't enough small numbers to meet the many demands made of them." In other words, any given small number appears in far more contexts than may seem...

:
• An odd perfect number is not divisible by 105.
• Every odd perfect number is of the form N = 1 mod 12 or N = 117 mod 468 or N = 81 mod 324.
• The only even perfect number of the form x3 + 1 is 28 (Makowski 1962).
• The reciprocals
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of the divisors of a perfect number N must add up to 2:
• For 6, we have ;
• For 28, we have , etc. (This is particularly easy to see, just by taking the definition of a perfect number, , and dividing both sides by n.)
• 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 number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

s. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and a class of numbers formed from Fermat primes 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.

## 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
Deficient number
In number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which...

, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology
Numerology
Numerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...

. A pair of numbers which are the sum of each other's proper divisors are called amicable
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...

, and larger cycles of numbers are called sociable
Sociable number
Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself...

. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number
Practical number
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n...

.

By definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

, a perfect number is a fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

of the restricted divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...

s(n) = σ(n) − n, and the aliquot sequence
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way:For example, the...

associated with a perfect number is a constant sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

.

All perfect numbers are also -perfect numbers, or Granville numbers.

• Perfection
• Superperfect number
Superperfect number
In mathematics a superperfect number is a positive integer n that satisfies\sigma^2=\sigma=2n\, ,where σ is the divisor function...

s
• Jan Brożek
Jan Brozek
Jan Brożek was a Polish polymath: a mathematician, astronomer, physician, poet, writer, musician and rector of the Kraków Academy.-Life:...

• List of perfect numbers