In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, a perfect number is a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
that is equal to the sum of its positive
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), 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
In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself.
That is,
:s(n)=\sum\nolimits_d.
It can be used to characterize the prime ...
, 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,
where
is the
sum-of-divisors function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includ ...
. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28.
This definition is ancient, appearing as early as
Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number'').
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
also proved a formation rule (IX.36) whereby
is an even perfect number whenever
is a prime of the form
for positive integer
—what is now called a
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...
. Two millennia later,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
proved that all even perfect numbers are of this form.
[Caldwell, Chris]
"A proof that all even perfect numbers are a power of two times a Mersenne prime"
This is known as the
Euclid–Euler theorem
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...
.
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
__NOTOC__
Year 496 ( CDXCVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Paulus without colleague (or, less frequently, ye ...
and
8128 .
History
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
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
, and the mathematician
Nicomachus
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was born i ...
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 (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.)
Philo of Alexandria
Philo of Alexandria (; grc, Φίλων, Phílōn; he, יְדִידְיָה, Yəḏīḏyāh (Jedediah); ), also called Philo Judaeus, was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt.
Philo's depl ...
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
Origen of Alexandria, ''Ōrigénēs''; Origen's Greek name ''Ōrigénēs'' () probably means "child of Horus" (from , "Horus", and , "born"). ( 185 – 253), also known as Origen Adamantius, was an early Christian scholar, ascetic, and theolo ...
, and by
Didymus the Blind
Didymus the Blind (alternatively spelled Dedimus or Didymous) (c. 313398) was a Christian theologian in the Church of Alexandria, where he taught for about half a century. He was a student of Origen, and, after the Second Council of Constantinopl ...
, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19). St Augustine defines perfect numbers in
City of God (Book 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. 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
Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) 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 conti ...
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.
Even perfect numbers
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
proved that 2
''p''−1(2
''p'' − 1) is an even perfect number whenever 2
''p'' − 1 is prime (Elements, Prop. IX.36).
For example, the first four perfect numbers are generated by the formula 2
''p''−1(2
''p'' − 1), with ''p'' a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, as follows:
:for ''p'' = 2: 2
1(2
2 − 1) = 2 × 3 = 6
:for ''p'' = 3: 2
2(2
3 − 1) = 4 × 7 = 28
:for ''p'' = 5: 2
4(2
5 − 1) = 16 × 31 = 496
:for ''p'' = 7: 2
6(2
7 − 1) = 64 × 127 = 8128.
Prime numbers of the form 2
''p'' − 1 are known as
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...
s, after the seventeenth-century monk
Marin Mersenne, who studied
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and perfect numbers. For 2
''p'' − 1 to be prime, it is necessary that ''p'' itself be prime. However, not all numbers of the form 2
''p'' − 1 with a prime ''p'' are prime; for example, 2
11 − 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,
2
''p'' − 1 is prime for only 47 of them.
Although
Nicomachus
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was born i ...
had stated (without proof) that all perfect numbers were of the form
where
is prime (though he stated this somewhat differently),
Ibn al-Haytham
Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
(Alhazen) circa AD 1000 conjectured only that every even perfect number is of that form. It was not until the 18th century that
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
proved that the formula 2
''p''−1(2
''p'' − 1) will yield all the even perfect numbers. Thus, there is a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...
.
An exhaustive search by the
GIMPS
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client a ...
distributed computing project has shown that the first 48 even perfect numbers are 2
''p''−1(2
''p'' − 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, 32582657, 37156667, 42643801, 43112609 and 57885161 .
Three higher perfect numbers have also been discovered, namely those for which ''p'' = 74207281, 77232917, and 82589933. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for ''p'' below 109332539. , 51 Mersenne primes are known,
and therefore 51 even perfect numbers (the largest of which is 2
82589932 × (2
82589933 − 1) with 49,724,095 digits). 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 2
''p''−1(2
''p'' − 1), each even perfect number is the
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
(and hence equal to the sum of the integers from 1 to ) and the
hexagonal number
A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...
. Furthermore, each even perfect number except for 6 is the
centered nonagonal number and is equal to the sum of the first odd cubes (odd cubes up to the cube of ):
:
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 , , T
42 = 903, T
2730 = 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
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit s ...
) 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 2
''p''−1(2
''p'' − 1) with odd prime ''p'' and, in fact, with all numbers of the form 2
''m''−1(2
''m'' − 1) for odd integer (not necessarily prime) ''m''.
Owing to their form, 2
''p''−1(2
''p'' − 1), every even perfect number is represented in binary form as ''p'' ones followed by ''p'' − 1 zeros; for example,
: 6
10 = 2
2 + 2
1 = 110
2,
: 28
10 = 2
4 + 2
3 + 2
2 = 11100
2,
: 496
10 = 2
8 + 2
7 + 2
6 + 2
5 + 2
4 = 111110000
2, and
: 8128
10 = 2
12 + 2
11 + 2
10 + 2
9 + 2
8 + 2
7 + 2
6 = 1111111000000
2.
Thus every even perfect number is a
pernicious number
In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1's when it is written as a binary number.
Examples
The first pernicious number i ...
.
Every even perfect number is also a
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. For example, 12 is a practical number because all the numbers from 1 ...
(cf.
Related concepts).
Odd perfect numbers
It is unknown whether any odd perfect numbers exist, 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
Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American 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 ha ...
has presented a
heuristic argument
A heuristic argument is an argument that reasons from the value of a method or principle that has been shown experimentally (especially through trial-and-error) to be useful or convincing in learning, discovery and problem-solving, but whose line ...
suggesting that indeed no odd perfect number should exist.
[Oddperfect.org](_blank)
All perfect numbers are also
Ore's harmonic number
In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are:
: 1, 6, 28, ...
s, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1. Many of the properties proved about odd perfect numbers also apply to
Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.
Any odd perfect number ''N'' must satisfy the following conditions:
* ''N'' > 10
1500.
* ''N'' is not divisible by 105.
* ''N'' is of the form ''N'' ≡ 1 (mod 12) or ''N'' ≡ 117 (mod 468) or ''N'' ≡ 81 (mod 324).
* ''N'' is of the form
::
:where:
:* ''q'', ''p''
1, ..., ''p''
''k'' are distinct odd primes (Euler).
:* ''q'' ≡ α ≡ 1 (
mod 4) (Euler).
:* The smallest prime factor of ''N'' is at most
:* Either ''q''
α > 10
62, or ''p''
''j''2''e''''j'' > 10
62 for some ''j''.
:*
:*
.
:*
.
* The largest prime factor of ''N'' is greater than 10
8 and less than
* The second largest prime factor is greater than 10
4,
and is less than