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[Number equal to the sum of its proper divisors] [the 2012 film]
In number theory, a PERFECT NUMBER is a positive integer that is equal to the
sum of its positive proper divisors, that is, divisors excluding the number
itself.[1] For instance, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6,
so 6 is a perfect number. The next perfect number is 28, because 1 + 2 + 4 +
7 + 14 = 28.
The first seven perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056,
and 137438691328.[2]
The sum of proper divisors of a number 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; in
symbols, <math>\sigma_1(n)=2n</math> where <math>\sigma_1</math> is the
sum-of-divisors function.
This definition is ancient, appearing as early as Euclid's _Elements_ (VII.22)
where it is called [grc] (_perfect_, _ideal_, or _complete number_). Euclid
also proved a formation rule (IX.36) whereby <math display="inline">\frac
{q(q+1)}{2}</math> is an even perfect number whenever <math>q</math> is a
prime of the form <math>2^p-1</math> for positive integer <math>p</math>—what
is now called a Mersenne prime. Two millennia later, Leonhard Euler proved
that all even perfect numbers are of this form.[3] 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.
** History
In about 300 BC Euclid showed that if 2<sup>_p_</sup> − 1 is prime then
2<sup>_p_−1</sup>(2<sup>_p_</sup> − 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.[4] In modern
language, Nicomachus states without proof that [every] perfect number is of
the form <math>2^{n-1}(2^n-1)</math> where <math>2^n-1</math> is prime.[5][6]
He seems to be unaware that [n] 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 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,[7] and by Didymus the Blind, who adds the observation that there are
only four perfect numbers that are less than 10,000. (Commentary on Genesis
1. 14–19).[8] Augustine of Hippo defines perfect numbers in _The 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.[9] The first known European
mention of the fifth perfect number is a manuscript written between 1456 and
1461 by an unknown mathematician.[10] In 1588, the Italian mathematician
Pietro Cataldi 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.[11][12][13]
** Even perfect numbers
[Euclid–Euler theorem] [mathematics] Euclid proved that
<math>2^{p-1}(2^p-1)</math> is an even perfect number whenever
<math>2^p-1</math> is prime (_Elements_, Prop. IX.36).
For example, the first four perfect numbers are generated by the formula
<math>2^{p-1}(2^p-1),</math> with [p] a prime number, as follows: <math
display=block>\begin{align} p = 2 &: \quad 2^1(2^2 - 1) = 2 \times 3 = 6 \\ p
= 3 &: \quad 2^2(2^3 - 1) = 4 \times 7 = 28 \\ p = 5 &: \quad 2^4(2^5 - 1) =
16 \times 31 = 496 \\ p = 7 &: \quad 2^6(2^7 - 1) = 64 \times 127 = 8128.
\end{align}</math>
Prime numbers of the form <math>2^p-1</math> are known as Mersenne primes,
after the seventeenth-century monk Marin Mersenne, who studied number theory
and perfect numbers. For <math>2^p-1</math> to be prime, it is necessary that
[p] itself be prime. However, not all numbers of the form <math>2^p-1</math>
with a prime [p] are prime; for example, [1=2 [11] − 1 = 2047 = 23 × 89] is
not a prime number.[All factors of <math> 2^p-1 </math> are congruent to [1
mod 2 _p_] . For example, [1=2 [11] − 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 prime —that is, [2_p_ + 1] is also prime—and [2_p_ + 1] is
congruent to 1 or 7 mod 8, then [2_p_ + 1] will be a factor of <math> 2^p-1,
</math> which is the case for [1= [p] = 11, 23, 83, 131, 179, 191, 239, 251,
....] [id=A002515] .] In fact, Mersenne primes are very rare: of the
approximately 4 million primes [p] up to 68,874,199, <math>2^p-1</math> is
prime for only 48 of them.[14]
While Nicomachus had stated (without proof) that [all] perfect numbers were of
the form <math>2^{n-1}(2^n-1)</math> where <math>2^n-1</math> is prime (though
he stated this somewhat differently), Ibn al-Haytham (Alhazen) circa AD 1000
was unwilling to go that far, declaring instead (also without proof) that the
formula yielded only every even perfect number.[15] It was not until the 18th
century that Leonhard Euler proved that the formula
<math>2^{p-1}(2^p-1)</math> 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 49 even perfect numbers are <math>2^{p-1}(2^p-1)</math> 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, 57885161, 74207281 [id=A000043] . <ref name="GIMPS Milestones" />
Three higher perfect numbers have also been discovered, namely those for which
[p] = 77232917, 82589933, and 136279841. 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 137750177. [2024], 52
Mersenne primes are known,[16] and therefore 52 even perfect numbers (the
largest of which is [2<sup> 136279840 </sup> × (2 <sup> 136279841 </sup> − 1)]
with 82,048,640 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 <math>2^{p-1}(2^p-1)</math>, each even perfect
number is the <math>(2^p-1)</math>-th triangular number (and hence equal to
the sum of the integers from 1 to <math>2^p-1</math>) and the
<math>2^{p-1}</math>-th hexagonal number. Furthermore, each even perfect
number except for 6 is the <math>\tfrac{2^p+1}{3}</math>-th centered nonagonal
number and is equal to the sum of the first <math>2^\frac{p-1}{2}</math> odd
cubes (odd cubes up to the cube of <math>2^\frac{p+1}{2}-1</math>):
<math display=block>\begin{alignat}{3}
6 &= 2^1(2^2 - 1) &&= 1 + 2 + 3, \\[8pt]
28 &= 2^2(2^3 - 1) &&= 1 + 2 + 3 + 4 + 5 + 6 + 7 \\
& &&= 1^3 + 3^3 \\[8pt]
496 &= 2^4(2^5 - 1) &&= 1 + 2 + 3 + \cdots + 29 + 30 + 31 \\
& &&= 1^3 + 3^3 + 5^3 + 7^3 \\[8pt]
8128 &= 2^6(2^7 - 1) &&= 1 + 2 + 3 + \cdots + 125 + 126 + 127 \\
& &&= 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3 \\[8pt]
33550336 &= 2^{12}(2^{13} - 1) &&= 1 + 2 + 3 + \cdots + 8189 + 8190 + 8191 \\
& &&= 1^3 + 3^3 + 5^3 + \cdots + 123^3 + 125^3 + 127^3
\end{alignat}</math>
Even perfect numbers (except 6) are of the form <math display=block>T_{2^p -
1} = 1 + \frac{(2^p - 2) \times (2^p + 1)}{2} = 1 + 9 \times T_{(2^p -
2)/3}</math>
with each resulting triangular number [T<sub> 7 </sub> [=] 28], [T<sub> 31
</sub> [=] 496], [T<sub> 127 </sub> [=] 8128] (after subtracting 1 from the
perfect number and dividing the result by 9) ending in 3 or 5, the sequence
starting with [T<sub> 2 </sub> [=] 3], [T[10] [=] 55], [1=T <sub> 42 </sub> =
903], [1=T <sub> 2730 </sub> = 3727815, ...][17] It follows that by 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 [1=8 + 1 + 2 + 8 = 19], [1=1 + 9 = 10], and
[1=1 + 0 = 1]. This works with all perfect numbers
<math>2^{p-1}(2^p-1)</math> with odd prime [p] and, in fact, with [all]
numbers of the form <math>2^{m-1}(2^m-1)</math> for odd integer (not
necessarily prime) [m].
Owing to their form, <math>2^{p-1}(2^p-1),</math> every even perfect number is
represented in binary form as [p] ones followed by [_p_ − 1] zeros; for
example:
<math display=block>\begin{array}{rcl} 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 \\ 8128_{10} =& \!\! 2^{12} + 2^{11} + 2^{10} + 2^9 + 2^8 + 2^7 +
2^6 \!\! &= 1111111000000_2 \end{array}</math>
Thus every even perfect number is a pernicious number.
Every even perfect number is also a practical number (cf. Related concepts).
==Odd perfect numbers ==<!-- This section is linked from Unsolved problems in
mathematics --> [mathematics] 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,[18] thus implying that no
odd perfect number exists, but Euler himself stated: "Whether ... there are
any odd perfect numbers is a most difficult question".[19] More recently, Carl
Pomerance has presented a heuristic argument suggesting that indeed no odd
perfect number should exist.[20] All perfect numbers are also harmonic divisor
numbers, and it has been conjectured as well that there are no odd harmonic
divisor 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.[21]
Any odd perfect number _N_ must satisfy the following conditions:
- _N_ > 10 <sup> 1500 </sup> . [22]
- _N_ is not divisible by 105. [23]
- _N_ is of the form _N_ ≡ 1 (mod 12) or _N_ ≡ 117 (mod 468) or _N_ ≡ 81 (mod 324). [24]
- The largest prime factor of _N_ is greater than 10 <sup> 8 </sup> , [25] and less than <math> \sqrt [ 3 ] {3N}. </math> [26]
- The second largest prime factor is greater than 10 <sup> 4 </sup> , [27] and is less than <math> \sqrt [ 5 ] {2N} </math> . [28]
- The third largest prime factor is greater than 100, [29] and less than <math> \sqrt [ 6 ] {2N}. </math> [30]
- _N_ has at least 101 prime factors and at least 10 distinct prime factors. <ref name="Ochem and Rao (2012)"/> [31] If 3 does not divide _N_ , then _N_ has at least 12 distinct prime factors. [32]
- _N_ is of the form
<math> N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k}, </math>
where:
* _q_ , _p_ <sub> 1 </sub> , ..., _p_ <sub> _k_ </sub> are distinct odd primes (Euler).
* _q_ ≡ α ≡ 1 ( mod 4) (Euler).
* The smallest prime factor of _N_ is at most <math display="inline"> \frac{k-1}{2}. </math> [33]
* At least one of the prime powers dividing _N_ exceeds 10 <sup> 62 </sup> . <ref name="Ochem and Rao (2012)"/>
* <math> N < 2^{(4^{k+1}-2^{k+1})} </math> [34] [35]
* <math display="inline"> \alpha + 2e_1 + 2e_2 + 2e_3 + \cdots + 2e_k \geq \frac{99k-224}{37} </math> . <ref name="Zelinsky 2021"/> [36] [37]
* <math> qp_1p_2p_3 \cdots p_k < 2N^{\frac{17}{26 }} </math> . [38]
* <math display="inline"> \frac{1}{q} + \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k} < \ln 2 </math> . [39] [40]
Furthermore, several minor results are known about the exponents
_e_<sub>1</sub>, ..., _e_<sub>_k_</sub>.
- Not all _e_ <sub> _i_ </sub> ≡ 1 ( mod 3). [41]
- Not all _e_ <sub> _i_ </sub> ≡ 2 ( mod 5). [42]
- If all _e_ <sub> _i_ </sub> ≡ 1 ( mod 3) or 2 ( mod 5), then the smallest prime factor of _N_ must lie between 10 <sup> 8 </sup> and 10 <sup> 1000 </sup> . <ref name="Fletcher, Nielsen and Ochem (2012)"/>
- More generally, if all 2 _e_ <sub> _i_ </sub> +1 have a prime factor in a given finite set _S_ , then the smallest prime factor of _N_ must be smaller than an effectively computable constant depending only on _S_ . <ref name="Fletcher, Nielsen and Ochem (2012)"/>
- If ( _e_ <sub> 1 </sub> , ..., _e_ <sub> _k_ </sub> ) = (1, ..., 1, 2, ..., 2) with _t_ ones and _u_ twos, then <math display="inline"> \frac {t-1}{4}\leq u \leq 2t+\sqrt{\alpha} </math> . [43]
- (_e_ <sub> 1 </sub> , ..., _e_ <sub> _k_ </sub> ) ≠ (1, ..., 1, 3), [44] (1, ..., 1, 5), (1, ..., 1, 6). [45]
- If [1= _e_ <sub> 1 </sub> = ... = _e_ <sub> _k_ </sub> = _e_] , then
- _e_ cannot be 3, [46] 5, 24, [47] 6, 8, 11, 14 or 18. <ref name="Cohen and Williams (1985)" />
- <math> k\leq 2e^2+8e+2 </math> and <math> N < 2^{4^{2e^2 + 8e+3 }} </math> . [48]
In 1888, Sylvester stated:[49] [... 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.]
On the other hand, several odd integers come close to being perfect. René
Descartes observed that the number [_D_ [=] 3 <sup> 2 </sup> ⋅ 7 <sup> 2
</sup> ⋅ 11 <sup> 2 </sup> ⋅ 13 <sup> 2 </sup> ⋅ 22021 [=] (3⋅1001) <sup> 2
</sup> ⋅ (22⋅1001 − 1) [=] 198585576189] would be an odd perfect number if
only [22021 ( [=] 19 <sup> 2 </sup> ⋅ 61)] were a prime number. The odd
numbers with this property (they would be perfect if one of their composite
factors were prime) are the Descartes numbers.
** 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 only even perfect number of the form _n_ <sup> 3 </sup> + 1 is 28 [Makowski] . [50]
- 28 is also the only even perfect number that is a sum of two positive cubes of integers [Gallardo] . [51]
- 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, <math> \sigma_1(n) = 2n </math> , and divide both sides by _n_ ):
- For 6, we have <math display="inline"> \frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{1}{1} = \frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{6}{6} = \frac{1+2+3+6}{6} = \frac{2\cdot 6}{6} = 2 </math> ;
- For 28, we have <math display="inline"> \frac {1}{28}+ \frac{1}{14}+ \frac{1}{7}+ \frac {1}{4}+ \frac{1}{2}+ \frac {1}{1}= 2 </math> , etc.
- The number of divisors of a perfect number (whether even or odd) must be even, because _N_ cannot be a perfect square. [52]
- From these two results it follows that every perfect number is an Ore's harmonic number .
- The even perfect numbers are not trapezoidal number s; that is, they cannot be represented as the difference of two positive non-consecutive triangular number s. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form <math> 2^{n-1}(2^n+1) </math> formed as the product of a Fermat prime <math> 2^n+1 </math> with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes. [53]
- The number of perfect numbers less than _n_ is less than <math> c\sqrt{n} </math> , where _c_ > 0 is a constant. [54] In fact it is <math> o(\sqrt{n}) </math> , using little-o notation . [55]
- Every even perfect number ends in 6 or 28 in base ten and, with the only exception of 6, ends in 1 in base 9. [56] [57] Therefore, in particular the digital root of every even perfect number other than 6 is 1.
- The only square-free perfect number is 6. [58]
** Related concepts
[Euler_diagram_numbers_with_many_divisors.svg] 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 [1= _s_ ( _n_ ) = _σ_ ( _n_ ) − _n_], and the aliquot sequence
associated with a perfect number is a constant sequence. All perfect numbers
are also <math>\mathcal{S}</math>-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.
** See also
- Harmonic divisor number
- Hyperperfect number
- Leinster group
- List of Mersenne primes and perfect numbers
- Multiply perfect number
- Superperfect number s
- Unitary perfect number
** Notes
[Notelist]
** References
[Reflist]
*** Sources
[refbegin]
- Euclid, _Elements_ , Book IX, Proposition 36. See D.E. Joyce's website (see http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html) for a translation and discussion of this proposition and its proof.
- [ last1 = Kanold ]
- [ last1 = Steuerwald ]
- [last=Tóth]
[refend]
** Further reading
<!-- From http://mathforum.org/library/drmath/view/51516.html -->
- Nankar, M.L.: "History of perfect numbers", Ganita Bharati 1, no. 1–2 (1979), 7–8.
- [ last1 = Hagis ]
- Riele, H.J.J. "Perfect Numbers and Aliquot Sequences" in H.W. Lenstra and R. Tijdeman (eds.): _Computational Methods in Number Theory_ , Vol. 154, Amsterdam, 1982, pp. 141–157.
- Riesel, H. _Prime Numbers and Computer Methods for Factorisation_ , Birkhauser, 1985.
- [ last1=Sándor ]
** External links
- [title=Perfect number]
- David Moews: Perfect, amicable, and sociable numbers (see http://djm.cc/amicable.html)
- Perfect numbers – History and Theory (see https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers/)
- [urlname=PerfectNumber]
- [sequencenumber=A000396]
- Great Internet Mersenne Prime Search (see https://www.mersenne.org/) (GIMPS)
- Perfect Numbers (see http://mathforum.org/dr.math/faq/faq.perfect.html) , math forum at Drexel
- [last=Grimes]
[Divisor classes] [Classes of natural numbers]
[Authority control] Category:Divisor function Category:Integer sequences
Category:Mersenne primes Category:Unsolved problems in number theory