View source for Perfect number

{{Short description|Number equal to the sum of its proper divisors}}
{{About||the 2012 film|Perfect Number (film){{!}}''Perfect Number'' (film)}}
[[File:Perfect number Cuisenaire rods 6 exact.svg|thumb|Illustration of the perfect number status of the number 6]]

In [[number theory]], a '''perfect number''' is a [[positive integer]] that is equal to the sum of its positive proper [[divisor]]s, that is, divisors excluding the number itself.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Perfect Number |url=https://mathworld.wolfram.com/PerfectNumber.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en |quote=Perfect numbers are positive integers n such that  n=s(n), where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), ...}}</ref> 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 (number)|6]], [[28 (number)|28]], [[496 (number)|496]], [[8128 (number)|8128]], 33550336, 8589869056, and 137438691328.<ref>{{Cite web |title=A000396 - OEIS |url=https://oeis.org/A000396 |access-date=2024-03-21 |website=oeis.org}}</ref>

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|Euclid's ''Elements'']] (VII.22) where it is called {{lang|grc|τέλειος ἀριθμός}} (''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.<ref name="The Euclid–Euler theorem">Caldwell, Chris, [https://primes.utm.edu/notes/proofs/EvenPerfect.html "A proof that all even perfect numbers are a power of two times a Mersenne prime"].</ref> 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&nbsp;BC Euclid showed that if 2<sup>''p''</sup>&nbsp;−&nbsp;1 is prime then 2<sup>''p''−1</sup>(2<sup>''p''</sup>&nbsp;−&nbsp;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&nbsp;100.<ref name="Dickinson LE (1919)">{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|page=4|year=1919|publisher=Carnegie Institution of Washington|location=Washington|url=https://archive.org/stream/historyoftheoryo01dick#page/4/}}</ref> In modern language, Nicomachus states without proof that {{em|every}} perfect number is of the form <math>2^{n-1}(2^n-1)</math> where <math>2^n-1</math> is prime.<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Perfect_numbers.html|title=Perfect numbers|website=www-groups.dcs.st-and.ac.uk|access-date=9 May 2018}}</ref><ref>In ''[https://archive.org/download/NicomachusIntroToArithmetic/nicomachus_introduction_arithmetic.pdf Introduction to Arithmetic]'', Chapter 16, he says of perfect numbers, "There is a method of producing them, neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way." He then goes on to explain a procedure which is equivalent to finding a [[triangular number]] based on a Mersenne prime.</ref> He seems to be unaware that {{mvar|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]],<ref>Commentary on the Gospel of John 28.1.1–4, with further references in the [[Sources Chrétiennes]] edition: vol. 385, 58–61.</ref> 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).<ref>{{cite conference|url=http://torreys.org/sblpapers2015/S22-05_philonic_arithmological_exegesis.pdf |first=Justin M.|last=Rogers|title=The Reception of Philonic Arithmological Exegesis in Didymus the Blind's ''Commentary on Genesis''|work=Society of Biblical Literature National Meeting, Atlanta, Georgia|year=2015}}</ref> [[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 [[Ibn Fallus|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.<ref>Roshdi Rashed, ''The Development of Arabic Mathematics: Between Arithmetic and Algebra'' (Dordrecht: Kluwer Academic Publishers, 1994), pp.&nbsp;328–329.</ref> The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.<ref>[[Bayerische Staatsbibliothek]], Clm 14908. See {{cite book|author=David Eugene Smith|author-link=David Eugene Smith|title=History of Mathematics: Volume II|year=1925|publisher=Dover|location=New York|isbn=0-486-20430-8|pages=21|url=https://archive.org/stream/historyofmathema031897mbp#page/n35/mode/2up}}</ref> 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.<ref>{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|year=1919|publisher=Carnegie Institution of Washington|location=Washington|page=10|url=https://archive.org/stream/historyoftheoryo01dick#page/10/}}</ref><ref name="Pickover C (2001)">{{cite book|last=Pickover|first=C|title=Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning|year=2001|publisher=Oxford University Press|location=Oxford|isbn=0-19-515799-0|pages=360|url=https://books.google.com/books?id=52N0JJBspM0C&pg=PA360}}</ref><ref name="Peterson I (2002)">{{cite book|last=Peterson|first=I|title=Mathematical Treks: From Surreal Numbers to Magic Circles|year=2002|publisher=Mathematical Association of America|location=Washington|isbn=88-8358-537-2|pages=132|url=https://books.google.com/books?id=4gWSAraVhtAC&pg=PA132}}</ref>

== Even perfect numbers ==
{{See also|Euclid–Euler theorem}}
{{Unsolved|mathematics|Are there infinitely many perfect numbers?}}
[[Euclid]] proved that <math>2^{p-1}(2^p-1)</math> is an even perfect number whenever <math>2^p-1</math> is prime (''[[Euclid's Elements|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 {{mvar|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 prime]]s, 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 {{mvar|p}} itself be prime. However, not all numbers of the form <math>2^p-1</math> with a prime {{mvar|p}} are prime; for example, {{nowrap|1=2{{sup|11}} − 1 = 2047 = 23 × 89}} is not a prime number.{{efn|All factors of <math>2^p-1</math> are congruent to {{math|1 [[Modular arithmetic|mod]] 2''p''}}. For example, {{nowrap|1=2{{sup|11}} − 1 = 2047 = 23 × 89}}, and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever {{mvar|p}} is a [[Sophie Germain prime]]—that is, {{math|2''p'' + 1}} is also prime—and {{math|2''p'' + 1}} is congruent to 1 or 7 mod 8, then {{math|2''p'' + 1}} will be a factor of <math>2^p-1,</math> which is the case for {{nowrap|1={{mvar|p}} = 11, 23, 83, 131, 179, 191, 239, 251, ...}} {{OEIS2C|id=A002515}}.}} In fact, Mersenne primes are very rare: of the approximately 4 million primes {{mvar|p}} up to 68,874,199, <math>2^p-1</math> is prime for only 48 of them.<ref name="GIMPS Milestones">{{Cite web |title=GIMPS Milestones Report |url=https://www.mersenne.org/report_milestones/ |access-date=28 July 2024 |website=[[Great Internet Mersenne Prime Search]]}}</ref>

While [[Nicomachus]] had stated (without proof) that {{em|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.<ref>{{MacTutor|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> It was not until the 18th century that [[Leonhard Euler]] proved that the formula <math>2^{p-1}(2^p-1)</math> indeed yields all the even perfect numbers. Thus, there is a [[bijection|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 50 even perfect numbers are <math>2^{p-1}(2^p-1)</math> for
: {{mvar|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, 77232917 {{OEIS2C|id=A000043}}.<ref name="GIMPS Milestones" />

Two higher perfect numbers have also been discovered, namely those for which {{mvar|p}} = 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 {{mvar|p}} below 138277717. {{As of|2024|10}}, 52 Mersenne primes are known,<ref name="mersenne">{{cite web |url=http://www.mersenne.org/ |title=GIMPS Home |publisher=Mersenne.org |access-date=2024-10-21}}</ref> and therefore 52 even perfect numbers (the largest of which is {{nowrap|2<sup>136279840</sup> × (2<sup>136279841</sup> − 1)}} with 82,048,640 digits). It is [[List of unsolved problems in mathematics|not known]] whether there are [[infinite set|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 {{nowrap|T<sub>7</sub> {{=}} 28}}, {{nowrap|T<sub>31</sub> {{=}} 496}}, {{nowrap|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 {{nowrap|T<sub>2</sub> {{=}} 3}}, {{nowrap|T{{sub|10}} {{=}} 55}}, {{nowrap|1=T<sub>42</sub> = 903}}, {{nowrap|1=T<sub>2730</sub> = 3727815, ...}}<ref name="mathworld">{{MathWorld|urlname=PerfectNumber|title=Perfect Number}}</ref> 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 {{nowrap|1=8 + 1 + 2 + 8 = 19}}, {{nowrap|1=1 + 9 = 10}}, and {{nowrap|1=1 + 0 = 1}}. This works with all perfect numbers <math>2^{p-1}(2^p-1)</math> with odd prime {{mvar|p}} and, in fact, with {{em|all}} numbers of the form <math>2^{m-1}(2^m-1)</math> for odd integer (not necessarily prime) {{mvar|m}}.

Owing to their form, <math>2^{p-1}(2^p-1),</math> every even perfect number is represented in binary form as {{mvar|p}} ones followed by {{math|''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|Related concepts]]).

== Odd perfect numbers ==<!-- This section is linked from [[Unsolved problems in mathematics]] -->
{{Unsolved|mathematics|Are there any odd perfect numbers?}}
It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, [[Jacques Lefèvre d'Étaples|Jacques Lefèvre]] stated that Euclid's rule gives all perfect numbers,<ref>{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|year=1919|publisher=Carnegie Institution of Washington|location=Washington|page=6|url=https://archive.org/stream/historyoftheoryo01dick#page/6/}}</ref> thus implying that no odd perfect number exists, but Euler himself stated: "Whether&nbsp;... there are any odd perfect numbers is a most difficult question".<ref>{{cite web|url=https://people.math.harvard.edu/~knill/seminars/perfect/handout.pdf|title=The oldest open problem in mathematics
|website=Harvard.edu|access-date=16 June 2023}}</ref> More recently, [[Carl Pomerance]] has presented a [[heuristic argument]] suggesting that indeed no odd perfect number should exist.<ref name="oddperfect">[http://oddperfect.org/pomerance.html Oddperfect.org]. {{Webarchive|url=https://web.archive.org/web/20061229094011/http://oddperfect.org/pomerance.html |date=2006-12-29 }}</ref> All perfect numbers are also [[harmonic divisor number]]s, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1.

Any odd perfect number ''N'' must satisfy the following conditions:
* ''N'' > 10<sup>1500</sup>.<ref name="Ochem and Rao (2012)">{{cite journal | last1=Ochem | first1=Pascal | last2=Rao | first2=Michaël | title=Odd perfect numbers are greater than 10<sup>1500</sup> | journal=[[Mathematics of Computation]] | year=2012 | volume=81 | issue=279 | doi=10.1090/S0025-5718-2012-02563-4 | url=http://www.lirmm.fr/~ochem/opn/opn.pdf | pages=1869–1877 | zbl=1263.11005 | issn=0025-5718 | doi-access=free }}</ref>
* ''N'' is not divisible by 105.<ref name="Kühnel U">{{cite journal|last=Kühnel|first=Ullrich|title=Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen|journal=Mathematische Zeitschrift|year=1950|volume=52|pages=202–211|doi=10.1007/BF02230691|s2cid=120754476|language=de}}</ref>
* ''N'' is of the form ''N'' ≡ 1 (mod 12) or ''N'' ≡ 117 (mod 468) or ''N'' ≡ 81 (mod 324).<ref name="Roberts T (2008)">{{cite journal|last=Roberts|first=T|title=On the Form of an Odd Perfect Number|journal=Australian Mathematical Gazette|year=2008|volume=35|issue=4|pages=244|url=http://www.austms.org.au/Publ/Gazette/2008/Sep08/CommsRoberts.pdf|archive-date=2013-05-14|access-date=2009-12-18|archive-url=https://web.archive.org/web/20130514033635/http://www.austms.org.au/Publ/Gazette/2008/Sep08/CommsRoberts.pdf|url-status=dead}}</ref>
* The largest [[prime power]] ''p<sup>a</sup>'' that divides ''N'' is greater than 10<sup>62</sup>.<ref name="Ochem and Rao (2012)"/>
* The largest prime factor of ''N'' is greater than 10<sup>8</sup>,<ref name="Goto and Ohno (2008)">{{cite journal|last=Goto|first=T|author2=Ohno, Y|title=Odd perfect numbers have a prime factor exceeding 10<sup>8</sup>|journal=Mathematics of Computation|year=2008|volume=77|issue=263|pages=1859–1868|doi=10.1090/S0025-5718-08-02050-9|url=http://www.ma.noda.tus.ac.jp/u/tg/perfect/perfect.pdf|access-date=30 March 2011|bibcode=2008MaCom..77.1859G|doi-access=free|archive-date=2011-08-07|archive-url=https://web.archive.org/web/20110807101906/http://www.ma.noda.tus.ac.jp/u/tg/perfect/perfect.pdf|url-status=dead}}</ref> and less than <math>\sqrt[3]{3N}.</math> <ref name="AK 2012">{{cite journal |last1=Konyagin |first1=Sergei |last2=Acquaah |first2=Peter |title=On Prime Factors of Odd Perfect Numbers |journal=International Journal of Number Theory |date=2012 |volume=8 |issue=6 |pages=1537–1540|doi=10.1142/S1793042112500935 }}</ref>
* The second largest prime factor is greater than 10<sup>4</sup>,<ref name="Ianucci DE (1999)">{{cite journal|last=Iannucci|first=DE|title=The second largest prime divisor of an odd perfect number exceeds ten thousand|journal=Mathematics of Computation|year=1999|volume=68|issue=228|pages=1749–1760|url=https://www.ams.org/journals/mcom/1999-68-228/S0025-5718-99-01126-6/S0025-5718-99-01126-6.pdf|access-date=30 March 2011|doi=10.1090/S0025-5718-99-01126-6|bibcode=1999MaCom..68.1749I|doi-access=free}}</ref> and is less than <math>\sqrt[5]{2N}</math>.<ref name="Zelinsky 2019">{{cite journal |last1=Zelinsky |first1=Joshua |title=Upper bounds on the second largest prime factor of an odd perfect number |journal=International Journal of Number Theory |date=July 2019 |volume=15 |issue=6 |pages=1183–1189 |doi=10.1142/S1793042119500659 |arxiv=1810.11734 |s2cid=62885986 }}.</ref>
* The third largest prime factor is greater than 100,<ref name="Ianucci DE (2000)">{{cite journal|last=Iannucci|first=DE|title=The third largest prime divisor of an odd perfect number exceeds one hundred|journal=Mathematics of Computation|year=2000|volume=69|issue=230|pages=867–879|url=https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01127-8/S0025-5718-99-01127-8.pdf|access-date=30 March 2011|doi=10.1090/S0025-5718-99-01127-8|bibcode=2000MaCom..69..867I|doi-access=free}}</ref> and less than <math>\sqrt[6]{2N}.</math><ref name="Zelinsky 2021a">{{cite journal |first1=Sean|last1=Bibby|first2=Pieter|last2=Vyncke|last3=Zelinsky |first3=Joshua |title=On the Third Largest Prime Divisor of an Odd Perfect Number |journal=Integers |date=23 November 2021 |volume=21 |url=http://math.colgate.edu/~integers/v115/v115.pdf |access-date=6 December 2021}}</ref>
* ''N'' has at least 101 prime factors and at least 10 distinct prime factors.<ref name="Ochem and Rao (2012)"/><ref name="Nielsen Pace P. (2015)">{{cite journal|last=Nielsen|first=Pace P.|title=Odd perfect numbers, Diophantine equations, and upper bounds|journal=Mathematics of Computation|year=2015|volume=84|issue=295|pages=2549–2567|url=https://math.byu.edu/~pace/BestBound_web.pdf|access-date=13 August 2015|doi=10.1090/S0025-5718-2015-02941-X|doi-access=free|archive-date=8 July 2015|archive-url=https://web.archive.org/web/20150708185554/https://math.byu.edu/~pace/BestBound_web.pdf|url-status=dead}}</ref> If 3 does not divide ''N'', then ''N'' has at least 12 distinct prime factors.<ref name="Nielsen Pace P. (2007)">{{cite journal|last=Nielsen|first=Pace P.|title=Odd perfect numbers have at least nine distinct prime factors|journal=Mathematics of Computation|year=2007|volume=76|pages=2109–2126|url=https://math.byu.edu/~pace/NotEight_web.pdf|access-date=30 March 2011|doi=10.1090/S0025-5718-07-01990-4|issue=260|arxiv=math/0602485|bibcode=2007MaCom..76.2109N|s2cid=2767519|archive-date=2021-11-03|archive-url=https://web.archive.org/web/20211103100630/https://math.byu.edu/~pace/NotEight_web.pdf|url-status=dead}}</ref>
* ''N'' is of the form
::<math>N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k}, </math>
:where:
:* ''q'',&nbsp;''p''<sub>1</sub>,&nbsp;...,&nbsp;''p''<sub>''k''</sub> are distinct odd primes (Euler).
:* ''q'' ≡&nbsp;α ≡&nbsp;1 ([[Modular arithmetic|mod]] 4) (Euler).
:* The smallest prime factor of ''N'' is at most <math display="inline">\frac{k-1}{2}.</math><ref name="Zelinsky 2021">{{cite journal |last1=Zelinsky |first1=Joshua |title=On the Total Number of Prime Factors of an Odd Perfect Number |journal=Integers |date=3 August 2021 |volume=21 |url=http://math.colgate.edu/~integers/v76/v76.pdf |access-date=7 August 2021}}</ref>
:* <math> N < 2^{(4^{k+1}-2^{k+1})}</math><ref name="Chen and Tang">{{cite journal |last1=Chen |first1=Yong-Gao |last2=Tang |first2=Cui-E |title=Improved upper bounds for odd multiperfect numbers. |journal=Bulletin of the Australian Mathematical Society |date=2014 |volume=89 |issue=3 |pages=353–359|doi=10.1017/S0004972713000488 |doi-access=free }}</ref><ref name="Nielsen (2003)">{{Cite journal |last=Nielsen |first=Pace P. |year=2003 |title=An upper bound for odd perfect numbers |url=https://math.colgate.edu/~integers/vol3.html |journal=Integers |volume=3 |doi=10.5281/zenodo.7607545 |access-date=14 August 2025}}</ref>
:* <math display="inline">\alpha + 2e_1 + 2e_2 + 2e_3 + \cdots + 2e_k \geq \frac{99k-224}{37} </math>.<ref name="Zelinsky 2021"/><ref name="Ochem and Rao (2014)">{{cite journal | last1=Ochem | first1=Pascal | last2=Rao | first2=Michaël | title=On the number of prime factors of an odd perfect number.  | journal=[[Mathematics of Computation]] | year=2014 | volume=83 | issue=289 | pages=2435–2439  | doi=10.1090/S0025-5718-2013-02776-7 | doi-access=free }}</ref><ref name="ClayotonHansen">{{cite journal |last1=Graeme Clayton, Cody Hansen |title=On inequalities involving counts of the prime factors of an odd perfect number |journal=Integers |date=2023 |volume=23 |arxiv=2303.11974 |url=http://math.colgate.edu/~integers/x79/x79.pdf |access-date=29 November 2023}}</ref>
:* <math> qp_1p_2p_3 \cdots p_k < 2N^{\frac{17}{26}}</math>.<ref name="LucaPomerance">{{cite journal |last1=Pomerance |first1=Carl |last2=Luca |first2=Florian |title=On the radical of a perfect number |journal=New York Journal of Mathematics |date=2010 |volume=16 |pages=23–30 |url=http://nyjm.albany.edu/j/2010/16-3.html |access-date=7 December 2018}}</ref>
:* <math display="inline"> \frac{1}{q} + \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k} < \ln 2</math>.<ref name="Cohen1978">{{cite journal |last1=Cohen |first1=Graeme |title=On odd perfect numbers |journal=Fibonacci Quarterly |date=1978 |volume=16 |issue=6 |page=523-527|doi=10.1080/00150517.1978.12430277 }}</ref><ref>{{cite journal |last1=Suryanarayana |first1=D. |title=On Odd Perfect Numbers II |journal=Proceedings of the American Mathematical Society |date=1963 |volume=14 |issue=6 |pages=896–904|doi=10.1090/S0002-9939-1963-0155786-8 }}</ref>

Furthermore, several minor results are known about the exponents
''e''<sub>1</sub>,&nbsp;...,&nbsp;''e''<sub>''k''</sub>.
* Not all ''e''<sub>''i''</sub>&nbsp;≡&nbsp;1 ([[Modular arithmetic|mod]] 3).<ref name="McDaniel (1970)">{{cite journal | last1=McDaniel | first1=Wayne L. | title=The non-existence of odd perfect numbers of a certain form | journal=Archiv der Mathematik | volume=21 | year=1970 | issue=1 | pages=52–53 | doi=10.1007/BF01220877 | mr=0258723 | s2cid=121251041 | issn=1420-8938 }}</ref>
* Not all ''e''<sub>''i''</sub>&nbsp;≡&nbsp;2 ([[Modular arithmetic|mod]] 5).<ref name="Fletcher, Nielsen and Ochem (2012)">{{cite journal | last1=Fletcher | first1=S. Adam | last2=Nielsen | first2=Pace P. | last3=Ochem | first3=Pascal | title=Sieve methods for odd perfect numbers | journal=[[Mathematics of Computation]] | volume=81 | year=2012 | issue=279 | pages=1753?1776  | doi=10.1090/S0025-5718-2011-02576-7 | url=http://www.lirmm.fr/~ochem/opn/OPNS_Adam_Pace.pdf | mr = 2904601 | issn=0025-5718 | doi-access=free }}</ref>
* If all ''e''<sub>''i''</sub>&nbsp;≡&nbsp;1 ([[Modular arithmetic|mod]] 3) or 2 ([[Modular arithmetic|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>,&nbsp;...,&nbsp;''e''<sub>''k''</sub>)&nbsp;=&nbsp; (1,&nbsp;...,&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;2) with ''t'' ones and ''u'' twos, then <math display="inline">\frac {t-1}{4}\leq u \leq 2t+\sqrt{\alpha}</math>.<ref name="Cohen (1987)">{{cite journal | last1=Cohen | first1=G. L. | title=On the largest component of an odd perfect number | journal=Journal of the Australian Mathematical Society, Series A | volume=42 | year=1987 | issue=2 | pages=280–286 | doi=10.1017/S1446788700028251 | mr = 0869751| issn=1446-8107 | doi-access=free }}</ref>
* (''e''<sub>1</sub>,&nbsp;...,&nbsp;''e''<sub>''k''</sub>) &ne; (1, ..., 1, 3),<ref name="Kanold (1950)">{{cite journal | last1=Kanold | author-link=:de:Hans-Joachim Kanold | first1=Hans-Joachim | title=Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II | journal=[[Journal für die reine und angewandte Mathematik]] | volume=188  | year=1950 | issue=1 | pages=129–146 | doi=10.1515/crll.1950.188.129 | mr=0044579 | s2cid=122452828 | issn=1435-5345 }}</ref> (1, ..., 1, 5), (1, ..., 1, 6).<ref name="Cohen and Williams (1985)">{{cite journal | last1=Cohen | first1=G. L. | last2=Williams | first2=R. J. | title=Extensions of some results concerning odd perfect numbers | journal=[[Fibonacci Quarterly]] | volume=23 | year=1985 | issue=1 | pages=70–76 | doi=10.1080/00150517.1985.12429857 | url=https://www.fq.math.ca/Scanned/23-1/cohen.pdf | mr=0786364 | issn=0015-0517 }}</ref>
* If {{math|1= ''e''<sub>1</sub> = ... = ''e''<sub>''k''</sub> = ''e''}}, then
** ''e'' cannot be 3,<ref name="Hagis and McDaniel (1972)">{{cite journal | last1=Hagis | first1=Peter Jr. | last2=McDaniel | first2=Wayne L. | title=A new result concerning the structure of odd perfect numbers | journal=Proceedings of the American Mathematical Society | volume=32 | year=1972 | issue=1 | pages=13–15 | doi=10.1090/S0002-9939-1972-0292740-5 | mr = 0292740 | issn=1088-6826 | doi-access=free }}</ref> 5, 24,<ref name="McDaniel and Hagis (1975)">{{cite journal | last1=McDaniel | first1=Wayne L. | last2=Hagis | first2=Peter Jr.  | title=Some results concerning the non-existence of odd perfect numbers of the form <math>p^{\alpha} M^{2\beta}</math> | journal=[[Fibonacci Quarterly]] | volume=13 | year=1975 | issue=1 | pages=25–28 | doi=10.1080/00150517.1975.12430680 | url=https://www.fq.math.ca/Scanned/13-1/mcdaniel.pdf | mr=0354538 | issn=0015-0517 }}</ref> 6, 8, 11, 14 or 18.<ref name="Cohen and Williams (1985)" />
** <math> k\leq 2e^2+8e+2</math>.<ref name="Yamada (2019)">{{cite journal | last1=Yamada | first1=Tomohiro | title=A new upper bound for odd perfect numbers of a special form | journal=Colloquium Mathematicum | volume=156 | year=2019 | issue=1 | pages=15–21 | doi=10.4064/cm7339-3-2018 | issn=1730-6302 | arxiv=1706.09341 | s2cid=119175632 }}</ref>

In 1888, [[James Joseph Sylvester|Sylvester]] stated:<ref>The Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton", ''Compte Rendu de l'Association Française'' (Toulouse, 1887), pp. 164–168.</ref>
{{blockquote|...&nbsp;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 {{math|''D'' {{=}} 3<sup>2</sup> ⋅ 7<sup>2</sup> ⋅ 11<sup>2</sup> ⋅ 13<sup>2</sup> ⋅ 22021 {{=}} (3⋅1001)<sup>2</sup>&thinsp;⋅&thinsp;(22⋅1001 − 1) {{=}} 198585576189}} would be an odd perfect number if only {{math|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 number]]s. Many of the properties proven about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of these numbers may lead to a proof that no odd perfect numbers exist.<ref>{{cite news |last1=Nadis |first1=Steve |title=Mathematicians Open a New Front on an Ancient Number Problem |url=https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/ |access-date=10 September 2020 |work=Quanta Magazine |date=10 September 2020}}</ref>

== 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 K. Guy|Richard Guy]]'s [[strong law of small numbers]]:
* The only even perfect number of the form ''n''<sup>3</sup>&nbsp;+&nbsp;1 is 28 {{harv|Makowski|1962}}.<ref>{{cite journal|first=A.|last=Makowski|title=Remark on perfect numbers|journal=[[Elem. Math.]]|volume=17|year=1962|issue=5|page=109}}</ref>
* 28 is also the only even perfect number that is a sum of two positive cubes of integers {{harv|Gallardo|2010}}.<ref>{{cite journal|first=Luis H.|last=Gallardo|title=On a remark of Makowski about perfect numbers|journal=[[Elem. Math.]]|volume=65|year=2010|issue=3 |pages=121–126|doi=10.4171/EM/149|doi-access=free}}.</ref>
* The [[multiplicative inverse|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.<ref>{{citation|title=Computational Number Theory and Modern Cryptography|first=Song Y.|last=Yan|publisher=John Wiley & Sons|year=2012|isbn=9781118188613|at=Section 2.3, Exercise 2(6)|url=https://books.google.com/books?id=eLAV586iF-8C&pg=PA30}}.</ref>
** 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.<ref>{{Cite journal|title=Characterising non-trapezoidal numbers|first1=Chris|last1=Jones|first2=Nick|last2=Lord|journal=The Mathematical Gazette|volume=83|issue=497|year=1999|pages=262–263|doi=10.2307/3619053|jstor=3619053|publisher=The Mathematical Association|s2cid=125545112 }}</ref>
* The number of perfect numbers less than ''n'' is less than <math>c\sqrt{n}</math>, where ''c'' > 0 is a constant.<ref name="Hornfeck (1955)">{{cite journal|last=Hornfeck|first=B|title=Zur Dichte der Menge der vollkommenen zahlen|journal=Arch. Math.|year=1955|volume=6|pages=442–443|doi=10.1007/BF01901120|issue=6|s2cid=122525522}}</ref> In fact it is <math>o(\sqrt{n})</math>, using [[little-o notation]].<ref>{{cite journal|last=Kanold|first=HJ|title=Eine Bemerkung ¨uber die Menge der vollkommenen zahlen|journal=Math. Ann.|year=1956|volume=131|pages=390–392|doi=10.1007/BF01350108|issue=4|s2cid=122353640}}</ref>
* 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.<ref>H. Novarese. ''Note sur les nombres parfaits'' Texeira J. VIII (1886), 11–16.</ref><ref>{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|year=1919|publisher=Carnegie Institution of Washington|location=Washington|page=25|url=https://archive.org/stream/historyoftheoryo01dick#page/25/}}</ref> Therefore, in particular the [[digital root]] of every even perfect number other than 6 is 1.
* The only [[Square-free integer|square-free]] perfect number is 6.<ref>{{cite book|title=Number Theory: An Introduction to Pure and Applied Mathematics|volume=201|series=Chapman & Hall/CRC Pure and Applied Mathematics|first=Don|last=Redmond|publisher=CRC Press|year=1996|isbn=9780824796969|at=Problem 7.4.11, p.&nbsp;428|url=https://books.google.com/books?id=3ffXkusQEC0C&pg=PA428}}.</ref>

== Related concepts ==
{{Euler_diagram_numbers_with_many_divisors.svg}}
The sum of [[proper divisor]]s gives various other kinds of numbers. Numbers where the sum is less than the number itself are called [[deficient number|deficient]], and where it is greater than the number, [[abundant 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 number|amicable]], and larger cycles of numbers are called [[sociable number|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 (mathematics)|fixed point]] of the [[restricted divisor function]] {{nowrap|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 number]]s.

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 number]]s.

== 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, ''[[Euclid's Elements|Elements]]'', Book IX, Proposition 36. See [http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html D.E. Joyce's website] for a translation and discussion of this proposition and its proof.
* {{cite journal | last1 = Kanold | first1 = H.-J. | year = 1941 | title = Untersuchungen über ungerade vollkommene Zahlen | journal = Journal für die Reine und Angewandte Mathematik | volume = 1941 | issue = 183 | pages = 98–109 | doi = 10.1515/crll.1941.183.98 | s2cid = 115983363 }}
* {{cite journal | last1 = Steuerwald | first1 = R. | title = Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl | journal = S.-B. Bayer. Akad. Wiss. | volume = 1937 | pages = 69–72 }}
* {{cite journal|last=Tóth|first=László|title=Odd Spoof Multiperfect Numbers|journal=Integers|volume=25|year=2025|issue=A19 |arxiv=2502.16954|url=https://math.colgate.edu/~integers/z19/z19.pdf}}
{{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.
* {{cite journal | last1 = Hagis | first1 = P. | year = 1973 | title = A Lower Bound for the set of odd Perfect Prime Numbers | journal = [[Mathematics of Computation]] | volume = 27 | issue = 124| pages = 951–953 | doi=10.2307/2005530| jstor = 2005530 | doi-access = free }}
* 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.&nbsp;141–157.
* Riesel, H. ''Prime Numbers and Computer Methods for Factorisation'', Birkhauser, 1985.
* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | url=https://archive.org/details/handbooknumberth02sand | url-access=limited | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 | pages=[https://archive.org/details/handbooknumberth02sand/page/n16 15]–98 }}

== External links ==
* {{SpringerEOM|title=Perfect number|id=p/p072090}}
* David Moews: [http://djm.cc/amicable.html Perfect, amicable, and sociable numbers]
* [https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers/ Perfect numbers – History and Theory]
* {{MathWorld|urlname=PerfectNumber|title=Perfect Number}}
* {{OEIS el|sequencenumber=A000396|name=Perfect numbers|formalname=Perfect numbers n: n is equal to the sum of the proper divisors of n}}
* [https://www.mersenne.org/ Great Internet Mersenne Prime Search] (GIMPS)
* [http://mathforum.org/dr.math/faq/faq.perfect.html Perfect Numbers], math forum at Drexel
* {{cite web|last=Grimes|first=James|title=8128: Perfect Numbers|url=https://www.numberphile.com/videos/8128.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2013-04-02|url-status=live|archive-url=https://web.archive.org/web/20130531000409/http://numberphile.com/videos/8128.html|archive-date=2013-05-31}}

{{Divisor classes}}
{{Classes of natural numbers}}

{{Authority control}}
[[Category:Perfect numbers| ]]
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Mersenne primes]]
[[Category:Unsolved problems in number theory]]