Perfect number

{{Short description|Integer 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

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.{{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), ...}} 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, since 1 + 2 + 4 + 7 + 14 = 28.

The first four perfect numbers are 6, 28, 496 and 8128.{{Cite web |title=A000396 - OEIS |url=https://oeis.org/A000396 |access-date=2024-03-21 |website=oeis.org}}

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, $\sigma_1(n)=2n$ where $\sigma_1$ is the sum-of-divisors function.

This definition is ancient, appearing as early as 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 $q(q+1)/2$ is an even perfect number whenever $q$ is a prime of the form $2^p-1$ for positive integer $p$ —what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.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"]. 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^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, and the mathematician Nicomachus noted 8128 as early as around AD 100.{{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/}} In modern language, Nicomachus states without proof that {{em|every}} perfect number is of the form $2^{n-1}(2^n-1)$ where $2^n-1$ is prime.{{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}}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. 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,Commentary on the Gospel of John 28.1.1–4, with further references in the Sources Chrétiennes edition: vol. 385, 58–61. 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).{{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}} 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.Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer Academic Publishers, 1994), pp. 328–329. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.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}} 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.{{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/}}{{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}}{{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}}

Even perfect numbers

{{Unsolved|mathematics|Are there infinitely many perfect numbers?}}

Euclid 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 {{mvar|p}} a prime number, as follows:

$\begin{align}
p = 2 &: \quad 2^1(2^2 - 1) = 2 \times 3 = 6 \\
p = 3 &: \quad 2^2(2^3 - 1) = 4 \times 7 = 28 \\
p = 5 &: \quad 2^4(2^5 - 1) = 16 \times 31 = 496 \\
p = 7 &: \quad 2^6(2^7 - 1) = 64 \times 127 = 8128.
\end{align}$

Prime numbers of the form $2^p-1$ are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For $2^p-1$ to be prime, it is necessary that {{mvar|p}} itself be prime. However, not all numbers of the form $2^p-1$ 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 $2^p-1$ are congruent to {{math|1 mod 2p}}. 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|2p + 1}} is also prime—and {{math|2p + 1}} is congruent to 1 or 7 mod 8, then {{math|2p + 1}} will be a factor of $2^p-1,$ which is the case for {{nowrap|1={{mvar|p}} = 11, 23, 83, 131, 179, 191, 239, 251, ...}} {{oeis|id=A002515}}.}} In fact, Mersenne primes are very rare: of the approximately 4 million primes {{mvar|p}} up to 68,874,199, $2^p-1$ is prime for only 48 of them.{{Cite web |title=GIMPS Milestones Report |url=https://www.mersenne.org/report_milestones/ |access-date=28 July 2024 |website=Great Internet Mersenne Prime Search}}

While Nicomachus had stated (without proof) that {{em|all}} perfect numbers were of the form $2^{n-1}(2^n-1)$ where $2^n-1$ 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.{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}} It was not until the 18th century that Leonhard Euler 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 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 48 even perfect numbers are $2^{p-1}(2^p-1)$ 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 and 57885161 {{OEIS|id=A000043}}.

Four higher perfect numbers have also been discovered, namely those for which {{mvar|p}} = 74207281, 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 {{mvar|p}} below 109332539. {{As of|2024|10}}, 52 Mersenne primes are known,{{cite web |url=http://www.mersenne.org/ |title=GIMPS Home |publisher=Mersenne.org |access-date=2024-10-21}} and therefore 52 even perfect numbers (the largest of which is {{nowrap|2^136279840 × (2^136279841 − 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 $2^{p-1}(2^p-1)$ , each even perfect number is the $(2^p-1)$ -th triangular number (and hence equal to the sum of the integers from 1 to $2^p-1$ ) and the $2^{p-1}$ -th hexagonal number. Furthermore, each even perfect number except for 6 is the $\tfrac{2^p+1}{3}$ -th centered nonagonal number and is equal to the sum of the first $2^\frac{p-1}{2}$ odd cubes (odd cubes up to the cube of $2^\frac{p+1}{2}-1$ ):

$\begin{alignat}{3}
6 &= 2^1(2^2 - 1) &&= 1 + 2 + 3, \\[8pt]
28 &= 2^2(2^3 - 1) &&= 1 + 2 + 3 + 4 + 5 + 6 + 7 \\
& &&= 1^3 + 3^3 \\[8pt]
496 &= 2^4(2^5 - 1) &&= 1 + 2 + 3 + \cdots + 29 + 30 + 31 \\
& &&= 1^3 + 3^3 + 5^3 + 7^3 \\[8pt]
8128 &= 2^6(2^7 - 1) &&= 1 + 2 + 3 + \cdots + 125 + 126 + 127 \\
& &&= 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3 \\[8pt]
33550336 &= 2^{12}(2^{13} - 1) &&= 1 + 2 + 3 + \cdots + 8189 + 8190 + 8191 \\
& &&= 1^3 + 3^3 + 5^3 + \cdots + 123^3 + 125^3 + 127^3
\end{alignat}$

Even perfect numbers (except 6) are of the form

$T_{2^p - 1} = 1 + \frac{(2^p - 2) \times (2^p + 1)}{2} = 1 + 9 \times T_{(2^p - 2)/3}$

with each resulting triangular number {{nowrap|T₇ {{=}} 28}}, {{nowrap|T₃₁ {{=}} 496}}, {{nowrap|T₁₂₇ {{=}} 8128}} (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with {{nowrap|T₂ {{=}} 3}}, {{nowrap|T{{sub|10}} {{=}} 55}}, {{nowrap|1=T₄₂ = 903}}, {{nowrap|1=T₂₇₃₀ = 3727815, ...}}{{Mathworld|urlname=PerfectNumber|title=Perfect Number}} 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 $2^{p-1}(2^p-1)$ with odd prime {{mvar|p}} and, in fact, with {{em|all}} numbers of the form $2^{m-1}(2^m-1)$ for odd integer (not necessarily prime) {{mvar|m}}.

Owing to their form, $2^{p-1}(2^p-1),$ every even perfect number is represented in binary form as {{mvar|p}} ones followed by {{math|p − 1}} zeros; for example:

$\begin{array}{rcl}
6_{10} =& 2^2 + 2^1 &= 110_2 \\
28_{10} =& 2^4 + 2^3 + 2^2 &= 11100_2 \\
496_{10} =& 2^8 + 2^7 + 2^6 + 2^5 + 2^4 &= 111110000_2 \\
8128_{10} =& \!\! 2^{12} + 2^{11} + 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 \!\! &= 1111111000000_2
\end{array}$

Thus every even perfect number is a pernicious number.

Every even perfect number is also a practical number (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,{{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/}} thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question".{{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}} More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.[http://oddperfect.org/pomerance.html Oddperfect.org]. {{Webarchive|url=https://web.archive.org/web/20061229094011/http://oddperfect.org/pomerance.html |date=2006-12-29 }} 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.{{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}}

Any odd perfect number N must satisfy the following conditions:

N > 10¹⁵⁰⁰.{{cite journal | last1=Ochem | first1=Pascal | last2=Rao | first2=Michaël | title=Odd perfect numbers are greater than 10¹⁵⁰⁰ | 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 }}
N is not divisible by 105.{{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}}
N is of the form N ≡ 1 (mod 12) or N ≡ 117 (mod 468) or N ≡ 81 (mod 324).{{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}}
The largest prime factor of N is greater than 10⁸,{{cite journal|last=Goto|first=T|author2=Ohno, Y|title=Odd perfect numbers have a prime factor exceeding 10⁸|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}} and less than $\sqrt[3]{3N}.$ {{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 }}
The second largest prime factor is greater than 10⁴,{{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}} and is less than $\sqrt[5]{2N}$ .{{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 }}.
The third largest prime factor is greater than 100,{{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}} and less than $\sqrt[6]{2N}.$ {{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}}
N has at least 101 prime factors and at least 10 distinct prime factors.{{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}} If 3 does not divide N, then N has at least 12 distinct prime factors.{{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}}
N is of the form

:: $N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k},$

:where:

:* q, p₁, ..., p_k are distinct odd primes (Euler).

:* q ≡ α ≡ 1 (mod 4) (Euler).

:* The smallest prime factor of N is at most $\frac{k-1}{2}.$ {{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}}

:* At least one of the prime powers dividing N exceeds 10⁶².

:* $N < 2^{(4^{k+1}-2^{k+1})}$ {{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 }}{{cite journal|last=Nielsen|first=Pace P.|title=An upper bound for odd perfect numbers|journal=Integers|year=2003|volume=3|pages=A14–A22|url=http://www.westga.edu/~integers/vol3.html|access-date=23 March 2021}}

:* $\alpha + 2e_1 + 2e_2 + 2e_3 + \cdots + 2e_k \geq \frac{99k-224}{37}$ .{{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 }}{{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}}

:* $qp_1p_2p_3 \cdots p_k < 2N^{\frac{17}{26}}$ .{{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}}

:* $\frac{1}{q} + \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k} < \ln 2$ .{{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 }}{{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 }}

Furthermore, several minor results are known about the exponents

e₁, ..., e_k.

Not all e_i ≡ 1 (mod 3).{{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 }}
Not all e_i ≡ 2 (mod 5).{{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 }}
If all e_i ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10⁸ and 10¹⁰⁰⁰.
More generally, if all 2e_i+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.
If (e₁, ..., e_k) = (1, ..., 1, 2, ..., 2) with t ones and u twos, then $(t-1)/4 \leq u \leq 2t+\sqrt{\alpha}$ .{{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 }}
(e₁, ..., e_k) ≠ (1, ..., 1, 3),{{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 }} (1, ..., 1, 5), (1, ..., 1, 6).{{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 }}
If {{math|1= e₁ = ... = e_k = e}}, then
e cannot be 3,{{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 }} 5, 24,{{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 $p^{\alpha} M^{2\beta}$ | 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 }} 6, 8, 11, 14 or 18.
$k\leq 2e^2+8e+2$ and $N<2^{4^{2e^2 + 8e+3}}$ .{{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 }}

In 1888, Sylvester stated: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.

{{blockquote|... 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 {{=}} 3² ⋅ 7² ⋅ 11² ⋅ 13² ⋅ 22021 {{=}} (3⋅1001)² ⋅ (22⋅1001 − 1) {{=}} 198585576189}} would be an odd perfect number if only {{math|22021 ({{=}} 19² ⋅ 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 n³ + 1 is 28 {{harv|Makowski|1962}}.{{cite journal|first=A.|last=Makowski|title=Remark on perfect numbers|journal=Elem. Math.|volume=17|year=1962|issue=5|page=109}}
28 is also the only even perfect number that is a sum of two positive cubes of integers {{harv|Gallardo|2010}}.{{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}}.
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, $\sigma_1(n) = 2n$ , and divide both sides by n):
For 6, we have $\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{1}{1} = \frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{6}{6} = \frac{1+2+3+6}{6} = \frac{2\cdot 6}{6} = 2$ ;
For 28, we have $1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1/1 = 2$ , etc.
The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.{{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}}.
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 numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form $2^{n-1}(2^n+1)$ formed as the product of a Fermat prime $2^n+1$ with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.{{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 }}
The number of perfect numbers less than n is less than $c\sqrt{n}$ , where c > 0 is a constant.{{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}} In fact it is $o(\sqrt{n})$ , using little-o notation.{{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}}
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.H. Novarese. Note sur les nombres parfaits Texeira J. VIII (1886), 11–16.{{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/}} Therefore, in particular the digital root of every even perfect number other than 6 is 1.
The only square-free perfect number is 6.{{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. 428|url=https://books.google.com/books?id=3ffXkusQEC0C&pg=PA428}}.

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, 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 {{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 $\mathcal{S}$ -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.

Notes

References

= Sources =

Euclid, 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}}

External links

{{springer|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://web.archive.org/web/20181106015226/http://oddperfect.org/ OddPerfect.org] A projected distributed computing project to search for odd perfect numbers.
[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=http://www.numberphile.com/videos/8128.html|work=Numberphile|publisher=Brady Haran|access-date=2013-04-02|archive-url=https://web.archive.org/web/20130531000409/http://numberphile.com/videos/8128.html|archive-date=2013-05-31|url-status=dead}}

Category:Divisor function

Category:Integer sequences

Category:Unsolved problems in number theory

Category:Mersenne primes