Editing Perfect number (section)

== 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. Many of the properties proved about odd perfect numbers also apply to [[Descartes number]]s, and Pace Nielsen has suggested that sufficient study of those 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>

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}}</ref>
* 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}}</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}}</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}}</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>\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>
:* 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><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.|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}}</ref>
:* <math>\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> \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>(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> and <math> N<2^{4^{2e^2 + 8e+3}}</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.