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Seriously rework lead to keep each paragraph focused on one idea. Mersenne primes, (even) perfect numbers, odd perfect numbers, density of Mersenne primes. I'm trying not to duplicate the Mersenne prime and perfect number articles, but one paragraph is hopefully enough intro.
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[[File:Perfect number Cuisenaire rods 6 exact.svg|alt=Cuisenaire rods showing the proper divisors of 6 (1, 2, and 3) adding up to 6|thumb|Visualization of 6 as a perfect number]]
[[File:Perfect number Cuisenaire rods 6 exact.svg|alt=Cuisenaire rods showing the proper divisors of 6 (1, 2, and 3) adding up to 6|thumb|Visualization of 6 as a perfect number]]
[[File:Digits in largest prime found as a function of time.svg|alt=A graph plotting years on the x-axis with the number of digits of the largest known prime logarithmically on the y-axis, with two trendlines|thumb|[[Logarithmic scale|Logarithmic graph]] of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes]]
[[File:Digits in largest prime found as a function of time.svg|alt=A graph plotting years on the x-axis with the number of digits of the largest known prime logarithmically on the y-axis, with two trendlines|thumb|[[Logarithmic scale|Logarithmic graph]] of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes]]
[[Mersenne prime]]s and [[perfect number]]s are two deeply interlinked types of [[natural number]]s in [[number theory]]. Mersenne primes, named after the friar [[Marin Mersenne]], are [[prime number]]s that can be expressed as {{math|1=2<sup>''p''</sup> − 1}} for some positive integer {{math|1=''p''}}. For example, {{math|3}} is a Mersenne prime as it is a prime number and is expressible as {{math|1=2<sup>2</sup> − 1}}.<ref name="Stillwell">{{Cite book |last=Stillwell |first=John |url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40 |title=Mathematics and Its History |publisher=[[Springer Science+Business Media]] |year=2010 |isbn=978-1-4419-6052-8 |series=Undergraduate Texts in Mathematics |pages=40 |author-link=John Stillwell |access-date=13 October 2021 |archive-url=https://web.archive.org/web/20211013184518/https://www.google.com/books/edition/Mathematics_and_Its_History/V7mxZqjs5yUC?hl=en&gbpv=1&pg=PA40&printsec=frontcover |archive-date=13 October 2021 |url-status=live}}</ref><ref name="CaldwellMP">{{Cite web |last=Caldwell |first=Chris K. |title=Mersenne Primes: History, Theorems and Lists |url=https://primes.utm.edu/mersenne/ |url-status=live |archive-url=https://web.archive.org/web/20211004054614/https://primes.utm.edu/mersenne/ |archive-date=4 October 2021 |access-date=4 October 2021 |website=[[PrimePages]]}}</ref> The numbers {{math|1=''p''}} corresponding to Mersenne primes must themselves be prime, although the vast majority of primes {{math|1=''p''}} do not lead to Mersenne primes—for example, {{math|1=2<sup>11</sup> − 1 = 2047 = 23 × 89}}.<ref>{{Cite web |last=Caldwell |first=Chris K. |title=If 2<sup>n</sup>-1 is prime, then so is n |url=https://primes.utm.edu/notes/proofs/Theorem2.html |url-status=live |archive-url=https://web.archive.org/web/20211005003814/https://primes.utm.edu/notes/proofs/Theorem2.html |archive-date=5 October 2021 |access-date=12 October 2021 |website=[[PrimePages]]}}</ref> Meanwhile, perfect numbers are [[natural numbers]] that equal the sum of their positive [[proper divisor]]s, which are divisors excluding the number itself. So, {{math|6}} is a perfect number because the proper divisors of {{math|6}} are {{math|1, 2}}, and {{math|3}}, and {{math|1=1 + 2 + 3 = 6}}.<ref name="CaldwellMP" /><ref name="prielipp">{{Cite journal |last=Prielipp |first=Robert W. |date=1970 |title=Perfect Numbers, Abundant Numbers, and Deficient Numbers |url=http://www.jstor.org/stable/27958492 |url-status=live |journal=The Mathematics Teacher |volume=63 |issue=8 |pages=692–96 |doi=10.5951/MT.63.8.0692 |jstor=27958492 |archive-url=https://web.archive.org/web/20211005010408/https://www.jstor.org/stable/27958492 |archive-date=5 October 2021 |access-date=13 October 2021 |via=JSTOR}}</ref>
[[Mersenne prime]]s and [[perfect number]]s are two deeply interlinked types of [[natural number]]s in [[number theory]]. Mersenne primes, named after the friar [[Marin Mersenne]], are [[prime number]]s that can be expressed as {{math|1=2<sup>''p''</sup> − 1}} for some positive integer {{mvar|p}}. For example, {{math|3}} is a Mersenne prime as it is a prime number and is expressible as {{math|1=2<sup>2</sup> − 1}}.<ref name="Stillwell">{{Cite book |last=Stillwell |first=John |url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40 |title=Mathematics and Its History |publisher=[[Springer Science+Business Media]] |year=2010 |isbn=978-1-4419-6052-8 |series=Undergraduate Texts in Mathematics |pages=40 |author-link=John Stillwell |access-date=13 October 2021 |archive-url=https://web.archive.org/web/20211013184518/https://www.google.com/books/edition/Mathematics_and_Its_History/V7mxZqjs5yUC?hl=en&gbpv=1&pg=PA40&printsec=frontcover |archive-date=13 October 2021 |url-status=live}}</ref><ref name="CaldwellMP">{{Cite web |last=Caldwell |first=Chris K. |title=Mersenne Primes: History, Theorems and Lists |url=https://primes.utm.edu/mersenne/ |url-status=live |archive-url=https://web.archive.org/web/20211004054614/https://primes.utm.edu/mersenne/ |archive-date=4 October 2021 |access-date=4 October 2021 |website=[[PrimePages]]}}</ref> The exponents {{math|1=''p''}} corresponding to Mersenne primes must themselves be prime, although the vast majority of primes {{math|1=''p''}} do not lead to Mersenne primes—for example, {{math|1=2<sup>11</sup> − 1 = 2047 = 23 × 89}}.<ref>{{Cite web |last=Caldwell |first=Chris K. |title=If 2<sup>n</sup>-1 is prime, then so is n |url=https://primes.utm.edu/notes/proofs/Theorem2.html |url-status=live |archive-url=https://web.archive.org/web/20211005003814/https://primes.utm.edu/notes/proofs/Theorem2.html |archive-date=5 October 2021 |access-date=12 October 2021 |website=[[PrimePages]]}}</ref>


There is a [[one-to-one correspondence]] between the Mersenne primes and the even perfect numbers, but it is unknown whether there exist odd perfect numbers. This is due to the [[Euclid–Euler theorem]], partially proved by [[Euclid]] and completed by [[Leonhard Euler]]: even numbers are perfect [[if and only if]] they can be expressed in the form {{math|1=2<sup>''p'' − 1</sup> × (2<sup>''p''</sup> − 1)}}, where {{math|1=2<sup>''p''</sup> − 1}} is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of {{math|1=''p'' = 2}}, {{math|1=2<sup>2</sup> − 1 = 3}} is prime, and {{math|1=2<sup>2 − 1</sup> × (2<sup>2</sup> − 1) = 2 × 3 = 6}} is perfect.<ref name="Stillwell" /><ref>{{Cite web |last=Caldwell |first=Chris K. |title=Characterizing all even perfect numbers |url=https://primes.utm.edu/notes/proofs/EvenPerfect.html |url-status=live |archive-url=https://web.archive.org/web/20141008023521/http://primes.utm.edu/notes/proofs/EvenPerfect.html |archive-date=8 October 2014 |access-date=12 October 2021 |website=[[PrimePages]]}}</ref><ref name="Crilly2007">{{Cite book |last=Crilly |first=Tony |url=https://books.google.com/books?id=f46JAwAAQBAJ |title=50 mathematical ideas you really need to know |publisher=Quercus Publishing |year=2007 |isbn=978-1-84724-008-8 |chapter=Perfect numbers |access-date=13 October 2021 |archive-url=https://web.archive.org/web/20211013184556/https://www.google.com/books/edition/50_Mathematical_Ideas_You_Really_Need_to/f46JAwAAQBAJ?hl=en&gbpv=1 |archive-date=13 October 2021 |url-status=live}}</ref>
Perfect numbers are [[natural numbers]] which equal the sum of their positive [[proper divisor]]s, which are divisors excluding the number itself. So, {{math|6}} is a perfect number because the proper divisors of {{math|6}} are {{math|1, 2}}, and {{math|3}}, and {{math|1=1 + 2 + 3 = 6}}.<ref name="CaldwellMP" /><ref name="prielipp">{{Cite journal |last=Prielipp |first=Robert W. |date=1970 |title=Perfect Numbers, Abundant Numbers, and Deficient Numbers |url=http://www.jstor.org/stable/27958492 |url-status=live |journal=The Mathematics Teacher |volume=63 |issue=8 |pages=692–96 |doi=10.5951/MT.63.8.0692 |jstor=27958492 |archive-url=https://web.archive.org/web/20211005010408/https://www.jstor.org/stable/27958492 |archive-date=5 October 2021 |access-date=13 October 2021 |via=JSTOR}}</ref> [[Euler]] proved {{circa|300 BCE}} that every Mersenne prime {{math|1=''M<sub>p</sub>'' = 2<sup>''p''</sup> − 1}} has a corresponding perfect number {{math|1=''M<sub>p</sub>'' × (''M<sub>p</sub>''+1)/2 = 2<sup>''p'' − 1</sup> × (2<sup>''p''</sup> − 1)}}. For example, the Mersenne prime {{math|1=2<sup>2</sup> − 1 = 3}} leads to the corresponding prefect number {{math|1=2<sup>2 − 1</sup> × (2<sup>2</sup> − 1) = 2 × 3 = 6}}. In 1747, [[Leonhard Euler]] completed what is now called the [[Euclid–Euler theorem]], showing that these are the ''only'' even perfect numbers.<ref name="Stillwell" /><ref>{{Cite web |last=Caldwell |first=Chris K. |title=Characterizing all even perfect numbers |url=https://primes.utm.edu/notes/proofs/EvenPerfect.html |url-status=live |archive-url=https://web.archive.org/web/20141008023521/http://primes.utm.edu/notes/proofs/EvenPerfect.html |archive-date=8 October 2014 |access-date=12 October 2021 |website=[[PrimePages]]}}</ref><ref name="Crilly2007">{{Cite book |last=Crilly |first=Tony |url=https://books.google.com/books?id=f46JAwAAQBAJ |title=50 mathematical ideas you really need to know |publisher=Quercus Publishing |year=2007 |isbn=978-1-84724-008-8 |chapter=Perfect numbers |access-date=13 October 2021 |archive-url=https://web.archive.org/web/20211013184556/https://www.google.com/books/edition/50_Mathematical_Ideas_You_Really_Need_to/f46JAwAAQBAJ?hl=en&gbpv=1 |archive-date=13 October 2021 |url-status=live}}</ref>


It is currently an [[open problem]] as to whether there are an infinite number of Mersenne primes and even perfect numbers.<ref name="CaldwellMP" /><ref name="Crilly2007" /> The frequency of Mersenne primes is the subject of the [[Lenstra–Pomerance–Wagstaff conjecture]], which states that the expected number of Mersenne primes less than some given {{math|''x''}} is {{math|(''e''<sup>γ</sup> / log 2) × log log ''x''}}, where {{math|''e''}} is [[Euler's number]], {{math|''γ''}} is [[Euler's constant]], and {{math|log}} is the [[natural logarithm]].<ref>{{Cite web |last=Caldwell |first=Chris K. |title=Heuristics Model for the Distribution of Mersennes |url=https://primes.utm.edu/mersenne/heuristic.html |url-status=live |archive-url=https://web.archive.org/web/20211005005313/https://primes.utm.edu/mersenne/heuristic.html |archive-date=5 October 2021 |access-date=13 October 2021 |website=[[PrimePages]]}}</ref><ref>{{Cite journal |last=Wagstaff |first=Samuel S. |author-link=Samuel S. Wagstaff Jr. |date=January 1983 |title=Divisors of Mersenne numbers |url=http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-1983-0679454-X |journal=Mathematics of Computation |language=en |volume=40 |issue=161 |pages=385–397 |doi=10.1090/S0025-5718-1983-0679454-X |issn=0025-5718|doi-access=free }}</ref><ref>{{Cite journal |last=Pomerance |first=Carl |author-link=Carl Pomerance |date=September 1981 |title=Recent developments in primality testing |url=https://link.springer.com/content/pdf/10.1007/BF03022861.pdf |journal=The Mathematical Intelligencer |language=en |volume=3 |issue=3 |pages=97–105 |doi=10.1007/BF03022861 |issn=0343-6993 |s2cid=121750836}}</ref> It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a [[lower bound]] of {{math|10<sup>1500</sup>}}.<ref>{{Cite journal |last1=Ochem |first1=Pascal |last2=Rao |first2=Michaël |date=30 January 2012 |title=Odd perfect numbers are greater than 10<sup>1500</sup> |url=http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-2012-02563-4 |journal=Mathematics of Computation |language=en |volume=81 |issue=279 |pages=1869–1877 |doi=10.1090/S0025-5718-2012-02563-4 |issn=0025-5718|doi-access=free }}</ref>
It is currently an [[open problem]] whether any [[Perfect number#Odd perfect numbers|odd perfect numbers]] exist (various conditions on possible odd perfect numbers have been proven, including a [[lower bound]] of {{math|10<sup>1500</sup>}}<ref>{{Cite journal |last1=Ochem |first1=Pascal |last2=Rao |first2=Michaël |date=30 January 2012 |title=Odd perfect numbers are greater than 10<sup>1500</sup> |url=http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-2012-02563-4 |journal=Mathematics of Computation |language=en |volume=81 |issue=279 |pages=1869–1877 |doi=10.1090/S0025-5718-2012-02563-4 |issn=0025-5718|doi-access=free }}</ref>), so at the moment a list of known Mersenne primes is also a list of known perfect numbers.


It is also an open problem whether there are an infinite number of Mersenne primes and even perfect numbers.<ref name="CaldwellMP" /><ref name="Crilly2007" /> The frequency of Mersenne primes is the subject of the [[Lenstra–Pomerance–Wagstaff conjecture]], which states that the expected number of Mersenne primes less than some given {{mvar|x}} is {{math|(''e<sup>γ</sup>'' / log 2) × log log ''x''}}, where {{mvar|e}} is [[Euler's number]], {{mvar|γ}} is [[Euler's constant]], and {{math|log}} is the [[natural logarithm]].<ref>{{Cite web |last=Caldwell |first=Chris K. |title=Heuristics Model for the Distribution of Mersennes |url=https://primes.utm.edu/mersenne/heuristic.html |url-status=live |archive-url=https://web.archive.org/web/20211005005313/https://primes.utm.edu/mersenne/heuristic.html |archive-date=5 October 2021 |access-date=13 October 2021 |website=[[PrimePages]]}}</ref><ref>{{Cite journal |last=Wagstaff |first=Samuel S. |author-link=Samuel S. Wagstaff Jr. |date=January 1983 |title=Divisors of Mersenne numbers |url=http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-1983-0679454-X |journal=Mathematics of Computation |language=en |volume=40 |issue=161 |pages=385–397 |doi=10.1090/S0025-5718-1983-0679454-X |issn=0025-5718|doi-access=free }}</ref><ref>{{Cite journal |last=Pomerance |first=Carl |author-link=Carl Pomerance |date=September 1981 |title=Recent developments in primality testing |url=https://link.springer.com/content/pdf/10.1007/BF03022861.pdf |journal=The Mathematical Intelligencer |language=en |volume=3 |issue=3 |pages=97–105 |doi=10.1007/BF03022861 |issn=0343-6993 |s2cid=121750836}}</ref>
⚫
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents {{math|''p''}}. {{As of|2024}}, there are 52 known Mersenne primes (and therefore perfect numbers), the largest 18 of which have been discovered by the [[distributed computing]] project [[Great Internet Mersenne Prime Search]], or GIMPS.<ref name="CaldwellMP" /> New Mersenne primes are found using the [[Lucas–Lehmer primality test|Lucas–Lehmer test]] (LLT), a [[primality test]] for Mersenne primes that is efficient for binary computers.<ref name="CaldwellMP" />


⚫
The following is a list of all 52 currently ({{As of|2024|lc=y}}) known Mersenne primes and correponding perfect numbers, along with their exponents {{mvar|p}}. The largest 18 of these have been discovered by the [[distributed computing]] project [[Great Internet Mersenne Prime Search]], or GIMPS;<ref name="CaldwellMP" /> their discoverers are listed as "GIMPS / ''name''", where the ''name'' is the person who supplied the computer that made the discovery. New Mersenne primes are found using the [[Lucas–Lehmer primality test|Lucas–Lehmer test]] (LLT), a [[primality test]] for Mersenne primes that is efficient for binary computers.<ref name="CaldwellMP" /> Because of this efficiency, the [[largest known prime number]] is usually a Mersenne prime.
⚫
All possible exponents up to the 48th ({{math|1=''p'' = 57,885,161}}) have been checked and verified by GIMPS as of October 2021.<ref name="mile">{{Cite web |title=GIMPS Milestones Report |url=https://www.mersenne.org/report_milestones/ |url-status=live |archive-url=https://web.archive.org/web/20211013062600/https://www.mersenne.org/report_milestones/ |archive-date=13 October 2021 |access-date=31 January 2024 |website=[[Great Internet Mersenne Prime Search]]}}</ref> Ranks 49 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Discoverers denoted as "GIMPS / ''name''" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last six digits of each number are shown.

⚫
All possible exponents up to the 48th ({{math|1=''p'' = 57,885,161}}) have been checked and verified by GIMPS as of October 2021.<ref name="mile">{{Cite web |title=GIMPS Milestones Report |url=https://www.mersenne.org/report_milestones/ |url-status=live |archive-url=https://web.archive.org/web/20211013062600/https://www.mersenne.org/report_milestones/ |archive-date=13 October 2021 |access-date=31 January 2024 |website=[[Great Internet Mersenne Prime Search]]}}</ref> Ranks 49 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Later entries are extremely long, so only the first and last six digits of each number are shown, along with the number of decimal digits.


{| class="wikitable sortable" style="font-size:95%"
{| class="wikitable sortable" style="font-size:95%"
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|-
|-
! class="unsortable" scope=col rowspan=2 | Rank
! class="unsortable" scope=col rowspan=2 | Rank
! scope=col rowspan=2 | {{math|''p''}}
! scope=col rowspan=2 | {{mvar|p}}
! class="unsortable" scope=col colspan=2 | Mersenne prime
! class="unsortable" scope=col colspan=2 | Mersenne prime
! class="unsortable" scope=col colspan=2 | Perfect number
! class="unsortable" scope=col colspan=2 | Perfect number
! scope=col rowspan=2 | Discovery
! class="unsortable" scope=col colspan=3 | Discovered
⚫
! class="unsortable" scope=col rowspan=2 | Discoverer
⚫
! class="unsortable" scope=col rowspan=2 | Method
! class="unsortable" scope=col rowspan=2 | {{Abbr|Ref.|References}}<ref>Sources applying to almost all entries:
! class="unsortable" scope=col rowspan=2 | {{Abbr|Ref.|References}}<ref>Sources applying to almost all entries:
* {{Cite web |title=List of Known Mersenne Prime Numbers |url=https://www.mersenne.org/primes/ |url-status=live |archive-url=https://web.archive.org/web/20200607033022/https://www.mersenne.org/primes/ |archive-date=7 June 2020 |access-date=4 October 2021 |website=[[Great Internet Mersenne Prime Search]] |ref=none}}
* {{Cite web |title=List of Known Mersenne Prime Numbers |url=https://www.mersenne.org/primes/ |url-status=live |archive-url=https://web.archive.org/web/20200607033022/https://www.mersenne.org/primes/ |archive-date=7 June 2020 |access-date=4 October 2021 |website=[[Great Internet Mersenne Prime Search]] |ref=none}}
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|-
|-
! Value || Digits || Value || Digits
! Value || Digits || Value || Digits
! scope=col | Date
⚫
! class="unsortable" scope=col | Discoverer
⚫
! class="unsortable" scope=col | Method
|-
|-
! scope=row align="right" | 1
! scope=row align="right" | 1
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| rowspan="4" | Known to [[Ancient Greek mathematicians]]
| rowspan="4" | Known to [[Ancient Greek mathematicians]]
| rowspan="4" | Unrecorded
| rowspan="4" | Unrecorded
| align="center" | <ref name="el">{{Cite web |last=Joyce |first=David E. |title=Euclid's Elements, Book IX, Proposition 36 |url=https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX36.html |url-status=live |archive-url=https://web.archive.org/web/20210617083328/https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX36.html |archive-date=17 June 2021 |access-date=13 October 2021 |website=mathcs.clarku.edu}}</ref><ref name="hist">{{Cite book |last=Dickson |first=Leonard Eugene |url=https://books.google.com/books?id=DQXvAAAAMAAJ |title=History of the Theory of Numbers, Vol. I |publisher=Carnegie Institution of Washington |year=1919 |pages=4–6 |author-link=Leonard Eugene Dickson |access-date=2023-03-19 |archive-date=2023-04-08 |archive-url=https://web.archive.org/web/20230408093614/https://books.google.com/books?id=DQXvAAAAMAAJ |url-status=live }}</ref><ref name="DES">{{Cite book |last=Smith |first=David Eugene |url=https://archive.org/stream/historyofmathema031897mbp#page/n35/mode/2up |title=History of Mathematics: Volume II |publisher=Dover |year=1925 |isbn=978-0-486-20430-7 |pages=21 |author-link=David Eugene Smith}}</ref>
| align="center" rowspan="4" | <ref name="el">{{Cite web |last=Joyce |first=David E. |title=Euclid's Elements, Book IX, Proposition 36 |url=https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX36.html |url-status=live |archive-url=https://web.archive.org/web/20210617083328/https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX36.html |archive-date=17 June 2021 |access-date=13 October 2021 |website=mathcs.clarku.edu}}</ref><ref name="hist">{{Cite book |last=Dickson |first=Leonard Eugene |url=https://books.google.com/books?id=DQXvAAAAMAAJ |title=History of the Theory of Numbers, Vol. I |publisher=Carnegie Institution of Washington |year=1919 |pages=4–6 |author-link=Leonard Eugene Dickson |access-date=2023-03-19 |archive-date=2023-04-08 |archive-url=https://web.archive.org/web/20230408093614/https://books.google.com/books?id=DQXvAAAAMAAJ |url-status=live }}</ref><ref name="DES">{{Cite book |last=Smith |first=David Eugene |url=https://archive.org/stream/historyofmathema031897mbp#page/n35/mode/2up |title=History of Mathematics: Volume II |publisher=Dover |year=1925 |isbn=978-0-486-20430-7 |pages=21 |author-link=David Eugene Smith}}</ref>
|-
|-
! scope=row align="right" | 2
! scope=row align="right" | 2
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| align="right" | [[28 (number)| 28]]
| align="right" | [[28 (number)| 28]]
| align="right" | 2
| align="right" | 2
| align="center" | <ref name="el" /><ref name="hist" /><ref name="DES" />
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| align="center" | <ref>{{Cite news |date=21 October 2024|title=GIMPS Discovers Largest Known Prime Number: 2<sup>136,279,841</sup>-1 |work=[[Great Internet Mersenne Prime Search]] |url=https://www.mersenne.org/primes/?press=M136279841 |url-status=live |access-date=21 October 2024}}</ref>
| align="center" | <ref>{{Cite news |date=21 October 2024|title=GIMPS Discovers Largest Known Prime Number: 2<sup>136,279,841</sup>-1 |work=[[Great Internet Mersenne Prime Search]] |url=https://www.mersenne.org/primes/?press=M136279841 |url-status=live |access-date=21 October 2024}}</ref>
|}
|}

Historically, the [[largest known prime number]] has often been a Mersenne prime.


==Notes==
==Notes==
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==External links==
==External links==
* {{OEIS el|A000043|Corresponding exponents {{math|''p''}}}}
* {{OEIS el|A000043|Corresponding exponents {{mvar|p}}}}
* {{OEIS el|A000396|Perfect numbers}}
* {{OEIS el|A000396|Perfect numbers}}
* {{OEIS el|A000668|Mersenne primes}}
* {{OEIS el|A000668|Mersenne primes}}

Revision as of 17:34, 3 November 2024

Cuisenaire rods showing the proper divisors of 6 (1, 2, and 3) adding up to 6
Visualization of 6 as a perfect number
A graph plotting years on the x-axis with the number of digits of the largest known prime logarithmically on the y-axis, with two trendlines
Logarithmic graph of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes

Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1.[1][2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89.[3]

Perfect numbers are natural numbers which equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.[2][4] Euler proved c. 300 BCE that every Mersenne prime Mp = 2p − 1 has a corresponding perfect number Mp × (Mp+1)/2 = 2p − 1 × (2p − 1). For example, the Mersenne prime 22 − 1 = 3 leads to the corresponding prefect number 22 − 1 × (22 − 1) = 2 × 3 = 6. In 1747, Leonhard Euler completed what is now called the Euclid–Euler theorem, showing that these are the only even perfect numbers.[1][5][6]

It is currently an open problem whether any odd perfect numbers exist (various conditions on possible odd perfect numbers have been proven, including a lower bound of 101500[7]), so at the moment a list of known Mersenne primes is also a list of known perfect numbers.

It is also an open problem whether there are an infinite number of Mersenne primes and even perfect numbers.[2][6] The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm.[8][9][10]

The following is a list of all 52 currently (as of 2024) known Mersenne primes and correponding perfect numbers, along with their exponents p. The largest 18 of these have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS;[2] their discoverers are listed as "GIMPS / name", where the name is the person who supplied the computer that made the discovery. New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.[2] Because of this efficiency, the largest known prime number is usually a Mersenne prime.

All possible exponents up to the 48th (p = 57,885,161) have been checked and verified by GIMPS as of October 2021.[11] Ranks 49 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Later entries are extremely long, so only the first and last six digits of each number are shown, along with the number of decimal digits.

Table of all 52 currently-known Mersenne primes and corresponding perfect numbers
Rank p Mersenne prime Perfect number Discovered Ref.[12]
Value Digits Value Digits Date Discoverer Method
1 2 3 1 6 1 Ancient
times[a] 		Known to Ancient Greek mathematicians Unrecorded [13][14][15]
2 3 7 1 28 2
3 5 31 2 496 3
4 7 127 3 8128 4
5 13 8191 4 33550336 8 13th century
or 1456[b] 		Ibn Fallus or anonymous[c] Trial division [14][15]
6 17 131071 6 8589869056 10 1588[b] Pietro Cataldi [2][18]
7 19 524287 6 137438691328 12 [2][18]
8 31 2147483647 10 230584...952128 19 1772 Leonhard Euler Trial division with modular restrictions [19][20]
9 61 230584...693951 19 265845...842176 37 Nov 1883 Ivan Pervushin Lucas sequences [21]
10 89 618970...562111 27 191561...169216 54 Jun 1911 Ralph Ernest Powers [22]
11 107 162259...288127 33 131640...728128 65 Jun 1, 1914 [23]
12 127 170141...105727 39 144740...152128 77 Jan 10, 1876 Édouard Lucas [24]
13 521 686479...057151 157 235627...646976 314 Jan 30, 1952 Raphael M. Robinson LLT on SWAC [25]
14 607 531137...728127 183 141053...328128 366 [25]
15 1,279 104079...729087 386 541625...291328 770 Jun 25, 1952 [26]
16 2,203 147597...771007 664 108925...782528 1,327 Oct 7, 1952 [27]
17 2,281 446087...836351 687 994970...915776 1,373 Oct 9, 1952 [27]
18 3,217 259117...315071 969 335708...525056 1,937 Sep 8, 1957 Hans Riesel LLT on BESK [28]
19 4,253 190797...484991 1,281 182017...377536 2,561 Nov 3, 1961 Alexander Hurwitz LLT on IBM 7090 [29]
20 4,423 285542...580607 1,332 407672...534528 2,663 [29]
21 9,689 478220...754111 2,917 114347...577216 5,834 May 11, 1963 Donald B. Gillies LLT on ILLIAC II [30]
22 9,941 346088...463551 2,993 598885...496576 5,985 May 16, 1963 [30]
23 11,213 281411...392191 3,376 395961...086336 6,751 Jun 2, 1963 [30]
24 19,937 431542...041471 6,002 931144...942656 12,003 Mar 4, 1971 Bryant Tuckerman LLT on IBM 360/91 [31]
25 21,701 448679...882751 6,533 100656...605376 13,066 Oct 30, 1978 Landon Curt Noll & Laura Nickel LLT on CDC Cyber 174 [32]
26 23,209 402874...264511 6,987 811537...666816 13,973 Feb 9, 1979 Landon Curt Noll [32]
27 44,497 854509...228671 13,395 365093...827456 26,790 Apr 8, 1979 Harry L. Nelson & David Slowinski LLT on Cray-1 [33][34]
28 86,243 536927...438207 25,962 144145...406528 51,924 Sep 25, 1982 David Slowinski [35]
29 110,503 521928...515007 33,265 136204...862528 66,530 Jan 29, 1988 Walter Colquitt & Luke Welsh LLT on NEC SX-2 [36][37]
30 132,049 512740...061311 39,751 131451...550016 79,502 Sep 19, 1983 David Slowinski et al. (Cray) LLT on Cray X-MP [38]
31 216,091 746093...528447 65,050 278327...880128 130,100 Sep 1, 1985 LLT on Cray X-MP/24 [39][40]
32 756,839 174135...677887 227,832 151616...731328 455,663 Feb 17, 1992 LLT on Harwell Lab's Cray-2 [41]
33 859,433 129498...142591 258,716 838488...167936 517,430 Jan 4, 1994 LLT on Cray C90 [42]
34 1,257,787 412245...366527 378,632 849732...704128 757,263 Sep 3, 1996 LLT on Cray T94 [43][44]
35 1,398,269 814717...315711 420,921 331882...375616 841,842 Nov 13, 1996 GIMPS / Joel Armengaud LLT / Prime95 on 90 MHz Pentium PC [45]
36 2,976,221 623340...201151 895,932 194276...462976 1,791,864 Aug 24, 1997 GIMPS / Gordon Spence LLT / Prime95 on 100 MHz Pentium PC [46]
37 3,021,377 127411...694271 909,526 811686...457856 1,819,050 Jan 27, 1998 GIMPS / Roland Clarkson LLT / Prime95 on 200 MHz Pentium PC [47]
38 6,972,593 437075...193791 2,098,960 955176...572736 4,197,919 Jun 1, 1999 GIMPS / Nayan Hajratwala LLT / Prime95 on IBM Aptiva with 350 MHz Pentium II processor [48]
39 13,466,917 924947...259071 4,053,946 427764...021056 8,107,892 Nov 14, 2001 GIMPS / Michael Cameron LLT / Prime95 on PC with 800 MHz Athlon T-Bird processor [49]
40 20,996,011 125976...682047 6,320,430 793508...896128 12,640,858 Nov 17, 2003 GIMPS / Michael Shafer LLT / Prime95 on Dell Dimension PC with 2 GHz Pentium 4 processor [50]
41 24,036,583 299410...969407 7,235,733 448233...950528 14,471,465 May 15, 2004 GIMPS / Josh Findley LLT / Prime95 on PC with 2.4 GHz Pentium 4 processor [51]
42 25,964,951 122164...077247 7,816,230 746209...088128 15,632,458 Feb 18, 2005 GIMPS / Martin Nowak [52]
43 30,402,457 315416...943871 9,152,052 497437...704256 18,304,103 Dec 15, 2005 GIMPS / Curtis Cooper & Steven Boone LLT / Prime95 on PC at University of Central Missouri [53]
44 32,582,657 124575...967871 9,808,358 775946...120256 19,616,714 Sep 4, 2006 [54]
45 37,156,667 202254...220927 11,185,272 204534...480128 22,370,543 Sep 6, 2008 GIMPS / Hans-Michael Elvenich LLT / Prime95 on PC [55]
46 42,643,801 169873...314751 12,837,064 144285...253376 25,674,127 Jun 4, 2009[d] GIMPS / Odd Magnar Strindmo LLT / Prime95 on PC with 3 GHz Intel Core 2 processor [56]
47 43,112,609 316470...152511 12,978,189 500767...378816 25,956,377 Aug 23, 2008 GIMPS / Edson Smith LLT / Prime95 on Dell OptiPlex PC with Intel Core 2 Duo E6600 processor [55][57][58]
48 57,885,161 581887...285951 17,425,170 169296...130176 34,850,340 Jan 25, 2013 GIMPS / Curtis Cooper LLT / Prime95 on PC at University of Central Missouri [59][60]
* 70,202,087 Lowest unverified milestone[e]
49[f] 74,207,281 300376...436351 22,338,618 451129...315776 44,677,235 Jan 7, 2016[g] GIMPS / Curtis Cooper LLT / Prime95 on PC with Intel Core i7-4790 processor [61][62]
50[f] 77,232,917 467333...179071 23,249,425 109200...301056 46,498,850 Dec 26, 2017 GIMPS / Jonathan Pace LLT / Prime95 on PC with Intel Core i5-6600 processor [63][64]
51[f] 82,589,933 148894...902591 24,862,048 110847...207936 49,724,095 Dec 7, 2018 GIMPS / Patrick Laroche LLT / Prime95 on PC with Intel Core i5-4590T processor [65][66]
* 124,817,431 Lowest untested milestone[e]
52[f] 136,279,841 881694...871551 41,024,320 388692...008576 82,048,640 Oct 12, 2024 GIMPS / Luke Durant LLT / PRPLL on Nvidia H100 GPU[h] [67]

Notes

  1. ^ The first four perfect numbers were documented by Nicomachus circa 100, and the concept was known (along with corresponding Mersenne primes) to Euclid at the time of his Elements. There is no record of discovery.
  2. ^ a b Islamic mathematicians such as Ismail ibn Ibrahim ibn Fallus (1194–1239) may have known of the fifth through seventh perfect numbers prior to European records.[16]
  3. ^ Found in an anonymous manuscript designated Clm 14908, dated 1456 and 1461. Ibn Fallus' earlier work in the 13th century also mentioned the prime, but was not widely distributed[14][17]
  4. ^ M42,643,801 was first reported to GIMPS on April 12, 2009, but was not noticed by a human until June 4, 2009, due to a server error.
  5. ^ a b As of 31 October 2024.[11] All exponents below the lowest unverified milestone have been checked more than once. All exponents below the lowest untested milestone have been checked at least once.
  6. ^ a b c d It has not been verified whether any undiscovered Mersenne primes exist between the 48th (M57,885,161) and the 52nd (M136,279,841) on this table; the ranking is therefore provisional.
  7. ^ M74,207,281 was first reported to GIMPS on September 17, 2015 but was not noticed by a human until January 7, 2016 due to a server error.
  8. ^ First detected as a probable prime using Fermat primality test on an Nvidia A100 GPU on October 11, 2024

References

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  53. ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 24 December 2005. Archived from the original on 14 March 2021. Retrieved 13 October 2021.
  54. ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 11 September 2006. Archived from the original on 26 January 2021. Retrieved 13 October 2021.
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  56. ^ "GIMPS Discovers 47th Mersenne Prime". Great Internet Mersenne Prime Search. 12 April 2009. Archived from the original on 19 February 2021. Retrieved 13 October 2021.
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  66. ^ Palca, Joe (21 December 2018). "The World Has A New Largest-Known Prime Number". NPR. Archived from the original on 30 July 2021. Retrieved 13 October 2021.
  67. ^ "GIMPS Discovers Largest Known Prime Number: 2136,279,841-1". Great Internet Mersenne Prime Search. 21 October 2024. Retrieved 21 October 2024.{{cite news}}: CS1 maint: url-status (link)
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