List of Mersenne primes and perfect numbers: Difference between revisions
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. Tag: Reverted |
Undid revision 1255197206 by 97.102.205.224 (talk) - It was better |
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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]] |
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[[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]] |
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[[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 {{ |
[[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> |
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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> |
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It is currently an [[open problem]] whether |
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> |
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⚫ | 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" /> |
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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> |
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⚫ | The displayed ranks are among indices currently known {{As of|2022|lc=y}}; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent {{math|1=''p'' = 57,885,161}} have been checked and verified {{as of|2024|1|lc=y}}.<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> The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. 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. |
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⚫ | The following is a list of all |
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⚫ | |||
{| 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 |
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! scope=col rowspan=2 | {{ |
! scope=col rowspan=2 | {{math|''p''}} |
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! class="unsortable" scope=col colspan=2 | Mersenne prime |
! class="unsortable" scope=col colspan=2 | Mersenne prime |
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! class="unsortable" scope=col colspan=2 | Perfect number |
! class="unsortable" scope=col colspan=2 | Perfect number |
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! |
! scope=col rowspan=2 | Discovery |
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⚫ | |||
⚫ | |||
! 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: |
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* {{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 |
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! scope=col | Date |
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! 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]] |
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| rowspan="4" | Unrecorded |
| rowspan="4" | Unrecorded |
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| align="center |
| 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> |
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! scope=row align="right" | 2 |
! scope=row align="right" | 2 |
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| align="right" | [[28 (number)| 28]] |
| align="right" | [[28 (number)| 28]] |
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| align="right" | 2 |
| align="right" | 2 |
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| align="center" | <ref name="el" /><ref name="hist" /><ref name="DES" /> |
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! scope=row align="right" | 3 |
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| align="right" | [[496 (number)| 496]] |
| align="right" | [[496 (number)| 496]] |
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| align="right" | 3 |
| align="right" | 3 |
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| align="center" | <ref name="el" /><ref name="hist" /><ref name="DES" /> |
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! scope=row align="right" | 4 |
! scope=row align="right" | 4 |
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| align="right" | [[8128 (number)| 8128]] |
| align="right" | [[8128 (number)| 8128]] |
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| align="right" | 4 |
| align="right" | 4 |
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| align="center" | <ref name="el" /><ref name="hist" /><ref name="DES" /> |
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! scope=row align="right" | 5 |
! scope=row align="right" | 5 |
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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> |
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Historically, the [[largest known prime number]] has often been a Mersenne prime. |
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==Notes== |
==Notes== |
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==External links== |
==External links== |
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* {{OEIS el|A000043|Corresponding exponents {{ |
* {{OEIS el|A000043|Corresponding exponents {{math|''p''}}}} |
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* {{OEIS el|A000396|Perfect numbers}} |
* {{OEIS el|A000396|Perfect numbers}} |
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* {{OEIS el|A000668|Mersenne primes}} |
* {{OEIS el|A000668|Mersenne primes}} |
Revision as of 23:00, 3 November 2024
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 numbers 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] Meanwhile, perfect numbers are natural numbers that 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]
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 2p − 1 × (2p − 1), where 2p − 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 p = 2, 22 − 1 = 3 is prime, and 22 − 1 × (22 − 1) = 2 × 3 = 6 is perfect.[1][5][6]
It is currently an open problem as to 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.[7][8][9] 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 101500.[10]
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2024[update], 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.[2] New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.[2]
The displayed ranks are among indices currently known as of 2022[update]; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of January 2024[update].[11] The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. 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.
Rank | p | Mersenne prime | Perfect number | Discovery | Discoverer | Method | Ref.[12] | ||
---|---|---|---|---|---|---|---|---|---|
Value | Digits | Value | Digits | ||||||
1 | 2 | 3 | 1 | 6 | 1 | Ancient times[a] |
Known to Ancient Greek mathematicians | Unrecorded | [13][14][15] |
2 | 3 | 7 | 1 | 28 | 2 | [13][14][15] | |||
3 | 5 | 31 | 2 | 496 | 3 | [13][14][15] | |||
4 | 7 | 127 | 3 | 8128 | 4 | [13][14][15] | |||
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] |
Historically, the largest known prime number has often been a Mersenne prime.
Notes
- ^ 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.
- ^ 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]
- ^ 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]
- ^ 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.
- ^ a b As of 31 October 2024[update].[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.
- ^ 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.
- ^ 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.
- ^ First detected as a probable prime using Fermat primality test on an Nvidia A100 GPU on October 11, 2024
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External links
- OEIS sequence A000043 (Corresponding exponents p)
- OEIS sequence A000396 (Perfect numbers)
- OEIS sequence A000668 (Mersenne primes)
- List on GIMPS, with the full values of large numbers Archived 2020-06-07 at the Wayback Machine
- A technical report on the history of Mersenne numbers, by Guy Haworth Archived 2021-10-13 at the Wayback Machine