List of Mersenne primes and perfect numbers information
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 not all primes p 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. 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 2023[update], there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 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.
Table of all 51 currently-known Mersenne primes and corresponding perfect numbers
Rank
p
Mersenne prime
Mersenne prime digits
Perfect number
Perfect number digits
Discovery
Discoverer
Method
Ref.[12]
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
1200s/c. 1456[b]
Multiple[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
November 1883
Ivan Pervushin
Lucas sequences
[21]
10
89
618970...562111
27
191561...169216
54
June 1911
Ralph Ernest Powers
[22]
11
107
162259...288127
33
131640...728128
65
June 1, 1914
[23]
12
127
170141...105727
39
144740...152128
77
January 10, 1876
Édouard Lucas
[24]
13
521
686479...057151
157
235627...646976
314
January 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
June 25, 1952
[26]
16
2,203
147597...771007
664
108925...782528
1,327
October 7, 1952
[27]
17
2,281
446087...836351
687
994970...915776
1,373
October 9, 1952
[27]
18
3,217
259117...315071
969
335708...525056
1,937
September 8, 1957
Hans Riesel
LLT on BESK
[28]
19
4,253
190797...484991
1,281
182017...377536
2,561
November 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
June 2, 1963
[30]
24
19,937
431542...041471
6,002
931144...942656
12,003
March 4, 1971
Bryant Tuckerman
LLT on IBM 360/91
[31]
25
21,701
448679...882751
6,533
100656...605376
13,066
October 30, 1978
Landon Curt Noll & Laura Nickel
LLT on CDC Cyber 174
[32]
26
23,209
402874...264511
6,987
811537...666816
13,973
February 9, 1979
Landon Curt Noll
[32]
27
44,497
854509...228671
13,395
365093...827456
26,790
April 8, 1979
Harry L. Nelson & David Slowinski
LLT on Cray-1
[33][34]
28
86,243
536927...438207
25,962
144145...406528
51,924
September 25, 1982
David Slowinski
[35]
29
110,503
521928...515007
33,265
136204...862528
66,530
January 29, 1988
Walter Colquitt & Luke Welsh
LLT on NEC SX-2
[36][37]
30
132,049
512740...061311
39,751
131451...550016
79,502
September 19, 1983
David Slowinski et al. (Cray)
LLT on Cray X-MP
[38]
31
216,091
746093...528447
65,050
278327...880128
130,100
September 1, 1985
LLT on Cray X-MP/24
[39][40]
32
756,839
174135...677887
227,832
151616...731328
455,663
February 17, 1992
LLT on Harwell Lab's Cray-2
[41]
33
859,433
129498...142591
258,716
838488...167936
517,430
January 4, 1994
LLT on Cray C90
[42]
34
1,257,787
412245...366527
378,632
849732...704128
757,263
September 3, 1996
LLT on Cray T94
[43][44]
35
1,398,269
814717...315711
420,921
331882...375616
841,842
November 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
August 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
January 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
June 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
November 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
November 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
February 18, 2005
GIMPS / Martin Nowak
[52]
43
30,402,457
315416...943871
9,152,052
497437...704256
18,304,103
December 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
September 4, 2006
[54]
45
37,156,667
202254...220927
11,185,272
204534...480128
22,370,543
September 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
June 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
August 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
January 25, 2013
GIMPS / Curtis Cooper
LLT / Prime95 on PC at University of Central Missouri
[59][60]
*
68,029,391
Lowest unverified milestone[e]
49[f]
74,207,281
300376...436351
22,338,618
451129...315776
44,677,235
January 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
December 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
December 7, 2018
GIMPS / Patrick Laroche
LLT / Prime95 on PC with Intel Core i5-4590T processor
[65][66]
*
116,167,187
Lowest untested milestone[e]
Historically, the largest known prime number has often been a Mersenne prime.
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This is a listof notable numbersand articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are...
integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbersof the form 4k(8m + 7). A positive integer...
If 2k + 1 is primeand k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023[update]...
divisible by the first k primes. If A ( k ) {\displaystyle A(k)} represents the smallest abundant number not divisible by the first k primes then for all ϵ >...
many Kummer primes? Are there infinitely many Kynea primes? Are there infinitely many Lucas primes? Are there infinitely many Mersenneprimes (Lenstra–Pomerance–Wagstaff...
the Mersenne numbersand the binary repunit primes are the Mersenneprimes. It is unknown whether there are infinitely many Brazilian primes. If the Bateman–Horn...
can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect. With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci...
pseudoprime Probable prime Baillie–PSW primality test Miller–Rabin primality test Lucas–Lehmer primality test Lucas–Lehmer test for Mersennenumbers AKS primality...
and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbersand Fibonacci numbers form...