Largest known prime number

The largest known prime number is 2136,279,841 − 1, a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant to the Great Internet Mersenne Prime Search (GIMPS).[1]

A plot of the number of digits in the largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.

Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. As of October 2024, the seven largest known primes are Mersenne primes.[2] The last eighteen record primes were Mersenne primes.[3][4] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.[5]

Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not the case.[6][7]

Current record

edit

The record is currently held by 2136,279,841 − 1 with 41,024,320 digits, found by GIMPS on October 12, 2024.[1] The first and last 120 digits of its value are:[8]

881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874 ...

(41,024,080 digits skipped)

... 852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

As of October 2024, the previously discovered prime M82589933, having 24,862,048 digits, held the record for almost 6 years, longer than any other prime since M19937 (which held the record for 7 years from 1971 to 1978).[citation needed]

Prizes

edit

There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[9] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[10] In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[9] Time called this prime the 29th top invention of 2008.[11]

Both of these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further US$250,000 prize is offered for the first prime with at least one billion digits.[9]

GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[12]

History

edit
 
Commemorative postmark used by the UIUC Math Department after proving that M11213 is prime

The following table lists the progression of the largest known prime number in ascending order.[3] Here Mp = 2p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.[citation needed]

GIMPS volunteers found the sixteen latest records, all of them Mersenne primes. They were found on ordinary personal computers until the most recent one, found by ex-Nvidia employee Luke Durant using a network of thousands of dedicated graphics processing units (GPUs).[1] Durant spent almost exactly one year and approximately US$2 million of his personal money on the hunt.[13] This achievement marks the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs), ushering in a new era in prime number searches.[14][15]

Number Digits First 120 digits Last 120 digits Year found Discoverer
M13 4 8191 8191 1456 Anonymous
M17 6 131071 131071 1588 Pietro Cataldi
M19 6 524287 524287 1588 Pietro Cataldi
  7 6700417 6700417 1732 Leonhard Euler?
Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.[16]
M31 10 2147483647 2147483647 1772 Leonhard Euler
  12 999999000001 999999000001 1851 Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record.
  14 67280421310721 67280421310721 1855 Thomas Clausen (but no proof was provided).
M127 39 17014118346046923173
1687303715884105727
1701411834604692317
31687303715884105727
1876 Édouard Lucas
  44 20988936657440586486
15126425661022259386
3921
2098
89366574405864861512
64256610222593863921
1951 Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1 79 52106440156792287940
60694325390955853335
89848390805645835218
3851018372555735221
5210644015679228794
06069432539095585333
58984839080564583521
83851018372555735221
1951 J. C. P. Miller & D. J. Wheeler[17]
Using Cambridge's EDSAC computer
M521 157 68647976601306097149
81900799081393217269
43530014330540939446
34591855431833976560
52122559640661454554
97729631139148085803
26943530014330540939
44634591855431833976
56052122559640661454
55497729631139148085
80371219879997166438
12574028291115057151
1952 Raphael M. Robinson
M607 183 53113799281676709868
95882065524686273295
93117727031923199444
13820040355986085224
27391625022652292856
68889329486246501015
20040355986085224273
91625022652292856688
89329486246501015346
57933765270723940951
99787665873519438312
70835393219031728127
1952 Raphael M. Robinson
M1279 386 10407932194664399081
92524032736408553861
52622472667048053191
12350403608059673360
29801223944173232418
48424216139542810077
82853841658502825560
46662248318909188018
47068222203140521026
69843548873295802887
80508697361869007147
20710555703168729087
1952 Raphael M. Robinson
M2203 664 14759799152141802350
84898622737381736312
06614533316977514777
12164785702978780789
49377407337049389289
38274850753149648047
51945258754287534997
65585726702296339625
75212637477897785501
55264652260998886991
40135404838098656812
50419497686697771007
1952 Raphael M. Robinson
M2281 687 44608755718375842957
11517064021018098862
08632412859901111991
21996340468579282047
33691125452690039890
26153245931124316702
95491713975879606122
38033935373810346664
94402951052059047968
69325538864793044092
51041868170096401717
64133172418132836351
1952 Raphael M. Robinson
M3217 969 25911708601320262777
62467679224415309418
18887553125427303974
92316187401926658636
20862012095168004834
06550695241733194177
59459944433523118828
01236604062624686092
12150349937584782292
23714433962885848593
82157388212323936870
46160677362909315071
1957 Hans Riesel
M4423 1,332 28554254222827961390
15635661021640083261
64238644702889199247
45660228440039060065
38759545715055398432
39754513915896150297
82106760176875097786
61004600146021384084
48021225053689054793
74200309572209673295
47507217181155318713
10231057902608580607
1961 Alexander Hurwitz
M9689 2,917 47822027880546120295
28392986600059097414
97172402236500851334
51099183789509426629
70278927686112707894
58682472098152425631
96502507081973046642
28261056975105642897
98951182192885976352
22905389894873761464
21399109115358645058
18992696826225754111
1963 Donald B. Gillies
M9941 2,993 34608828249085121524
29603957674133167226
28668900238547790489
28344500622080983411
44643643755441537075
33664486747635050186
85925083476189478888
95525278984009881962
00014868575640233136
50914562812719135485
82750839078914699790
19426224883789463551
1963 Donald B. Gillies
M11213 3,376 28141120136973731333
93152975842584191818
66238201360078789241
93493455151766822763
13810715094745633257
07419878930853507153
87566914032072497856
85867185275866024396
02335283513944980064
32703027810422414497
18836805416897847962
67391476087696392191
1963 Donald B. Gillies
M19937 6,002 43154247973881626480
55235516337919839053
93504322671150516525
05414033306801376580
91130451362931858466
55452699382576488353
60727895549548774214
07535706212171982521
92978869786916734625
61843017545490386411
15854295045699209056
36741539030968041471
1971 Bryant Tuckerman
M21701 6,533 44867916611904333479
49514103615917787272
09023729388613010364
80447512785609158053
63716201839592018310
86891496139730355336
33369896693354436162
93913110417309565016
94662754558875644345
19126927960069355180
92719564502642940928
57410828353511882751
1978 Laura A. Nickel and Landon Curt Noll[18]
M23209 6,987 40287411577898877818
18733290715917677224
38506891622420041029
96357869459524088740
08676398614614665371
03833299413586592359
49990785611757500951
57465578625397647565
74427752110896827606
78602528203915287605
50508545118172938900
36743355523779264511
1979 Landon Curt Noll[18]
M44497 13,395 85450982430363380319
33007053184030365099
01591304021058343269
25828229006478216763
58562005000144576458
61481315295253223674
19107442963978359909
48993204100398635759
46472558059877105808
94247177392297739634
54976377895623405368
44867686961011228671
1979 David Slowinski and Harry L. Nelson[18]
M86243 25,962 53692799550275632152
23382779929453006110
20994042124005915678
63944335346298210347
98964395551413140596
01329696868637207994
57351862519228939958
84693761059056977054
15089600178032945914
35320137691545632232
02509608679061957196
99857021709433438207
1982 David Slowinski[18]
M132049 39,751 51274027626932072381
27857636203402218800
46586227069926831240
38418582312743056203
61077749499092908732
12555709320045159618
89256188390637660219
36832367367308227116
78956149432532644153
24079640048510932988
33786316447035663398
52138578455730061311
1983 David Slowinski[18]
M216091 65,050 74609310306466134368
73395794005114895402
28754084977328805113
30497779366272527096
87806643956351409557
30008364494154882757
41796441616213691597
66435268814054587246
91315195450691201831
18538411805217750684
69327867645141118776
91336204103815528447
1985 David Slowinski[18]
391581×2216193−1 65,087 14814063237640662751
89896116681502152616
14869061837067878963
23169460093384999355
40035564748752481896
29946106929509682950
82819868449333023401
04392759176586303336
22389718952919899041
01638046268529515895
76118449880787230436
89626791836387377151
1989 A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[19][20]
Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839 227,832 17413590682008709732
51635992459033278907
79363690507030974654
73553838272156206625
76319147974364224616
10635130071368293660
19619724789014565809
44396409267168409183
49113692649241768590
51134272012692706848
76804040558133428809
02603793328544677887
1992 David Slowinski and Paul Gage[18]
M859433 258,716 12949812560420764966
65334852555620733841
62019917416569370190
66267567814724084952
96919893191078354681
55567280151644798137
70366138430104674404
17291687756716831654
19536906002518061544
66211087607689521384
87432526245965721589
02414267243500142591
1994 David Slowinski and Paul Gage[18]
M1257787 378,632 41224577362142867472
53232184669789600527
87185654659469380413
20489580405544505611
40313191552792105979
05669363277683158359
92352317328348412624
08558666851703702032
47995651850069878600
72644421009952433369
54631641051358552671
31257188976089366527
1996 David Slowinski and Paul Gage[18]
M1398269 420,921 81471756441257307514
26772643891354260153
13783085022271032114
51048469938030899616
08340980239948586278
86398792156198534051
70112944662406744358
62878919205295726467
35633955407734562739
68427460950363262807
77790674776834625319
85532025868451315711
1996 GIMPS, Joel Armengaud
M2976221 895,932 62334007624857864988
60414411708927450502
70498680527705762010
44980837228500531612
87552386408711765558
35347026816848251160
12689188858968205493
08475288306533381326
50949313652525946734
18989311375605582078
15564860085353060451
76506256743729201151
1997 GIMPS, Gordon Spence
M3021377 909,526 12741168303009336743
35542151767349261473
65409710390533367899
30486889243847834725
96446989025955854374
97756265138125839679
47478189918377204959
69880392336860732039
11214513449538158982
93606342963753971823
36558874582102617702
25422631973024694271
1998 GIMPS, Roland Clarkson
M6972593 2,098,960 43707574412708137883
33232912069460708676
24770574851606631018
13181519232482250706
53865555856672485830
59003027082699320939
73675389080631004085
08543235704913317476
87718276359853562553
41815592459312082762
45050174988400346151
35366526142924193791
1999 GIMPS, Nayan Hajratwala
M13466917 4,053,946 92494773800670132224
77583825476640519253
54401079958299021030
93608029565658055961
00476131215557305846
49024542650476541902
22828849378011781756
76448390574570798287
48568541687729337577
30752297148385814257
76644015462093334911
30073855470256259071
2001 GIMPS, Michael Cameron
M20996011 6,320,430 12597689545033010502
04943095748243114559
93416085351835952254
67012565498768908351
56022124009680282853
61325441271583233254
53656018582721448133
13954215503264848667
10969127787170820477
53340930097294847523
19834716766530781632
94714065762855682047
2003 GIMPS, Michael Shafer
M24036583 7,235,733 29941042940415717208
90489263404469382573
67722975418473547677
34860009764022110074
10262658651099123208
58493344156415212635
97367931835649549332
62413429503748554259
55207718464378183256
42314252685868703980
05560312691184129150
67436921882733969407
2004 GIMPS, Josh Findley
M25964951 7,816,230 12216463006127794810
77539640312884392673
61424223075246409537
66046996455809056861
56907748512690404182
46405468474387100505
82841605918218299877
77039869777444372767
13026360619053009303
03992810433168520775
07113305351596265166
98933257280577077247
2005 GIMPS, Martin Nowak
M30402457 9,152,052 31541647561884608093
63030286645451701265
19656262323870316323
71079513538744900693
46209438629475170296
63623614229944506869
29904518450254170958
38942393049606751896
53422547853529862010
43713583091577749950
02748822185508467086
11134297411652943871
2005 GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M32582657 9,808,358 12457502601536945540
08555015747995031227
95985151151842843670
47566259111523599739
73805597596066168459
39100419886882111308
72660495893732258251
20726126214431145356
41869584273577446330
45746582133321244573
71046356920000926590
11752880154053967871
2006 GIMPS, Curtis Cooper and Steven Boone
M43112609 12,978,189 31647026933025592314
34537239493375160541
06188475264644140304
17673281124749306936
86920431851216118378
56726816539985465097
15927979190839813022
33048240831190931959
98014562456347941202
19590092807967072944
79216164918874782657
80022181166697152511
2008 GIMPS, Edson Smith
M57885161 17,425,170 58188726623224644217
51002121132323686363
70852325421589325781
70448058449276170744
23164282813494233769
42979071335489886655
19696440089898189117
97158303938275980625
06665259086044516822
49493774541094283332
30952037056456587257
46141988071724285951
2013 GIMPS, Curtis Cooper
M74207281 22,338,618 30037641808460618205
29860983591660500568
75863030301484843941
69334554772321906799
42968936553007726883
20448214882399426727
71777401476291246211
36468794258014451073
93100212927181629335
93149423901821387921
76711649562871904986
87010073391086436351
2016 GIMPS, Curtis Cooper
M77232917 23,249,425 46733318335923109998
83355855611155212513
21102817714495798582
33859356792348052117
72074843110997402088
49621368090038049317
28537600451878605540
22233766729256792821
31965467343395945397
37047636927989462799
99396146592173711365
82730618069762179071
2017 GIMPS, Jonathan Pace
M82589933 24,862,048 14889444574204132554
78064584723979166030
26273992795324185271
28942521323936106447
53103099711321803371
74752834401423587560
06210755794795829753
15952088071926936765
21782184472526640076
91211435530831196948
76337664578236950740
37951210325217902591
2018 GIMPS, Patrick Laroche
M136279841 41,024,320 88169432750383326555
39391003781173589712
07354509066041067156
37641242263069475684
14417259903477232831
08837509739959776874
85280651793145941256
79575682842282881240
96109707961148305849
34976608576417071506
04094045096221046655
55076706219486871551
2024 GIMPS, Luke Durant

Twenty largest

edit

A list of the 5,000 largest known primes is maintained by the PrimePages,[21] of which the twenty largest are listed below.[22]

Rank Number Discovered Digits First 120 digits Last 120 digits Form Ref
1 2136279841 − 1 2024-10-12 41,024,320 88169432750383326555
39391003781173589712
07354509066041067156
37641242263069475684
14417259903477232831
08837509739959776874
85280651793145941256
79575682842282881240
96109707961148305849
34976608576417071506
04094045096221046655
55076706219486871551
Mersenne [1]
2 282589933 − 1 2018-12-07 24,862,048 14889444574204132554
78064584723979166030
26273992795324185271
28942521323936106447
53103099711321803371
74752834401423587560
06210755794795829753
15952088071926936765
21782184472526640076
91211435530831196948
76337664578236950740
37951210325217902591
Mersenne [23]
3 277232917 − 1 2017-12-26 23,249,425 46733318335923109998
83355855611155212513
21102817714495798582
33859356792348052117
72074843110997402088
49621368090038049317
28537600451878605540
22233766729256792821
31965467343395945397
37047636927989462799
99396146592173711365
82730618069762179071
Mersenne [24]
4 274207281 − 1 2016-01-07 22,338,618 30037641808460618205
29860983591660500568
75863030301484843941
69334554772321906799
42968936553007726883
20448214882399426727
71777401476291246211
36468794258014451073
93100212927181629335
93149423901821387921
76711649562871904986
87010073391086436351
Mersenne [25]
5 257885161 − 1 2013-01-25 17,425,170 58188726623224644217
51002121132323686363
70852325421589325781
70448058449276170744
23164282813494233769
42979071335489886655
19696440089898189117
97158303938275980625
06665259086044516822
49493774541094283332
30952037056456587257
46141988071724285951
Mersenne [26]
6 243112609 − 1 2008-08-23 12,978,189 31647026933025592314
34537239493375160541
06188475264644140304
17673281124749306936
86920431851216118378
56726816539985465097
15927979190839813022
33048240831190931959
98014562456347941202
19590092807967072944
79216164918874782657
80022181166697152511
Mersenne [27]
7 242643801 − 1 2009-06-04 12,837,064 16987351645274162247
02898707511764713591
03325776997255365512
60020505373109218621
22599292756037678425
64017793851584510263
89793266835248591744
64060649185927134914
73117475647591955485
69867927456113537511
49133460978428956443
84101954765562314751
Mersenne [28]
8 Φ3(−5166931048576) 2023-10-02 11,981,518 13402906796489222357
52246822000881801252
41118044574855268822
40787049468713337605
50197597945996229191
43176765531862533944
45102449632978070416
89341970562017911020
84113168162771694298
54415779073874568943
91416059782334617095
67178301964288000001
Generalized unique [29]
9 Φ3(−4658591048576) 2023-05-31 11,887,192 17395442163066427324
04095947530927014429
23721230469791611973
13180378592661492867
58297267063261966785
92548252101237137788
85278914675748208502
55226473801289095503
68054401147310815004
88562918084974370698
95163405490252252372
63508838734878474241
Generalized unique [30]
10 237156667 − 1 2008-09-06 11,185,272 20225440689097733553
41881522631568299468
46602582743182989551
05736054751457975812
50846721390095896345
30142096674488997709
14728787551899048539
16991622232001005966
66765048100145151363
48394299744493358135
21893866570487429610
21340265022308220927
Mersenne [27]
11 232582657 − 1 2006-09-04 9,808,358 12457502601536945540
08555015747995031227
95985151151842843670
47566259111523599739
73805597596066168459
39100419886882111308
72660495893732258251
20726126214431145356
41869584273577446330
45746582133321244573
71046356920000926590
11752880154053967871
Mersenne [31]
12 10223 × 231172165 + 1 2016-10-31 9,383,761 50625026920996343077
76282032439067604835
90666966114515920950
45640633412043430359
88815895056171116175
51873728066666193155
91892134918826938976
55779680218334368800
88050529917153697492
60915967379870147035
24878105802550394137
86610918915347316737
Proth [32]
13 230402457 − 1 2005-12-15 9,152,052 31541647561884608093
63030286645451701265
19656262323870316323
71079513538744900693
46209438629475170296
63623614229944506869
29904518450254170958
38942393049606751896
53422547853529862010
43713583091577749950
02748822185508467086
11134297411652943871
Mersenne [33]
14 4 × 511786358 + 1 2024-10-01 8,238,312 20156998396261662175
28359889367930265681
50456335975784718728
38256327105334814872
69679155318722963484
77606435567527820548
96709057831042893049
21132697813111998064
54292933825858091969
62763705692893998269
73241403038628050126
13534927368164062501
Generalized Proth [34]
15 225964951 − 1 2005-02-18 7,816,230 12216463006127794810
77539640312884392673
61424223075246409537
66046996455809056861
56907748512690404182
46405468474387100505
82841605918218299877
77039869777444372767
13026360619053009303
03992810433168520775
07113305351596265166
98933257280577077247
Mersenne [35]
16 69 × 224612729 − 1 2024-08-13 7,409,102 34913857494942645537
77528193541070245743
51335040706255350040
22702450446821700067
28827887453950698207
63928288182629713589
05695137582497488595
79121604235601653208
59352298128248331223
49160290220193535509
71657492602305174873
93807281434214268927
Riesel [36]
17 224036583 − 1 2004-05-15 7,235,733 29941042940415717208
90489263404469382573
67722975418473547677
34860009764022110074
10262658651099123208
58493344156415212635
97367931835649549332
62413429503748554259
55207718464378183256
42314252685868703980
05560312691184129150
67436921882733969407
Mersenne [37]
18 107347 × 223427517 − 1 2024-08-04 7,052,391 23535192646535179116
38946094063474658764
68924164622481357963
62977099077527159960
22049070416163357350
57403900382750381230
94089016871571688757
09838699794575028002
54586750694611329151
02052885568916854511
29696212376296097359
46366182881423785983
Riesel [38]
19 3 × 222103376 − 1 2024-09-30 6,653,780 45557575201836797391
77924694750863521479
98028844478065239917
78652383915500660790
31585797936442894001
65356897435998223877
92420137288192690503
91322027063289672125
39967518049052118055
78416884567426783748
98800257594371326293
13552387699174801407
Thabit [39]
20 19637361048576 + 1 2022-09-24 6,598,776 80651637087363405038
34361791568727797701
58073966695963388273
29711850319512278452
77262669720071754000
30028539895066510961
56553963377729301657
14959666076810377165
24885272924136913514
08208351325498302161
88609841060749286375
74080313425433460737
Generalized Fermat [40]

See also

edit

References

edit
  1. ^ a b c d "GIMPS Project Discovers Largest Known Prime Number: 2136,279,841-1". Mersenne Research, Inc. 21 October 2024. Retrieved 21 October 2024.
  2. ^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
  3. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
  4. ^ The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell.
  5. ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
  6. ^ McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.
  7. ^ Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.
  8. ^ "List of known Mersenne prime numbers - PrimeNet". www.mersenne.org. "41024320" link is to a zip file with the digits. Retrieved 2024-10-21.
  9. ^ a b c "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
  10. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
  11. ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
  12. ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
  13. ^ Numberphile (2024-10-22). The Man Who Found the World's Biggest Prime - Numberphile. Retrieved 2024-11-28 – via YouTube.
  14. ^ Bragg, Julianna (2024-11-01). "World's largest known prime number found by former Nvidia programmer". CNN. Retrieved 2024-11-28.
  15. ^ McRae, Mike (2024-10-25). "Amateur Discovers The Largest Known Prime Number And It's Huge". ScienceAlert. Retrieved 2024-11-28.
  16. ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. The Mathematical Association of America. ISBN 9780883855843.
  17. ^ Miller, J. C. P. (1951). "Large Prime Numbers". Nature. 168 (4280): 838. Bibcode:1951Natur.168..838M. doi:10.1038/168838b0.
  18. ^ a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
  19. ^ Brown, John; Noll, Landon Curt; Parady, B. K.; Smith, Joel F.; Zarantonello, Sergio E.; Smith, Gene Ward; Robinson, Raphael M.; Andrews, George E. (1990). "Letters to the Editor". The American Mathematical Monthly. 97 (3): 214–215. doi:10.1080/00029890.1990.11995576. JSTOR 2324686.
  20. ^ Proof-code: Z, The Prime Pages.
  21. ^ "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
  22. ^ "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
  23. ^ "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
  24. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
  25. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
  26. ^ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
  27. ^ a b "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
  28. ^ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
  29. ^ "PrimePage Primes: Phi(3, - 516693^1048576)". t5k.org.
  30. ^ "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
  31. ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
  32. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
  33. ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
  34. ^ "4 × 511786358 + 1". t5k.org. PrimePages. 1 October 2024. Retrieved 5 October 2024.
  35. ^ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
  36. ^ "69 × 224612729 − 1". t5k.org. PrimePages. 13 August 2024. Retrieved 29 August 2024.
  37. ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
  38. ^ "107347 × 223427517 − 1". t5k.org. PrimePages. 4 August 2024. Retrieved 25 August 2024.
  39. ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com.[dead link]
  40. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.
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