For which positive integers p is 2p-1 a prime number?

Such prime numbers are called now-a-days Mersenne Primes and p is called its (prime) exponent.
A table ordered by discovery date follows:

Rank exponent p discovery date discoverer (credited persons) needed real time to prove supporting help
1 2< 0
2 3< 0
3 5< 0
4 7< 0
5 131456
6 171588Pietro Cataldi
7 191588Pietro Cataldi
8 311772Leonhard Euler
12 1271876Francois Edouard A. Lucas
9 611883Ivane M. Pervouchine
10 891911R. E. Powers
11 1071914R. E. Powers
13 5211952-01-30 Raphael M. Robinson~ 80 secSWAC
14 6071952-01-30 Raphael M. Robinson~ 2 minSWAC
15 12791952-06-25 Raphael M. Robinson13.5 minSWAC
16 22031952-10-07 Raphael M. Robinson~ 55 minSWAC
17 22811952-10-09 Raphael M. Robinson1 hourSWAC
18 32171957-09-08 Hans Riesel5.5 hoursBESK
19 42531961-11-03 Alexander Hurwitz & John L. Selfridge~ 50 min IBM 7090
20 44231961-11-03 Alexander Hurwitz & John L. Selfridge50 min IBM 7090
21 96891963-05-11 Donald B. Gillies83 minILLIAC II
22 99411963-05-16 Donald B. Gillies90 minILLIAC II
23 112131963-06-02 Donald B. Gillies135 minILLIAC II
24 199371971-03-04 Bryant Tuckerman35 minIBM 360/91
25 217011978-10-30 Landon Curt Noll & Laura Nickel7h 40m 20sCDC Cyber 174
26 232091979-02-09 Landon Curt Noll & Laura Nickel8h 39m 37s CDC Cyber 174
27 444971979-04-08 Harry Nelson & David Slowinski Cray 1
28 862431982-09-25 David Slowinski Cray 1
30 1320491983-09-19 David Slowinski Cray X-MP
31 2160911985-09-01 David Slowinski Cray X-MP/24
29 1105031988-01-28 Walt Colquitt & Luke Welsh 686 secNEC SX/2
32 7568391992-04-01 David Slowinski & Paul Gage ~ 16 hours Cray-2
33 8594331994-02-01 David Slowinski & Paul Gage 7.2 hours Cray C90
34 12577871996-09-03 David Slowinski & Paul Gage ~ 6 hoursCray T90
35 13982691996-11-23 Joel Armengaud, Woltman, et. al. 88 hours 90 MHz Pentium PC
36 29762211997-09-01 Gordon Spence, Woltman, et. al. 15 days100 MHz Pentium PC
37 30213771998-01-27 Roland Clarkson, Woltman, Kurowski et. al. 46 days200 MHz Pentium PC
38 69725931999-06-01 Nayan Hajratwala, Woltman, Kurowski et. al. 111 days350 MHz Pentium II IBM Aptiva PC
39134669172001-11-14 Michael Cameron, Woltman, Kurowski et. al. 45 days800 MHz AMD T-Bird PC
40209960112003-11-17 Michael Shafer, Georg Woltman, Scott Kurowski et. al. 19 days2 GHz Pentium 4 Dell Dimension PC
41240365832004-05-15 Josh Findley, Georg Woltman, Scott Kurowski et. al. 14 days2.4 GHz Pentium 4 PC
42259649512005-02-18 Martin Nowak, Georg Woltman, Scott Kurowski et. al. 50 days2.4 GHz Pentium 4 PC
43304024572005-12-15 Curtis Cooper and Steven Boone, Georg Woltman, Scott Kurowski et. al. ~ 50 daysPC network of 700 PC
44325826572006-09-04 Curtis Cooper and Steven Boone, Georg Woltman, Scott Kurowski et. al. ~ 9 monthsPC network of 700 PC
47431126092008-08-23 Edson Smith, Georg Woltman, Scott Kurowski et. al. 35 days 2.4 GHz Intel Core 2 Duo E6600 Dell Optiplex 745
45371566672008-09-06 Hans-Michael Elvenich, Georg Woltman, Scott Kurowski et. al. 218 days 2.83 GHz Intel Core 2 Duo E8300
46426438012009-04-12 Odd Magnar Strindmo, Georg Woltman, Scott Kurowski et. al. 29 days 3.0 GHz Intel Core 2
48578851612013-01-25 Curtis Cooper, Georg Woltman, Scott Kurowski et. al. 39 daysPC network of 700 PC
time rangewhohowwhyimportants
Antiquity philosophs (?) reflecting by mind philosophy / culture discovery of meaning
Middle Ages hobby numberists trial divisions by hand calculations culture/fashion in Italy disproof of tradition 211-1 must be prime
18-th century until mid 20-th century (~ 220 years) first professional mathematicians hand calculations (Lucas sequences) arise of scientific precise culture important theoretical insights achieved
1951-1954a mathematician Lucas-Lehmer test used scientific culture first reliable electronic calculators
1954 - 1979 (27 years) different mathematicians LL-test on date-typical computers proud of record breaking
1979 - 1996 (17 years) mainly a single programmer LL-test on special vector-computers (Cray) proud of record breaking (making history)
January 1996 - today (17 years) mainly one, later two, fixed programmer(s) and random people coordinated distributed search on common computers for interested people (LL-test with FFT-multiplication) proud of record breaking (making history) social/info culture (GIMPS)

Historical information about Mersenne number discoveries before 1963.
More details on History of Mersenne Primes
GIMPS (Distributed search for Mersenne Primes


Achim Flammenkamp
2013-02-12 12:10 UTC+1