A Mersenne prime is a Mersenne number,
i.e., a number of the form
that is prime. In order for  to be prime,
 must itself be prime.
This is true since for composite  with factors  and  ,  . Therefore,
 can be written as  , which
is a binomial number that always
has a factor  .
The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (Sloane's A000668)
corresponding to indices  , 3, 5, 7, 13,
17, 19, 31, 61, 89, ... (Sloane's A000043).
Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect
number. L. Welsh maintains an extensive bibliography and history of Mersenne
numbers.
It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line fixed to pass through the origin to the asymptotic number of Mersenne primes
 with  for the
first 47 Mersenne primes gives a best-fit line with  . If the
line is not restricted to pass through the origin, the best fit is  .
It has been conjectured (without any particularly strong evidence) that the constant
is given by  , where  is the Euler-Mascheroni constant (Havil 2003, p. 116; Caldwell),
a result related to Wagstaff's
conjecture
However, finding Mersenne primes is computationally very challenging. For example, the 1963 discovery that  is prime was heralded by a special
postal meter design, illustrated above, issued in Urbana, Illinois.
G. Woltman has organized a distributed search program via the Internet known as GIMPS (Great Internet Mersenne Prime Search) in which hundreds of volunteers use
their personal computers to perform pieces of the search. The efforts of GIMPS volunteers
make this distributed computing project the discoverer of all twelve of the largest
known Mersenne primes. In fact, as of Jun. 2009, GIMPS participants have tested
and double-checked all exponents below  and tested
all exponents below  at least once (GIMPS).
The table below gives the index  of known Mersenne
primes (Sloane's A000043)
 , together with the number of digits, discovery
years, and discoverer. A similar table has been compiled by C. Caldwell. Note
that the region after the 40th known Mersenne prime has not been completely searched,
so identification of "the"  th Mersenne prime
is tentative for  .
Discovery of the 47th known Mersenne prime occurred in April 2009, but was not noticed due to a configuration problem with the server that prevented a notification email
being sent to the search organizers. The prime was subsequently verified and official
announcement of the value  , which
has  decimal digits, was made on Jun. 12,
2009, making it the 46th known Mersenne prime ranked by size and hence only the second
(not the first) largest.
| # |  | digits | year | discoverer (reference) | value | | 1 | 2 | 1 | antiquity | | 3 | | 2 | 3 | 1 | antiquity | | 7 | | 3 | 5 | 2 | antiquity | | 31 | | 4 | 7 | 3 | antiquity | | 127 | | 5 | 13 | 4 | 1461 | Reguis (1536), Cataldi (1603) | 8191 | | 6 | 17 | 6 | 1588 | Cataldi (1603) | 131071 | | 7 | 19 | 6 | 1588 | Cataldi (1603) | 524287 | | 8 | 31 | 10 | 1750 | Euler (1772) | 2147483647 | | 9 | 61 | 19 | 1883 | Pervouchine (1883), Seelhoff (1886) | 2305843009213693951 | | 10 | 89 | 27 | 1911 | Powers
(1911) | 618970019642690137449562111 | | 11 | 107 | 33 | 1913 | Powers (1914) | 162259276829213363391578010288127 | | 12 | 127 | 39 | 1876 | Lucas (1876) | 170141183460469231731687303715884105727 | | 13 | 521 | 157 | Jan. 30, 1952 | Robinson (1954) | 68647976601306097149...12574028291115057151 | | 14 | 607 | 183 | Jan. 30, 1952 | Robinson (1954) | 53113799281676709868...70835393219031728127 | | 15 | 1279 | 386 | Jun. 25, 1952 | Robinson (1954) | 10407932194664399081...20710555703168729087 | | 16 | 2203 | 664 | Oct. 7, 1952 | Robinson (1954) | 14759799152141802350...50419497686697771007 | | 17 | 2281 | 687 | Oct. 9, 1952 | Robinson (1954) | 44608755718375842957...64133172418132836351 | | 18 | 3217 | 969 | Sep. 8, 1957 | Riesel | 25911708601320262777...46160677362909315071 | | 19 | 4253 | 1281 | Nov. 3, 1961 | Hurwitz | 19079700752443907380...76034687815350484991 | | 20 | 4423 | 1332 | Nov. 3, 1961 | Hurwitz | 28554254222827961390...10231057902608580607 | | 21 | 9689 | 2917 | May 11, 1963 | Gillies (1964) | 47822027880546120295...18992696826225754111 | | 22 | 9941 | 2993 | May 16, 1963 | Gillies (1964) | 34608828249085121524...19426224883789463551 | | 23 | 11213 | 3376 | Jun. 2, 1963 | Gillies (1964) | 28141120136973731333...67391476087696392191 | | 24 | 19937 | 6002 | Mar. 4, 1971 | Tuckerman (1971) | 43154247973881626480...36741539030968041471 | | 25 | 21701 | 6533 | Oct. 30, 1978 | Noll and Nickel (1980) | 44867916611904333479...57410828353511882751 | | 26 | 23209 | 6987 | Feb. 9,
1979 | Noll (Noll and Nickel 1980) | 40287411577898877818...36743355523779264511 | | 27 | 44497 | 13395 | Apr. 8, 1979 | Nelson and Slowinski | 85450982430363380319...44867686961011228671 | | 28 | 86243 | 25962 | Sep. 25, 1982 | Slowinski | 53692799550275632152...99857021709433438207 | | 29 | 110503 | 33265 | Jan. 28, 1988 | Colquitt and Welsh (1991) | 52192831334175505976...69951621083465515007 | | 30 | 132049 | 39751 | Sep. 20, 1983 | Slowinski | 51274027626932072381...52138578455730061311 | | 31 | 216091 | 65050 | Sep. 6, 1985 | Slowinski | 74609310306466134368...91336204103815528447 | | 32 | 756839 | 227832 | Feb. 19, 1992 | Slowinski and Gage | 17413590682008709732...02603793328544677887 | | 33 | 859433 | 258716 | Jan. 10, 1994 | Slowinski and Gage | 12949812560420764966...02414267243500142591 | | 34 | 1257787 | 378632 | Sep. 3, 1996 | Slowinski and Gage | 41224577362142867472...31257188976089366527 | | 35 | 1398269 | 420921 | Nov. 12, 1996 | Joel Armengaud/GIMPS | 81471756441257307514...85532025868451315711 | | 36 | 2976221 | 895832 | Aug. 24, 1997 | Gordon Spence/GIMPS | 62334007624857864988...76506256743729201151 | | 37 | 3021377 | 909526 | Jan. 27, 1998 | Roland Clarkson/GIMPS | 12741168303009336743...25422631973024694271 | | 38 | 6972593 | 2098960 | Jun. 1, 1999 | Nayan Hajratwala/GIMPS | 43707574412708137883...35366526142924193791 | | 39 | 13466917 | 4053946 | Nov. 14, 2001 | Michael Cameron/GIMPS | 92494773800670132224...30073855470256259071 | | 40 | 20996011 | 6320430 | Nov. 17, 2003 | Michael Shafer/GIMPS | 12597689545033010502...94714065762855682047 | | 41? | 24036583 | 7235733 | May 15, 2004 | Josh Findley/GIMPS | 29941042940415717208...67436921882733969407 | | 42? | 25964951 | 7816230 | Feb. 18, 2005 | Martin Nowak/GIMPS | 12216463006127794810...98933257280577077247 | | 43? | 30402457 | 9152052 | Dec. 15, 2005 | Curtis Cooper and Steven Boone/GIMPS | 31541647561884608093...11134297411652943871 | | 44? | 32582657 | 9808358 | Sep. 4, 2006 | Curtis Cooper and Steven Boone/GIMPS | 12457502601536945540...11752880154053967871 | | 45? | 37156667 | 11185272 | Sep. 6, 2008 | Hans-Michael Elvenich/GIMPS | 20225440689097733553...21340265022308220927 | | 46? | 42643801 | 12837064 | Jun. 12, 2009 | Odd Magnar Strindmo/GIMPS | 16987351645274162247...84101954765562314751 | | 47? | 43112609 | 12978189 | Aug. 23, 2008 | Edson Smith/GIMPS | 31647026933025592314...80022181166697152511 |
Trial division is often used to establish the compositeness
of a potential Mersenne prime. This test immediately shows  to be composite for  , 23, 83, 131,
179, 191, 239, and 251 (with small factors 23, 47, 167, 263, 359, 383, 479, and 503,
respectively). A much more powerful primality test for  is the Lucas-Lehmer test.
If  is a prime, then  divides  iff  is prime. It is also true that prime
divisors of  must have the form  where  is a positive
integer and simultaneously of either the form  or  (Uspensky and
Heaslet 1939).
A prime factor  of a Mersenne number
 is a Wieferich
prime iff  . Therefore,
Mersenne primes are not Wieferich
primes.
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