Mersenne Prime
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, ... (OEIS A000668) corresponding to indices
, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS 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 through the origin to the asymptotic number of Mersenne primes 
with 
for the
first 49 (known) Mersenne primes gives a best-fit line with 
,
illustrated above. 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 Jan. 23, 2016, 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 (OEIS 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 45th known Mersenne prime has
not been completely searched, so identification of "the" 
th Mersenne prime
is tentative for 
.
| # |  | 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 |
| 48? | 57885161 | 17425170 | Jan. 25, 2013 | Curtis Cooper/GIMPS | 58188726623224644217...46141988071724285951 |
| 49? | 74207281 | 22338618 | Jan. 7, 2016 | Curtis Cooper/GIMPS | 30037641808460618205...87010073391086436351 |
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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