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# 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 51 (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 of the Mersenne primes discovered since late 1996. As of Feb. 19, 2024, GIMPS participants have tested and verified all exponents below 67 million and tested all exponents below 115 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 sequential indexing of "the" th Mersenne prime is tentative for until all exponents between and (namely up to ) have been verified to be composite (and therefore also tentative for the other known Mersenne primes and ).

 # 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 50? 77232917 23249425 Dec. 26, 2017 Jonathan Pace/GIMPS 46733318335923109998...82730618069762179071 51? 82589933 24862048 Dec. 7, 2018 Patrick Laroche/GIMPS 14889444574204132554...37951210325217902591

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.

Catalan-Mersenne Number, Cullen Number, Cunningham Number, Double Mersenne Number, Eberhart's Conjecture, Fermat-Lucas Number, Fermat Number, Fermat Polynomial, Integer Sequence Primes, Lucas-Lehmer Test, Mersenne Number, New Mersenne Prime Conjecture, Perfect Number, Repunit, Superperfect Number, Titanic Prime, Wagstaff's Conjecture, Woodall Prime

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## References

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Mersenne Prime

## Cite this as:

Weisstein, Eric W. "Mersenne Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MersennePrime.html