Mersenne prime

A Mersenne number is a number of the form 2^n - 1. Specifically, M_{n}=2^{n}-1denotes the nth Mersenne number. The first seven Mersenne numbers M_{n}are

{\displaystyle 1,3,7,15,31,63,127}(sequence A000225 in OEIS).

The primes among the Mersenne numbers are called Mersenne primes. The first eight Mersenne primes M_{p}are

{\displaystyle 3,7,31,127,8191,131071,524287,2147483647}(sequence A000668 in OEIS)

for exponents {\displaystyle p=2,3,5,7,13,17,19,31}(sequence A000043 in OEIS).

When represented in the dual system, Mersenne numbers show up as columns of ones, i.e. numbers that consist exclusively of ones. The nnth Mersenne number in the dual system is a number with n(example: {\displaystyle M_{3}=7=111_{2}}). In the binary, Mersenne numbers belong to the number palindromes, Mersenne prime numbers accordingly to the prime number palindromes.

These prime numbers take their name from the French monk and priest Marin Mersenne (1588-1648), who claimed in the preface to his Cogitata Physico-Mathematica that for {\displaystyle p=2,3,5,7,13,17,19,31,67,127}and {\displaystyle 257}the number M_{p}is a prime number.

However, he was mistaken about the numbers M_{{67}}and M_{{257}}overlooked the Mersenne primes M_{{61}}, M_{{89}}and M_{{107}}. That M_{{67}}is not a prime number was shown by Édouard Lucas in 1876, but it was not until 1903 that the mathematician Frank Nelson Cole was able to name the prime factors of this number. To prove that M_{{257}}is not a prime number, an early calculator was used in 1932. The number M_{{67}}possibly a reading error on the part of Mersenne from his correspondence with Bernard Frénicle de Bessy and Pierre de Fermat, where he p=67confused p=61with

Mersenne numbers also occur in the Mersenne Twister, a pseudo-random number generator.

Postmark with the 23rd Mersenne prime found at UIUC by Donald B. Gillies in 1963.Zoom
Postmark with the 23rd Mersenne prime found at UIUC by Donald B. Gillies in 1963.

History

Mersenne numbers were first studied in antiquity in connection with perfect numbers. A natural number is called perfect if it is equal to the sum of its real divisors (example: {\displaystyle 6=1+2+3}). Euclid had already shown that the number {\displaystyle 2^{n-1}(2^{n}-1)}is perfect if 2^n-1a prime number ( n=2yields the number 6). 2000 years later, the inverse was shown by Euler for even perfect numbers: every even perfect number is of the form {\displaystyle 2^{n-1}(2^{n}-1)}where {\displaystyle 2^{n}-1}is a prime number.

Odd perfect numbers have not yet been found, but their existence has not yet been proven or disproven.

The first four perfect numbers {\displaystyle 6,28,496}and {\displaystyle 8128}were already known in antiquity. The search for more perfect numbers motivated the search for more Mersenne primes. The most important property to note is the following:

If na compositenumber, then M_{n}also a composite number. That is divided {\displaystyle 2^{a\cdot b}-1}by {\displaystyle 2^{a}-1}and by {\displaystyle 2^{b}-1}without remainder can be shown by polynomial division if aand are bnatural numbers without the zero.

It follows immediately that the exponent pof a Mersenne prime M_{p}=2^{p}-1is itself a prime. This property facilitates the search for Mersenne primes, since only Mersenne numbers with prime exponent need to be considered.

However, the converse conclusion that {\displaystyle M_{p}}if {p}prime is false because, for example, {\displaystyle M_{11}'=2047=23\cdot 89}is not a prime number.

Mersenne primes are rare: so far (December 2018), only 51 of them have been found. Since there is a particularly efficient prime test for them, the largest known primes are Mersenne primes.

Year

Event

until 1536

It is believed that for all prime numbers p, 2p-1 is prime.

1536

The German arithmetician Ulrich Rieger (Latin Hudalrichus Regius) was the first to publish the fifth perfect number 212-(213-1) = 4096 - 8191 = 33550336 in print in his arithmetical book Utriusque Arithmetices epitome. Since the numbers 511 and 2047 do not appear in his tabular overview, one may assume that he recognised 211-1 = 2047 = 23 - 89 as a composite, although he does not mention this specifically.

1555

Johann Scheubel publishes in his German translation of books VII-IX of Euclid's Elements the next two perfect numbers 216-(217-1) = 65536 - 131071 = 8589869056 and 218-(219-1) = 262144 - 524287 = 137438691328. The second factors are Mersenne's primes M17 and M19. However, he did not recognise 211-1 = 2047 = 23 - 89, as well as 215-1 = 32767 = 7 - 31 - 151 as composite, but 221-1 = 2097151 = 72 - 127 - 337. (However, he does not give the decompositions at this point.) He thus erroneously obtains nine perfect numbers in his work, instead of the correct seven.

1603

Pietro Cataldi (1548-1626) shows that 2p-1 is prime for p = 17, 19 and correctly conjectures this for p = 31. Incorrectly, he also believes it for p = 23, 29 and 37.

1640

Fermat refutes Cataldi for p = 23 and p = 37: 223-1 = 47 - 178481 and 237-1 = 223 - 616318177 are not prime numbers.

1644

Mersenne claims that 2p-1 is prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, but not prime for all other natural numbers smaller than 257 (Preface to his work Cogitata Physico-Mathematica). We now know that this assertion is false, however, because 2p-1 is prime for p = 61 (Pervushin, 1883) as well as for p = 89 (Powers, 1911) and p = 107 (Powers and Fauquembergue, 1914); moreover, 267-1 is composite (Lucas, 1876; Cole 1903).

1738

Euler refutes Cataldi for p = 29: 229-1 = 233 - 1103 - 2089.

1750

Euler confirms that Cataldi was right for p = 31: 231-1 is prime.

1870

Édouard Lucas (1842-1891) formulates the theoretical basis for the Lucas-Lehmer test.

1876

Lucas confirms Mersenne: 2127-1 is prime and contradicts: 267-1 is not prime, factors remain unknown.

1883

Ivan Michejowitsch Pervuschin (1827-1900), a Russian mathematician and Orthodox priest from Perm/Russia, shows that 261-1 is prime (contradiction to Mersenne).

1903

Frank Nelson Cole names the prime factors of 267-1 = 193707721 - 761838257287.

1911

Ralph Ernest Powers contradicts Mersenne for p = 89: 2p-1 is prime.

1914

Powers also contradicts Mersenne for p = 107: 2p-1 is prime. Almost simultaneously, E. Fauquembergue also comes to this statement.

1930

Derrick Henry Lehmer (1905-1991) formulates the Lucas-Lehmer test.

1932

Lehmer shows: M(149) and M(257) are not prime, he calculates for this two hours a day on a desk calculator for a year.

1934

Powers shows: M(241) is not prime.

1944

Horace S. Uhler shows: M(157) and M(167) are not prime.

1945

Uhler shows: M(229) is not prime.

1947

Uhler shows: M(199) is not prime.

1947

The range from 1 to 257 is now completely checked. One now knows the Mersenne primes M(p) for p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

1951

Start of the use of computers. The length of the largest known prime number increases from 39 digits to 687 decimal places by 1952.

1963

Donald Gillies discovers M(11,213) with 3,376 digits.

1996

Joel Armengaud and George Woltman discover with GIMPS M(1,398,269) with 420,921 digits.

1999

With M(6,972,593), which has 2,098,960 digits, a prime number with more than 1 million digits is known for the first time on 1 June.

2004

On 15 May, it is proved that M(24,036,583), a number with 7,235,733 digits, is prime.

2005

On 18 February, the GIMPS project discovers the 42nd Mersenne prime: M(25,964,951) has 7,816,230 digits.

Also by the GIMPS project, the 43rd Mersenne prime number is discovered on 15 December: M(30,402,457) has 9,152,052 digits.

2006

On 4 September, the GIMPS project announces the discovery of the 44th Mersenne prime number M(32,582,657) with 9,808,358 digits.

2008

On 16 September, the 45th and 46th known Mersenne prime numbers are published by the GIMPS project: M(37,156,667) (discovered on 6 September) with 11,185,272 digits and M(43,112,609) (discovered on 23 August) with 12,978,189 digits.

2009

The 47th known Mersenne prime number M(42,643,801) is discovered by the GIMPS project on 12 April and published on 12 June.

2013

The 48th known Mersenne prime number M(57,885,161) is discovered by the GIMPS project on 25 January.

2016

The 49th known Mersenne prime number M(74,207,281) is discovered by the GIMPS project on 7 January.

2017

The 50th known Mersenne prime number M(77,232,917) is discovered by the GIMPS project on 26 December.

2018

The 51st known Mersenne prime number M(82,589,933) is discovered by the GIMPS project on 7 December.

Divisibility properties of the Mersenne numbers

In the course of their long history, many results have been found about Mersenne numbers. Besides the basic divisibility property already mentioned (rthe number nthen M_{r}is divisor of M_{n}), there are, for example, the following results:

  • If nis an even number and n+1is prime, then n+1is a divisor of M_{n}e.g. {\displaystyle M_{10}=1023=3\cdot 11\cdot 31,M_{12}=4095=3^{2}\cdot 5\cdot 7\cdot 13}.
  • If nis an odd prime number and qa prime factor of Mn, then {\displaystyle q\equiv 1\ mod\ 2n}and {\displaystyle q\equiv \pm 1\ mod\ 8}. Example: {\displaystyle M_{11}=2047=23\cdot 89}and {\displaystyle 23=2\cdot 11+1;89=4\cdot 2\cdot 11+1}.
  • If pa prime number with {\displaystyle p\equiv 3\ mod\ 4}, then the following equivalence holds: {\displaystyle 2p+1}divides the Mersenne number M_{p}exactly when {\displaystyle 2p+1}prime. Example: 11is prime and leaves a remainder of 3when divided by 4. Since {\displaystyle 23}(as the result of {\displaystyle 2\cdot 11+1}) is prime, it follows: 23divides the Mersenne number {\displaystyle M_{11}=2047}. This statement was formulated by Leonhard Euler, but only later proved by Joseph-Louis Lagrange (see also Sophie-Germain prime).
  • If {\displaystyle M_{p}>3}prime number, then M p + {\displaystyle M_{p}+2}prime number (namely 3by 3 {\displaystyle ). Mersenne primes are therefore not suitable as the smaller prime of a prime twin.
  • If n=2^{m}with m > 0then M_{n}the product of the Fermat numbers F_0to {\displaystyle F_{m-1}}. Example: {\displaystyle M_{16}=F_{0}\cdot F_{1}\cdot F_{2}\cdot F_{3}=3\cdot 5\cdot 17\cdot 257}.

The search for Mersenne primes

For obtaining prime records, Mersenne primes are particularly well suited in several respects, because (a) composite exponents can be disregarded because they do not generate primes, and therefore a list of candidate exponents peasily be generated with prime generators, (b) from this list, as described above, the Sophie-Germain primes with {\displaystyle p\equiv 3\ mod\ 4}out (such as. e.g. p = 11 → divisor 23), (c) due to the functional connection, the magnitude of the prime number pgrows exponentially - namely to the base two - with the argument thus one quickly obtains very large numbers, (d) with the Lucas-Lehmer test described below, a simple and effective prime number test is available.

The Lucas-Lehmer test

Main article: Lucas-Lehmer test

This test is a prime number test specifically tailored to Mersenne numbers, based on work done by Édouard Lucas in the 1870-1876 period and supplemented in 1930 by Derrick Henry Lehmer.

It works as follows:

be podd and prime. {\displaystyle S(k)}Let the sequence be defined by {\displaystyle S(1)=4,S(k+1)=S(k)^{2}-2}.

Then: M_{p}=2^{p}-1is a prime number if is M_{p}divisible {\displaystyle S(p-1)}by

GIMPS: The Great Internet Mersenne Prime Number Search

Main article: Great Internet Mersenne Prime Search

As of December 2018, 51 Mersenne primes are known. With computer help, they are trying to find more Mersenne primes. Since these are very large numbers - the 51st Mersenne prime has more than 24 million digits in the decimal system - the calculations are time-consuming and organisationally complex. Calculations with such large numbers are not inherently supported by computers. You have to store the numbers in large fields and programme the basic arithmetic operations required with them. This leads to long programme runtimes.

GIMPS (Great Internet Mersenne Prime Search) therefore tries to involve as many computers as possible in the calculations. The software required for this (Prime95) was created by George Woltman and Scott Kurowski and is available for a number of computer platforms (Windows, Unix, Linux ...).

Anyone can participate, provided they have a computer with (temporarily) free CPU capacity. To do this, you have to download the software from the website and then install it. Then you register with GIMPS and ask for a number to examine. When the calculations are done (usually after several months), you report the result back to GIMPS. In the course of the computer analysis, you are also told probabilities: the probability of finding a prime number (very small, less than 1 %); the probability of finding a factor (greater). This probability is supposed to make clear the chances of success for finding a prime number.

List of all known Mersenne primes

No.

p

Number of
digits of M(p)

Year

Discoverer

1

2

1

- –

- –

2

3

1

- –

- –

3

5

2

- –

- –

4

7

3

- –

- –

5

13

4

1456

- –

6

17

6

1555

Johann Scheubel

7

19

6

1555

Johann Scheubel

8

31

10

1772

Leonhard Euler

9

61

19

1883

Ivan Pervushin

10

89

27

1911

Ralph E. Powers

11

107

33

1914

Powers

12

127

39

1876

Édouard Lucas

13

521

157

1952

Raphael M. Robinson

14

607

183

1952

Robinson

15

1279

386

1952

Robinson

16

2203

664

1952

Robinson

17

2281

687

1952

Robinson

18

3217

969

1957

Hans Riesel

19

4253

1281

1961

Alexander Hurwitz

20

4423

1332

1961

Hurwitz

21

9689

2917

1963

Donald B. Gillies

22

9941

2993

1963

Gillies

23

11.213

3376

1963

Gillies

24

19.937

6002

1971

Bryant Tuckerman

25

21.701

6533

1978

Landon Curt Noll, Laura Nickel

26

23.209

6987

1979

Noll

27

44.497

13.395

1979

David Slowinski, Harry L. Nelson

28

86.243

25.962

1982

Slowinski

29

110.503

33.265

1988

Walter Colquitt, Luther Welsh Jr.

30

132.049

39.751

1983

Slowinski

31

216.091

65.050

1985

Slowinski

32

756.839

227.832

1992

Slowinski, Paul Gage

33

859.433

258.716

1994

Slowinski, Paul Gage

34

1.257.787

378.632

1996

Slowinski, Paul Gage

35

1.398.269

420.921

1996

GIMPS / Joel Armengaud

36

2.976.221

895.932

1997

GIMPS / Gordon Spence

37

3.021.377

909.526

1998

GIMPS / Roland Clarkson

38

6.972.593

2.098.960

1999

GIMPS / Nayan Hajratwala

39

13.466.917

4.053.946

2001

GIMPS / Michael Cameron

40

20.996.011

6.320.430

2003

GIMPS / Michael Shafer

41

24.036.583

7.235.733

2004

GIMPS / Josh Findley

42

25.964.951

7.816.230

2005

GIMPS / Martin Nowak

43

30.402.457

9.152.052

2005

GIMPS / Curtis Cooper, Steven Boone

44

32.582.657

9.808.358

2006

GIMPS / Curtis Cooper, Steven Boone

45

37.156.667

11.185.272

2008

GIMPS / Hans-Michael Elvenich

46

42.643.801

12.837.064

2009

GIMPS / Odd M. Strindmo

47

43.112.609

12.978.189

2008

GIMPS / Edson Smith

48?

57.885.161

17.425.170

2013

GIMPS / Curtis Cooper

49?

74.207.281

22.338.618

2016

GIMPS / Curtis Cooper

50?

77.232.917

23.249.425

2017

GIMPS / Jonathan Pace

51?

82.589.933

24.862.048

2018

GIMPS / Patrick Laroche

As of 13 March 2020, it cannot be ruled out that there are other, as yet undiscovered Mersenne primes between p = 43,112,609 and p = 82,589,933; hence the numbering from No. 48 onwards is still uncertain (and marked with a "?").

Graph of the number of digits in the largest known Mersenne primes versus year, from 1950, the most recent era of electronic calculating machines. Note: The vertical scale is a double logarithmic scale of the value of the Mersenne prime.Zoom
Graph of the number of digits in the largest known Mersenne primes versus year, from 1950, the most recent era of electronic calculating machines. Note: The vertical scale is a double logarithmic scale of the value of the Mersenne prime.

Open questions

As is so often the case in number theory, there are also unsolved problems concerning Mersenne numbers that are very easy to formulate:

  • Are there infinitely many Mersenne primes? Based on plausible heuristics, one suspects that there are {\displaystyle c\cdot \ln(x)}many Mersenne primes M_{p}with {\displaystyle p<x}a positive constant c If this were true, there would indeed be infinitely many Mersenne primes.
  • More precisely, is the conjecture made by H. W. Lenstra and C. Pomerance independently made, correct, that asymptotically e^{\gamma }\log _{2}(\log _{2}(x))+O(1)are many Mersenne primes less than or equal to x?
  • Conversely, are there infinitely many Mersenne numbers M_{p}with pprime that are not prime numbers? Again, one suspects the answer is yes. This would follow, for example, from the conjecture that there are infinitely many Sophie-Germain primes that are congruent 3 modulo 4.
  • Are all Mersenne numbers M_{p}with psquare-free, i.e. does each prime factor occur exactly once in the prime factorisation of the number? So far, it has not even been possible to prove that this is true for infinitely many Mersenne numbers.
  • Does the "new Mersenne conjecture" apply? The sequence of Mersenne primes given by Mersenne suggests that he meant that a Mersenne number M_{p}with }p prime is prime exactly when {\displaystyle p=2^{k}\pm 1}or {\displaystyle p=4^{k}\pm 3}. Since this statement does not hold, P. Bateman, J. Selfridge and S. Wagstaff established the new Mersenne conjecture.

This states that from two of the following three statements the third already follows:

    1. {\displaystyle n=2^{k}\pm 1}or {\displaystyle n=4^{k}\pm 3},
    2. 2^n-1is a (Mersenne) prime number,
    3. {\displaystyle (2^{n}+1)/3}is a prime number (it is called a Wagstaff prime).
  • Are all members of the sequence {\displaystyle C(0)=2,C(k+1)=2^{C(k)}-1}prime numbers? The stronger conjecture that all numbers {\displaystyle M_{M_{p}}}are primes for which M_{p}a prime was disproved by Raphael Robinson in 1957. (e.g. {\displaystyle M_{M_{13}}=M_{8191}}not prime) These latter numbers are called double Mersenne numbers (OEIS, A077585). So far, double Mersenne primes are only {\displaystyle p=2,3,5,7}known for (OEIS, A077586); small factors have been found for 31{\displaystyle p=13,17,19}and Whether there are further or even infinitely many double Mersenne primes remains unknown.

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