Mersenne prime

Mersenne prime

Named after	Marin Mersenne
Publication year	1536
Author of publication	Regius, H.
No. of known terms	49
First terms	3, 7, 31, 127
Largest known term	249,318,327-1
OEIS index	A000668

In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two:

$M_{p}=2^{p}-1.\,$

Some definitions of Mersenne numbers require that the exponent p be prime, since the associated number must be composite if p is.

A Mersenne prime is a Mersenne number that is prime. It is known that if 2^p − 1 is prime then p is prime, so it makes no difference which Mersenne number definition is used. As of October 2009, only 49 Mersenne primes are known. The largest known prime number (2^43,112,609 − 1) is a Mersenne prime. Since 1997, all newly-found Mersenne primes have been discovered by the "Great Internet Mersenne Prime Search" (GIMPS), a distributed computing project on the Internet.

About Mersenne primes

Unsolved problem in mathematics:

Are there infinitely many Mersenne primes?

(more unsolved problems in mathematics)

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite. The Lenstra–Pomerance–Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.

A basic theorem about Mersenne numbers states that in order for M_p to be a Mersenne prime, the exponent p itself must be a prime number. This rules out primality for numbers such as M₄ = 2⁴ − 1 = 15: since the exponent 4 = 2×2 is composite, the theorem predicts that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are

M₂ = 3, M₃ = 7, M₅ = 31.

While it is true that only Mersenne numbers M_p, where p = 2, 3, 5, … could be prime, often M_p is not prime even for a prime exponent p. The smallest counterexample is the Mersenne number

M₁₁ = 2¹¹ − 1 = 2047 = 23 × 89,

which is not prime, even though 11 is a prime number. The lack of an obvious rule to determine whether a given Mersenne number is prime makes the search for Mersenne primes an interesting task, which becomes difficult very quickly, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.

Mersenne primes are used in pseudorandom number generators such as the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG.

Searching for Mersenne primes

The identity

$2^{ab}-1=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\$

shows that M_p can be prime only if p itself is prime—that is, the primality of p is necessary but not sufficient for M_p to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that M_p is necessarily prime if p is prime, is false. The smallest counterexample is 2¹¹ − 1 = 2,047 = 23 × 89, a composite number.

Fast algorithms for finding Mersenne primes are available, and the largest known prime numbers as of 2009 are Mersenne primes.

The first four Mersenne primes M₂ = 3, M₃ = 7, M₅ = 31 and M₇ = 127 were known in antiquity. The fifth, M₁₃ = 8191, was discovered anonymously before 1461; the next two (M₁₇ and M₁₉) were found by Cataldi in 1588. After nearly two centuries, M₃₁ was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was M₁₂₇, found by Lucas in 1876, then M₆₁ by Pervushin in 1883. Two more (M₈₉ and M₁₀₇) were found early in the 20th century, by Powers in 1911 and 1914, respectively.

The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, M_p = 2^p − 1 is prime if and only if M_p divides S_p−2, where S₀ = 4 and, for k > 0,

$S_{k}=S_{k-1}^{2}-2.\$

Graph of number of digits in largest known Mersenne prime by year - electronic era. Note that the vertical scale, the number of digits, is a double logarithmic scale of the value of the prime.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949. But the first successful identification of a Mersenne prime, M₅₂₁, by this means was achieved at 10:00 P.M. on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of Lehmer, with a computer search program written and run by Prof. R.M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M₆₀₇, was found by the computer a little less than two hours later. Three more — M₁₂₇₉, M₂₂₀₃, M₂₂₈₁ — were found by the same program in the next several months. M₄₂₅₃ is the first Mersenne prime that is titanic, M₄₄₄₉₇ is the first gigantic, and M_6,972,593 was the first megaprime to be discovered, being a prime with at least 1,000,000 digits. All three were the first known prime of any kind of that size.

In September 2008, mathematicians at UCLA participating in GIMPS won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This is the eighth Mersenne prime discovered at UCLA.

On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. This report was apparently overlooked until June 4, 2009. The find was verified on June 12, 2009. The prime is 2^42,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is less than the largest known which was the 45th to be discovered.

Theorems about Mersenne numbers

1. If a and p are natural numbers such that a^p − 1 is prime, then a = 2 or p = 1.
   • Proof: a ≡ 1 (mod a − 1). Then a^p ≡ 1 (mod a − 1), so a^p − 1 ≡ 0 (mod a − 1). Thus a − 1 | a^p − 1. However, a^p − 1 is prime, so a − 1 = a^p − 1 or a − 1 = ±1. In the former case, a = a^p, hence a = 0,1 (which is a contradiction, as neither 1 nor 0 is prime) or p = 1. In the latter case, a = 2 or a = 0. If a = 0, however, 0^p − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2.
2. If 2^p − 1 is prime, then p is prime.
   • Proof: suppose that p is composite, hence can be written p = a⋅b with a and b > 1. Then (2^a)^b − 1 is prime, but b > 1 and 2^a > 2, contradicting statement 1.
3. If p is an odd prime, then any prime q that divides 2^p − 1 must be 1 plus a multiple of 2p. This holds even when 2^p − 1 is prime.
   • Examples: Example I: 2⁵ − 1 = 31 is prime, and 31 is 1 plus a multiple of 2×5. Example II: 2¹¹ − 1 = 23×89, where 23 = 1 + 2×11, and 89 = 1 + 8×11.
   • Proof: If q divides 2^p − 1 then 2^p ≡ 1 (mod q). By Fermat's Little Theorem, 2^(q − 1) ≡ 1 (mod q). Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that (q − 1)^(p − 1) ≡ 1 (mod p). Thus there is a number x ≡ (q − 1)^(p − 2) for which (q − 1)·x ≡ 1 (mod p), and therefore a number k for which (q − 1)·x − 1 = kp. Since 2^{(q − 1)} ≡ 1 (mod q), raising both sides of the congruence to the power x gives 2^{(q − 1)x} ≡ 1, and since 2^p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2^kp ≡ 1. Thus 2^{(q − 1)x}/2^kp = 2^{(q − 1)x − kp} ≡ 1 (mod q). But by definition, (q − 1)x − kp = 1, implying that 2¹ ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p and q − 1 are relatively prime is untenable. Since p is prime q − 1 must be a multiple of p. Of course, if the number m = ^(q − 1)⁄_p is odd, then q will be even, since it is equal to mp + 1. But q is prime and cannot be equal to 2; therefore, m is even.
   • Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2^p − 1 must be larger than p.
4. If p is an odd prime, then any prime q that divides $2^{p}-1$ must be congruent to $\pm 1{\pmod {8}}$.
   • Proof: $2^{p+1}=2{\pmod {q}}$, so $2^{(p+1)/2}$ is a square root of 2 modulo $q$. By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to $\pm 1{\pmod {8}}$.
5. A Mersenne prime cannot be a Wieferich prime.
   • Proof: We show if $p=2^{m}-1$ is a Mersenne prime, then the congruence $2^{p-1}\equiv 1{\pmod {p^{2}}}$ does not satisfy. By Fermat's Little theorem, $m|p-1$. Now write, $p-1=m\lambda$. If the given congruence satisfies, then $p^{2}|2^{m\lambda }-1$,therefore $0\equiv (2^{m\lambda }-1)/(2^{m}-1)=1+2^{m}+2^{2m}+...+2^{(\lambda -1)m}\equiv -\lambda {\pmod {2^{m}-1}}$. Hence $2^{m}-1|\lambda$,and therefore λ ≥ 2^m − 1. This leads to p − 1  ≥ m(2^m − 1), which is impossible since m ≥ 2.

History

Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers. They are named after 17th-century French scholar Marin Mersenne, who compiled a list of Mersenne primes with exponents up to 257. His list was only partially correct, as Mersenne mistakenly included M₆₇ and M₂₅₇ (which are composite), and omitted M₆₁, M₈₉, and M₁₀₇ (which are prime). Mersenne gave little indication how he came up with his list, and its rigorous verification was completed more than two centuries later.

List of known Mersenne primes

The table below lists all known Mersenne primes (sequence A000668 in the OEIS):

#	p	Mp	Mp digits	Date of discovery	Discoverer	Method used
1	2	3	1	c. 500 BC	Ancient Greek mathematicians
2	3	7	1	c. 500 BC	Ancient Greek mathematicians
3	5	31	2	c. 275 BC	Ancient Greek mathematicians
4	7	127	3	c. 275 BC	Ancient Greek mathematicians
5	13	8191	4	1456	Anonymous	Trial division
6	17	131071	6	1588	Pietro Cataldi	Trial division
7	19	524287	6	1588	Pietro Cataldi	Trial division
8	31	2147483647	10	1772	Leonhard Euler	Enhanced trial division
9	61	2305843009213693951	19	1883	I. M. Pervushin	Lucas sequences
10	89	618970019…449562111	27	1911	R. E. Powers	Lucas sequences
11	107	162259276…010288127	33	1914	R.E. Powers	Lucas sequences
12	127	170141183…884105727	39	1876	Edouard Lucas	Lucas sequences
13	521	686479766…115057151	157	1952 January 30	Raphael M. Robinson	LLT / SWAC
14	607	531137992…031728127	183	1952 January 30	Raphael M. Robinson	LLT / SWAC
15	1,279	104079321…168729087	386	1952 June 25	Raphael M. Robinson	LLT / SWAC
16	2,203	147597991…697771007	664	1952 October 7	Raphael M. Robinson	LLT / SWAC
17	2,281	446087557…132836351	687	1952 October 9	Raphael M. Robinson	LLT / SWAC
18	3,217	259117086…909315071	969	1957 September 8	Hans Riesel	LLT / BESK
19	4,253	190797007…350484991	1,281	1961 November 3	Alexander Hurwitz	LLT / IBM 7090
20	4,423	285542542…608580607	1,332	1961 November 3	Alexander Hurwitz	LLT / IBM 7090
21	9,689	478220278…225754111	2,917	1963 May 11	Donald B. Gillies	LLT / ILLIAC II
22	9,941	346088282…789463551	2,993	1963 May 16	Donald B. Gillies	LLT / ILLIAC II
23	11,213	281411201…696392191	3,376	1963 June 2	Donald B. Gillies	LLT / ILLIAC II
24	19,937	431542479…968041471	6,002	1971 March 4	Bryant Tuckerman	LLT / IBM 360/91
25	21,701	448679166…511882751	6,533	1978 October 30	Landon Curt Noll & Laura Nickel	LLT / CDC Cyber 174
26	23,209	402874115…779264511	6,987	1979 February 9	Landon Curt Noll	LLT / CDC Cyber 174
27	44,497	854509824…011228671	13,395	1979 April 8	Harry Lewis Nelson & David Slowinski	LLT / Cray 1
28	86,243	536927995…433438207	25,962	1982 September 25	David Slowinski	LLT / Cray 1
29	110,503	521928313…465515007	33,265	1988 January 28	Walter Colquitt & Luke Welsh	LLT / NEC SX-2
30	132,049	512740276…730061311	39,751	1983 September 19	David Slowinski	LLT / Cray X-MP
31	216,091	746093103…815528447	65,050	1985 September 1	David Slowinski	LLT / Cray X-MP/24
32	756,839	174135906…544677887	227,832	1992 February 19	David Slowinski & Paul Gage	LLT / Maple on Harwell Lab Cray-2
33	859,433	129498125…500142591	258,716	1994 January 4	David Slowinski & Paul Gage	LLT / Cray C90
34	1,257,787	412245773…089366527	378,632	1996 September 3	David Slowinski & Paul Gage	LLT / Cray T94
35	1,398,269	814717564…451315711	420,921	1996 November 13	GIMPS / Joel Armengaud	LLT / Prime95 on 90 MHz Pentium PC
36	2,976,221	623340076…729201151	895,932	1997 August 24	GIMPS / Gordon Spence	LLT / Prime95 on 100 MHz Pentium PC
37	3,021,377	127411683…024694271	909,526	1998 January 27	GIMPS / Roland Clarkson	LLT / Prime95 on 200 MHz Pentium PC
38	6,972,593	437075744…924193791	2,098,960	1999 June 1	GIMPS / Nayan Hajratwala	LLT / Prime95 on 350 MHz IBM Aptiva
39	13,466,917	924947738…256259071	4,053,946	2001 November 14	GIMPS / Michael Cameron	LLT / Prime95 on 800 MHz AMD T-Bird
40	20,996,011	125976895…855682047	6,320,430	2003 November 17	GIMPS / Michael Shafer	LLT / Prime95 on 2 GHz Dell Dimension
41[*]	24,036,583	299410429…733969407	7,235,733	2004 May 15	GIMPS / Josh Findley	LLT / Prime95 on 2.4 GHz Pentium 4 PC
42[*]	25,964,951	122164630…577077247	7,816,230	2005 February 18	GIMPS / Martin Nowak	LLT / Prime95 on 2.4 GHz Pentium 4 PC
43[*]	30,402,457	315416475…652943871	9,152,052	2005 December 15	GIMPS / Curtis Cooper & Steven Boone	LLT / Prime95 on 2 GHz Pentium 4 PC
44[*]	32,582,657	124575026…053967871	9,808,358	2006 September 4	GIMPS / Curtis Cooper & Steven Boone	LLT / Prime95 on 3 GHz Pentium 4 PC
45[*]	37,156,667	202254406…308220927	11,185,272	2008 September 6	GIMPS / Hans-Michael Elvenich	LLT / Prime95 on 2.83 GHz Core 2 Duo PC
46[*]	42,643,801	169873516…562314751	12,837,064	2009 April 12[**]	GIMPS / Odd M. Strindmo	LLT / Prime95 on 3 GHz Core 2 PC
47[*]	43,112,609	316470269…697152511	12,978,189	2008 August 23	GIMPS / Edson Smith	LLT / Prime95 on Dell Optiplex 745

 ^* It is not known whether any undiscovered Mersenne primes exist between the 40th (M_20,996,011) and the 47th (M_43,112,609) on this chart; the ranking is therefore provisional. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, the current record holder was followed by two smaller Mersenne primes, first 2 weeks later and then 8 months later.

 ^** M_42,643,801 was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad.

To help visualize the size of the 47th known Mersenne prime, it would require 3,461 pages to display the number in base 10 with 75 digits per line and 50 lines per page.

The largest known Mersenne prime (2^49,318,327 − 1) is also the largest known prime number, and was the first discovered prime number with more than 10 million base-10 digits.

In modern times, the largest known prime has almost always been a Mersenne prime.

Factorization of Mersenne numbers

The factorization of a prime number is by definition the number itself. This section is about composite numbers. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of March 2007, 2¹⁰³⁹ − 1 is the record-holder, after a calculation taking about a year on a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland. See integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor. As of 2011, the composite Mersenne number with largest proven prime factors is 2²⁰⁸⁸⁷ − 1, which is known to have a factor p with 6229 digits that was proven prime with ECPP. The largest with probable prime factors allowed is 2^1168183 − 1 = 54763676838381762583 × q, where q is a probable prime.

Perfect numbers

Mersenne primes are interesting to many for their connection to perfect numbers. In the 4th century BC, Euclid demonstrated that if M_p is a Mersenne prime then

$2^{p-1}\cdot (2^{p}-1)=M_{p}(M_{p}+1)/2\$

is an even perfect number (which is also the M_pth triangular number and the 2^p−1th hexagonal number). In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers, but it appears unlikely that there is one.

Generalization

The binary representation of 2^p − 1 is the digit 1 repeated p times, for example, 2⁵ − 1 = 11111₂ in the binary notation. A Mersenne number is therefore a repunit in base 2, and Mersenne primes are the base 2 repunit primes.

The base 2 representation of a Mersenne number shows the factorization pattern for composite exponent. For example:

$M_{35}=2^{35}-1=(11111111111111111111111111111111111)_{2}\,$

$=(11111)_{2}\cdot (1000010000100001000010000100001)_{2}=M_{5}\cdot (1000010000100001000010000100001)_{2}\,$

$=(1111111)_{2}\cdot (10000001000000100000010000001)_{2}=M_{7}\cdot (10000001000000100000010000001)_{2}\,$

$=(11111)_{2}\cdot (1111111)_{2}\cdot [(1000010100101010010100001)_{2}-(0100001010010100101000010)_{2}]\,$

$=M_{5}\cdot M_{7}\cdot (100001010010101101011111)_{2}.\,$

Mersenne numbers in nature and elsewhere

In computer science, unsigned p-bit integers can be used to express numbers up to M_p.

In the mathematical problem Tower of Hanoi, solving a puzzle with a p-disc tower requires at least M_p steps.

The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime (3 Juno, 7 Iris, 31 Euphrosyne and 127 Johanna having been discovered and named during the nineteenth century).

See also

• Repunit
• Fermat prime
• Erdős–Borwein constant
• Mersenne conjectures
• Mersenne Twister
• Prime95 / MPrime
• Largest known prime number
• Double Mersenne number
• Wieferich prime
• Wagstaff prime
• Solinas prime

Notes

1.^{^} Mersenne primes have already been described in Regius, H. (1536). Utrisque Arithmetices Epitome

External links

• GIMPS home page
• Mersenne Primes: History, Theorems and Lists — explanation
• GIMPS status — status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 40–47
• M_q = (8x)² − (3qy)² Mersenne proof (pdf)
• M_q = x² + d·y² math thesis (ps)
• Mersenne prime bibliography with hyperlinks to original publications
• Template:De icon report about Mersenne primes — detection in detail
• GIMPS wiki
• Will Edgington's Mersenne Page — contains factors for small Mersenne numbers
• a file containing the smallest known factors of all tested Mersenne numbers (requires program to open)
• Decimal digits and English names of Mersenne primes

MathWorld links

• Weisstein, Eric W. "Mersenne number". MathWorld.
• Weisstein, Eric W. "Mersenne prime". MathWorld.
• 47th Mersenne Prime Found