Mersenne prime

In mathematics, a Mersenne number, named after Marin Mersenne, 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.

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 47 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 were 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 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 based on the computation of a recurring sequence, as developed originally by Lucas in 1856 and improved by Lehmer in the 1930s, now known as the Lucas–Lehmer primality test. Specifically, it can be shown that (for 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 logarithhmic 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 2^p − 1 is prime, then p is prime.
   • Proof: suppose that p is composite, hence can be written $p=a\cdot b$ with a and b > 1. As stated above, $2^{ab}-1=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)$. (To check this formula, just compute the right-hand product: most terms will cancel out.) We have thus written $2^{ab}-1$ as a product of integers > 1, Q.E.D.
2. 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.
   • 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.
3. 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 $\pm 1{\pmod {8}}$.
4. 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 $\lambda \geq 2^{m}-1$. This leads to $p-1\geq m(2^{m}-1)$, which is impossible since $m\geq 2$.
5. If 2^k-1 is prime, then n=2^k-1(2^k-1) is perfect and every even perfect number is of this form.
   • Proof: Assume that n is an even perfect number. We may write n as n=2^k-1m where m is an odd integer. Then, $\sigma (n)=(2^{k}-1)\sigma (m)$. The assumption gives $\sigma (n)=2^{k}m$. These yield $m=2^{k}-1$ and m is a prime number. The converse is easy. Q.E.D.

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

 ^* It is not known whether any undiscovered Mersenne primes exist between the 39th (M_13,466,917) 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 2 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 (2^43,112,609 − 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 2010, 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 2⁶⁸⁴¹²⁷ − 1 = 23765203727 × 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). 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 the fifth Mersenne prime. The asteroids with the previous four numbers corresponding to Mersenne primes (3 Juno, 7 Iris, 31 Euphrosyne, 127 Johanna) were already named after others.

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

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.
• 44th Mersenne Prime Found