Mersenne prime

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:

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

A Mersenne prime is a Mersenne number that is prime. It is known^[2] 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^[ref], 47 Mersenne primes are known. The largest known prime number (2^43,112,609 – 1) is a Mersenne prime.^[3] 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

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 congruent to 3 (mod 4).

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, M_p is very rarely 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

shows that M_p can be prime only if p itself is 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,

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.^[4] 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.^[5] 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.^[6]

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 = 1 + 3×2×5. Example II: 2¹¹ − 1 = 23×89, where 23 = 1 + 2×11, and 89 = 1 + 4×2×11.
   • 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 must be congruent to .
   • Proof: , so is a square root of 2 modulo . By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to .
5. A Mersenne prime cannot be a Wieferich prime.
   • Proof: We show if is a Mersenne prime, then the congruence does not satisfy. By Fermat's Little theorem, . Now write, . If the given congruence satisfies, then ,therefore . Hence ,and therefore λ ≥ 2^m − 1. This leads to p − 1  ≥ m(2^m − 1), which is impossible since m ≥ 2.
6. A prime number divides at most one prime-exponent Mersenne number^[7]

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 what was supposed to be a list of Mersenne primes with exponents up to 257. His list was largely incorrect, 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.^[8] A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries later. M₆₁ was determined to be prime in 1883 by Ivan Mikheevich Pervushin and for this reason it is sometimes called Pervushin's number. At the time of Pervushin's proof it was the second-largest known prime number, (Édouard Lucas having shown earlier that M₁₂₇ is also prime), and it remained so until 1911.

List of known Mersenne primes

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

 ^* It is not known whether any undiscovered Mersenne primes exist between the 41st (M_24,036,583) 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.^[9]

The largest known Mersenne prime (2^43,112,609 − 1) is also the largest known prime number,^[3] 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.^[32]

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^[update], 2¹⁰³⁹ − 1 is the record-holder,^[33] 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 December 2011^[update], the composite Mersenne number with largest proven prime factors is 2²⁶⁹⁰³ − 1 = 1113285395642134415541632833178044793 × p, where p has 8063 digits and was proven prime with ECPP.^[34] The largest factorization with probable prime factors allowed is 2^1168183 − 1 = 54763676838381762583 × q, where q is a 351639-digit probable prime.^[35]

Perfect numbers

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

is an even perfect number (Elements ix.36)—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.

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:

Mersenne numbers in nature and elsewhere

In computer science, unsigned p-bit (or signed p+1-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).^[36]

See also

References

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
• (German) 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
• Prime curios: 2305843009213693951

MathWorld links

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