Mersenne prime

A Mersenne prime is a Mersenne number that is a prime number.

In mathematics, a Mersenne number is a number that is one less than a power of two,

M_n = 2ⁿ - 1.

As of August 2007, only 44 Mersenne primes are known; the largest known prime number (2^32,582,657−1) is a Mersenne prime and in modern times the largest known prime has nearly always been a Mersenne prime^[1]. Like several previous Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS).

About Mersenne primes

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether there is a largest Mersenne prime, which would mean that 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_n to be a Mersenne prime, the exponent n 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 says 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, it may nevertheless turn out that 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 a Mersenne 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 soon, since Mersenne numbers grow very fast. The Lucas–Lehmer test for Mersenne numbers is an efficient primality test that greatly aids this task. 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.

Searching for Mersenne primes

The identity

shows that M_n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. (This follows very simply from the Mersenne property of the sequence of numbers of the form xⁿ - yⁿ. This states that x^a - y^a | x^b - y^b if and only if a|b.) The converse statement, namely that M_n is necessarily prime if n is prime, is false. The smallest counterexample is 2¹¹−1 = 23×89, a composite number.

Fast algorithms for finding Mersenne primes are available, and the largest known prime numbers as of 2007 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 [1][2] and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test for Mersenne numbers. Specifically, it can be shown that (for n > 2) M_n = 2ⁿ - 1 is prime if and only if M_n divides S_n-2, where S₀ = 4 and for k > 0, .

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

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. 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 is the first megaprime, meaning a prime with at least 1,000,000 digits.[3] All three were the first known prime of any kind of that size.

Theorems about Mersenne numbers

• 1) If n is a positive integer, by the binomial theorem we can write:

,

or

by setting c = 2^a, d = 1, and n = b

proof

= aⁿ - bⁿ

• 2) If 2ⁿ - 1 is prime, then n is prime.

proof

By

If n is not prime, or n = ab where 1 < a,b < n. Therefore, 2^a - 1 would divide 2ⁿ - 1, or 2ⁿ - 1 is not prime.

• 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 of course even when 2^p - 1 is prime. Example I: 2⁵ - 1 = 31

is prime, and 31 is 1 plus a multiple of 2*5. Example II: 2¹¹ - 1=23*89, 23=1+2*11, and 89=1+8*11, and also 23*89=1+186*11.

proof

If q divides 2^p - 1 then 2^p is congruent to 1 mod q, so p divides the order of the multiplicative group mod q, by Lagrange's Theorem. This group has order q-1, so q-1=kp for some k, and q=1+kp. But q must be odd, and p is odd,(except for p=2) so k is even.

• 4) If p is an odd prime, then any prime q that divides 2^p - 1 must be ±1(mod 8). Proof: 2^{p + 1} = 2(mod q), so 2^{(p + 1) / 2} is a square root of 2 modulo q. By quadratic reciprocity, any prime modulo which two has a square root is ±1(mod 8).

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 no 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 OEIS):

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^*It is not known whether any undiscovered Mersenne primes exist between the 39th (M_13,466,917) and the 44th (M_32,582,657) on this chart; the ranking is therefore provisional.

To help visualize the size of the 44th known Mersenne Prime, a standard word processor layout (12pt Times New Roman, 1" margins) would require 2,769 pages to display the number in base 10. ^{[citation needed]}

Factorization of Mersenne numbers

Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorised 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.

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×(2^p-1) = M_p(M_p+1)/2

is an even perfect 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.

Generalization

The binary representation of 2ⁿ − 1 is the digit 1 repeated n times, for example, 2⁵ − 1 = 11111₂ in the binary notation. The Mersenne primes are therefore the base-2 repunit primes.

Notes

1. ^ The largest prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "The Largest Known Prime by Year: A Brief History", from The Prime Pages website, U of Tennessee at Martin.

See also

• Repunit
• Fermat prime
• Erdős–Borwein constant
• Mersenne conjectures
• Prime95 / MPrime
• Lucas–Lehmer test for Mersenne numbers
• Double Mersenne number
• Mersenne twister
• Wieferich prime

External links

• Great Internet Mersenne Prime Search (GIMPS) Orlando Florida - home page of mersenne.org
• prime Mersenne Numbers - History, Theorems and Lists Explanation
• GIMPS Mersenne Prime - status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 40-44
• Eric W. Weisstein, Mersenne number at MathWorld.
• Eric W. Weisstein, Mersenne prime at MathWorld.
• 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
• dpa - reportage about prime Mersenne number - detection in detail (German)
• Mersenne prime Wiki
• 44th Mersenne Prime Found article at MathWorld

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