PrimeCount

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

M_n = 2ⁿ − 1.

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

Mersenne primes are used in pseudorandom number generators such as Mersenne Twister and Park–Miller RNG.

Searching for Mersenne primes

The identity

shows that M_n can be prime only if n itself is prime—that is the primeness of n is necessary but not sufficient for M_n to be 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 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, .

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 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.

Reference: http://wikipedia.org

Tags: Largest, Mersenne primes
Posted in General | Comments (0)