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Home > Mersenne prime

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence,

M_n = 2ⁿ − 1.

Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist.

It is currently unknown whether there is an infinite number of Mersenne primes.

The calculation

shows that M_n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; M_n may be composite even though n is prime. For example, 2¹¹ − 1 = 23 · 89.

Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.

The first four Mersenne primes M₂, M₃, M₅, M₇ were known in antiquity. The fifth, M₁₃, was discovered anonymously before 1461; the next two (M₁₇ and M₁₉) were found by Cataldi in 1588. After more than a century M₃₁ was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was M₁₂₇, found by LucasFrancois Edouard Anatole Lucas ( April 4, 1842 October 3, 1891) was a French mathematician who was educated at the Ecole Normale Superieure. He worked in the Paris observatory and later became a professor of mathematics in Paris. In the meantime he served in 1876Events January events January 31 The United States orders all Native Americans to move into reservations. February events February 2 The National League of Professional Baseball Clubs of Major League Baseball is formed. February 14 Alexander Graham Bell a, then M₆₁ by Pervushin in 1883Events January January 16 The Pendleton Civil Service Reform Act, establishing the United States Civil service, is passed January 19 The first electric lighting system employing overhead wires begins service ( Roselle, New Jersey) It was built by Thomas E. Two more - M₈₉ and M₁₀₇ - were found early in the 20th century19th century 20th century 21st century more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901- 2000 in the sense of the Gre, by Powers in 19111911 is a common year starting on Sunday (click on link for calendar). Events January-June January 1 Northern Territory is separated from South Australia January 3 In London, a shootout between Russian anarchists and the Scots Guard January 10 Major Jimmi and 1914Events January 4 77 seal hunters freeze to death on ice near Labrador January 5 Ford Motor Company announces an eight-hour workday and a minimum wage of $5 for a day's labor February 13 Copyright: In New York City the ASCAP (for American Society of Compos, respectively.

The numbers are named after 17th century16th century 17th century 18th century more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601- 1700. During this period, the power of England and the United Provinces increased; while that of FrenchThe French Republic or France ( French: Republique francaise or France is a country whose metropolitan territory is located in western Europe, and which is further made up of a collection of overseas islands and territories located in other continents. mathematician Marin Mersenne, who provided a list of Mersenne primes with exponents up to 257; unfortunately, his list was not correct, though, as he mistakenly included M₆₇ and M₂₅₇, and omitted M₆₁, M₈₉ and M₁₀₇.

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 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test. Specifically, it can be shown that M_n = 2ⁿ − 1 is prime if and only if M_n evenly divides S_n-2, where S₀ = 4 and for k > 0, S_k = S_{k − 1}² − 2.

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.

^*It is not known whether any undiscovered Mersenne primes exist between the 38th (M_6972593) and the 41st (M_24036583) on this chart; the ranking is therefore provisional.

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Mersenne primeIn mathematics, a Mersenne prime is a prime number that is one less than a power of two. For example, 3 4 − 1 22 − 1 is a Mersenne prime; so is 7 8 − 1 23 − 1. On the other hand, 15 16 − 1 24 − 1, for example, is not a Mersenne twisterThe Mersenne twister is a pseudorandom number generator that was developed in 1997 by Makoto Matsumoto and Takuji Nishimura . It provides for fast generation of very high quality random numbers having been designed specifically to rectify many of the flaw