# Mersenne prime

**Mersenne numberMersenne numbersMersenne primes8191Gordon Spence13107123058430092136939515242872 n -18,191**

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

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### Marin Mersenne

**MersenneMersenne, MarinFather Mersenne**

. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.

He is perhaps best known today among mathematicians for Mersenne prime numbers, those which can be written in the form

### 31 (number)

**31thirty-oneXXXI**

which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

31 is the third Mersenne prime (2 5 − 1) and the eighth Mersenne prime exponent, as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime.

### Great Internet Mersenne Prime Search

**GIMPSDr. Steven BooneGreat Internet Mersenne Prime Search (GIMPS)**

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.

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

### 127 (number)

**127**

which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

### Power of two

**powers of twopower of 2powers of 2**

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

A prime number that is one less than a power of two is called a Mersenne prime.

### Prime number

**primeprime factorprime numbers**

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

Particularly fast methods are available for numbers of special forms, such as Mersenne numbers.

### Mersenne Twister

**Mersenne Twister MT19937MT19937**

Mersenne primes are used in pseudorandom number generators such as the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG.

Its name derives from the fact that its period length is chosen to be a Mersenne prime.

### Lucas–Lehmer primality test

**Lucas–Lehmer testLucas-Lehmer testLucas-Lehmer primality test**

The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size.

In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers.

### Largest known prime number

**largest known primelargest knownlargest prime numbers known**

The largest known prime number, 2 82,589,933 − 1, is a Mersenne prime.

Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two.

### Manchester Mark 1

**Manchester Mark IMark 1Manchester computer**

Alan Turing searched for them on the Manchester Mark 1 in 1949, but the first successful identification of a Mersenne prime,

Work began in August 1948, and the first version was operational by April 1949; a program written to search for Mersenne primes ran error-free for nine hours on the night of 16/17 June 1949.

### Perfect number

**perfect numbersperfectodd perfect number**

are also noteworthy due to their connection with perfect numbers.

Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p -1 for prime p—what is now called a Mersenne prime.

### Derrick Henry Lehmer

**D. H. LehmerLehmerD.H. Lehmer**

, by this means was achieved at 10:00 pm 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.

Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991) was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes.

### Euclid–Euler theorem

This is known as the Euclid–Euler theorem.

The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes.

### Lehmer random number generator

**LehmerLehmer generatorLehmer RNG**

Mersenne primes are used in pseudorandom number generators such as the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG.

In 1988, Park and Miller suggested a Lehmer RNG with particular parameters m = 2 − 1 = 2,147,483,647 (a Mersenne prime M) and a = 7 = 16,807 (a primitive root modulo M), now known as MINSTD.

### Ivan Pervushin

**Ivan Mikheevich PervushinPervushinPervushin, Ivan**

was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number.

He discovered the ninth perfect number and its odd prime factor, the ninth Mersenne prime.

### University of Central Missouri

**Central Missouri State UniversityCentral MissouriCentral Missouri State**

On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, 2 57,885,161 − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.

1 contributor to that project, and is the only GIMPS team that has discovered four Mersenne primes: M43 2 30402457 - 1 with 9,152,052 digits, M44 2 32582657 - 1 with 9,808,358 digits, M48 2 57,885,161 -1 with 17,425,170 digits, and M49 2 74,207,281 -1 with 22,338,618 digits.

### Wieferich prime

**Lucas–Wieferich primeWieferich's criterionFermat's last theorem**

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime.

### 7

**seven七7 (number)**

which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

### Euclid

**Euclid of AlexandriaEuklidGreek Mathematician**

In the 4th century BC, Euclid proved that if

It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

### Édouard Lucas

**LucasFrançois Édouard Anatole LucasLucas, Édouard**

Édouard Lucas proved in 1876 that

In 1876, after 19 years of testing, he finally proved that 2 127 − 1 was prime; this would remain the largest known Mersenne prime for three-quarters of a century.

### Sophie Germain prime

**Maximally periodic reciprocalsSophie Germain conjecture**

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).

If a Sophie Germain prime p is congruent to 3 (mod 4), then its matching safe prime 2p + 1 will be a divisor of the Mersenne number 2 p − 1.

### Curtis Cooper (mathematician)

**Curtis CooperCooperCooper, Curtis**

On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, 2 57,885,161 − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.

Using software from the GIMPS project, Cooper and Steven Boone found the 43rd known Mersenne prime on their 700 PC cluster on December 15, 2005.

### 3

**三threenumber 3**

### 2,147,483,647

**21474836472 31 − 12147483647 (number)**

The number 2,147,483,647 is the eighth Mersenne prime, equal to 2 31 − 1.

### Triangular number

**triangular numberstriangulartriangle number**

th triangular number and the

is a Mersenne prime.