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

seven7 (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

threenumber 3
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, ...

2,147,483,647

21474836472 31 − 12147483647 (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, ...
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.