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# Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. [1]

## AKS primality test

The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P".

## Allan J. C. Cunningham

Allan Joseph Champneys Cunningham (1842 – 1928) was a British mathematician.

## André Gérardin

André Gérardin (1879, Nancy, Meurthe-et-Moselle – 1953, Nancy, Meurthe-et-Moselle) was a French mathematician, specializing in number theory and calculating machines used in factoring large positive integers, finding primes, and calculating quadratic residues modulo a given positive integer.

## Édouard Lucas

François Édouard Anatole Lucas (4 April 1842 – 3 October 1891) was a French mathematician.

## Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.

## BESK

BESK (Binär Elektronisk SekvensKalkylator, Swedish for "Binary Electronic Sequence Calculator") was Sweden's first electronic computer, using vacuum tubes instead of relays.

## Binomial number

In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms.

## Bryant Tuckerman

Louis Bryant Tuckerman, III (November 28, 1915 – May 19, 2002) was an American mathematician, born in Lincoln, Nebraska.

## Carol number

A Carol number is an integer of the form 4^n - 2^ - 1.

## Centered nonagonal number

A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers.

## Circular prime

A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime.

## CN-group

In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of: are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvable.

## Cornwall Colts

The Cornwall Colts are a Canadian junior ice hockey team from Cornwall, Ontario, Canada.

## Cunningham number

In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.

## Cunningham project

The Cunningham project is a project, started in 1925, to factor numbers of the form bn ± 1 for b.

## Curtis Cooper (mathematician)

Curtis Niles Cooper is an American mathematician.

## Data compaction

In telecommunication, data compaction is the reduction of the number of data elements, bandwidth, cost, and time for the generation, transmission, and storage of data without loss of information by eliminating unnecessary redundancy, removing irrelevancy, or using special coding.

## David Slowinski

David Slowinski is a mathematician involved in prime numbers.

## Derrick Henry Lehmer

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.

## Donald B. Gillies

Donald Bruce Gillies (October 15, 1928 – July 17, 1975) was a Canadian mathematician and computer scientist, known for his work in game theory, computer design, and minicomputer programming environments.

## Double Mersenne number

In mathematics, a double Mersenne number is a Mersenne number of the form where p is a prime exponent.

## Duodecimal

The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base.

## Eisenstein prime

In mathematics, an Eisenstein prime is an Eisenstein integer that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units, itself and its associates.

## Elementary cellular automaton

In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors.

## Elliptic curve primality

In mathematics elliptic curve primality testing techniques are among the quickest and most widely used methods in primality proving.

## Elliptic-curve cryptography

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

## Entropia, Inc.

Entropia, Inc. was a company founded in 1997 that sold distributed computing software for CPU scavenging.

## Erdős–Borwein constant

The Erdős–Borwein constant is the sum of the reciprocals of the Mersenne numbers.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclid–Euler theorem

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

## Experimental mathematics

Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.

## Fürer's algorithm

Fürer's algorithm is an integer multiplication algorithm for extremely large integers with very low asymptotic complexity.

## Fermat number

In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer.

## Fermat pseudoprime

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

## FLOPS

In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations.

## Frank Nelson Cole

Frank Nelson Cole (September 20, 1861 – May 26, 1926) was an American mathematician, born in Ashland, Massachusetts, and educated at Harvard, where he lectured on mathematics from 1885 to 1887.

## Friendly number

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself.

## Generating primes

In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently.

## George Woltman

George Woltman (born November 10, 1957) is the founder of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project researching Mersenne prime numbers using his software Prime95 and MPrime.

## Gigantic prime

A gigantic prime is a prime number with at least 10,000 decimal digits.

## Gillies' conjecture

In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes.

## Great Internet Mersenne Prime Search

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

## Hans Riesel

Hans Ivar Riesel (May 28, 1929 in Stockholm &ndash; December 21, 2014) was a Swedish mathematician who discovered the 18th known Mersenne prime in 1957, using the computer BESK: this prime is 23217-1 and consists of 969 digits.

## Happy number

A happy number is defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1.

## Harry L. Nelson

Harry Lewis Nelson (born January 8, 1932) is an American mathematician and computer programmer.

## Hexagonal number

A hexagonal number is a figurate number.

## History of computing hardware

The history of computing hardware covers the developments from early simple devices to aid calculation to modern day computers.

## Holographic algorithm

In computer science, a holographic algorithm is an algorithm that uses a holographic reduction.

## IBM 7090

The IBM 7090 is a second-generation transistorized version of the earlier IBM 709 vacuum tube mainframe computers that was designed for "large-scale scientific and technological applications".

## ILLIAC

ILLIAC (Illinois Automatic Computer) was a series of supercomputers built at a variety of locations, some at the University of Illinois at Urbana-Champaign.

## ILLIAC II

The ILLIAC II was a revolutionary super-computer built by the University of Illinois that became operational in 1962.

## Indian Institute of Technology Roorkee

Indian Institute of Technology Roorkee (abbreviated IIT Roorkee or IITR), formerly University of Roorkee and Thomason College of Civil Engineering, is a public engineering university located in Roorkee, Uttarakhand, India.

## Integer sequence prime

In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence.

## Ivan Pervushin

Ivan Mikheevich Pervushin (Иван Михеевич Первушин, sometimes transliterated as Pervusin or Pervouchine) (—) was a Russian clergyman and mathematician of the second half of the 19th century, known for his achievements in number theory.

## Jacobi symbol

Jacobi symbol for various k (along top) and n (along left side).

## Jens Franke

Jens Franke (born June 29, 1964) is a German mathematician.

## June 1949

The following events occurred in June 1949.

## Kynea number

A Kynea number is an integer of the form An equivalent formula is This indicates that a Kynea number is the nth power of 4 plus the (n + 1)th Mersenne number.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Landon Curt Noll

Landon Curt Noll (born October 28, 1960) is an American computer scientist, co-discoverer of the 25th Mersenne prime and discoverer of the 26th, which he found while still enrolled at Hayward High School and concurrently at California State University, Hayward.

## Large numbers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions.

## Largest known prime number

The largest known prime number is 277,232,917 − 1, a number with 23,249,425 digits.

## Lehmer random number generator

The Lehmer random number generator (named after D. H. Lehmer), sometimes also referred to as the Park–Miller random number generator (after Stephen K. Park and Keith W. Miller), is a type of linear congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is: where the modulus n is a prime number or a power of a prime number, the multiplier g is an element of high multiplicative order modulo n (e.g., a primitive root modulo n), and the seed X is coprime to n.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Liber Abaci

Liber Abaci (1202, also spelled as Liber Abbaci) is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.

## Linear congruential generator

A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.

## List of distributed computing projects

This is a list of distributed computing and grid computing projects.

## List of Hockey Heritage North Honoured Members

This is a list of the Honoured Members of Hockey Heritage North.

## List of logarithmic identities

In mathematics, there are many logarithmic identities.

## List of long mathematical proofs

This is a list of unusually long mathematical proofs.

## List of OEIS sequences

This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their own English Wikipedia entries.

## List of perfect numbers

The following is a list of the known perfect numbers, and the exponents p that can be used to generate them (using the expression 2p−1× (2p − 1)) whenever 2p − 1 is a Mersenne prime.

## List of prime numbers

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

## List of things named after Leonhard Euler

Leonhard Euler (1707–1783)In mathematics and physics, there are a large number of topics named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations.

## List of unsolved problems in mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still remain unsolved.

## Lucas sequence

In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation where P and Q are fixed integers.

## Lucas test

Lucas test may refer to.

## Lucas–Lehmer primality test

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

## Lucas–Lehmer–Riesel test

In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N.

## M36

M36, M-36 or M.36 may refer to.

## M40

M40 or M-40 may refer to: In transportation.

## M41

M41, M-41, or M.41 may refer to.

## M42

M42 or M-42 may refer to: In science.

## M43

M43 may refer to.

## M44

M44 or M-44 may refer to.

## M45

M45 or M-45 may refer to: In science.

## M46

M46 or M-46 may refer to.

## M47

M47 or M-47 may refer to.

## M48

M48 or M-48 may refer to.

## M49

M49, M.49 or M-49 may be.

## Manchester Mark 1

The Manchester Mark 1 was one of the earliest stored-program computers, developed at the Victoria University of Manchester from the Manchester Baby (operational in June 1948).

## Marin Mersenne

Marin Mersenne, Marin Mersennus or le Père Mersenne (8 September 1588 – 1 September 1648) was a French polymath, whose works touched a wide variety of fields.

## Megaprime

A megaprime is a prime number with at least one million decimal digits (whereas titanic prime is a prime number with at least 1,000 digits, and gigantic prime has at least 10,000 digits).

## Mersenne conjectures

In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.

## Mersenne Twister

The Mersenne Twister is a pseudorandom number generator (PRNG).

## Multiplication algorithm

A multiplication algorithm is an algorithm (or method) to multiply two numbers.

## On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences.

## Orders of magnitude (numbers)

This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities.

## Palindromic number

A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed.

## Palindromic prime

A palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number.

## Pandigital number

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once.

## Paul Gage

Paul Gage is a research computer scientist who works at Cray Supercomputers.

## Paul T. Bateman

Paul Trevier Bateman (June 6, 1919 – December 26, 2012) was an American number theorist, known for formulating the Bateman–Horn conjecture on the density of prime number values generated by systems of polynomials and the New Mersenne conjecture relating the occurrences of Mersenne primes and Wagstaff primes.

## Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum).

## Permutable prime

A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number.

## Pernicious number

In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.

## Peter Barlow (mathematician)

Peter Barlow (13 October 1776 – 1 March 1862)Lance Day and Ian McNeil, Biographical dictionary of the history of technology, Routledge, 1995, page 42.

## Pierpont prime

A Pierpont prime is a prime number of the form for some nonnegative integers and.

## Polite number

In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers.

## Power of two

In mathematics, a power of two is a number of the form where is an integer, i.e. the result of exponentiation with number two as the base and integer as the exponent.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime95

Prime95 is the freeware application written by George Woltman that is used by GIMPS, a distributed computing project dedicated to finding new Mersenne prime numbers.

## Primecoin

Primecoin (sign: Ψ; code: XPM) is a peer-to-peer open source cryptocurrency that implements a unique scientific computing proof-of-work system.

## Primitive polynomial (field theory)

In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF(pm).

## Proth number

In number theory, a Proth number, named after the mathematician François Proth, is a number of the form where k is an odd positive integer and n is a positive integer such that 2^n > k. Without the latter condition, all odd integers greater than 1 would be Proth numbers.

## Proth's theorem

In number theory, Proth's theorem is a primality test for Proth numbers.

## Quasi-opportunistic supercomputing

Quasi-opportunistic supercomputing is a computational paradigm for supercomputing on a large number of geographically disperse computers.

## Ralph Ernest Powers

Ralph Ernest Powers (April 27, 1875 – January 31, 1952) was an American amateur mathematician who worked on prime numbers.

## Ramanujan–Nagell equation

In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two.

## Raphael M. Robinson

Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an American mathematician.

## Repdigit

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal).

## Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 &mdash; a more specific type of repdigit.

## Richard P. Brent

Richard Peirce Brent (born 20 April 1946, Melbourne) is an Australian mathematician and computer scientist.

## Safe prime

A safe prime is a prime number of the form 2p + 1, where p is also a prime.

## Scholz conjecture

In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains.

## Scott Kurowski

Scott Kurowski is an entrepreneurial software technologist and inventor.

## Self number

A self number, Colombian number or Devlali number is an integer that cannot be written as the sum of any other integer n and the individual digits of n. This property is specific to the base used to represent the integers.

## Semiperfect number

In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors.

## Seventeen or Bust

Seventeen or Bust was a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.

## Solinas prime

In mathematics, a Solinas prime, or generalized mersenne prime, is a prime number that has the form f(2^m), where f(x) is a low-degree polynomial with small integer coefficients.

## Sophie Germain prime

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime.

## Special number field sieve

In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm.

## Strobogrammatic number

A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees.

## Supercomputer

A supercomputer is a computer with a high level of performance compared to a general-purpose computer.

## Superperfect number

In mathematics, a superperfect number is a positive integer n that satisfies where σ is the divisor summatory function.

## SWAC (computer)

The SWAC (Standards Western Automatic Computer) was an early electronic digital computer built in 1950 by the U.S. National Bureau of Standards (NBS) in Los Angeles, California.

## The Housekeeper and the Professor

(literally "The Professor's Beloved Equation") is a novel by Yoko Ogawa set in modern-day Japan.

## The Intimate P. D. Q. Bach

The Intimate P. D. Q. Bach is "a live recording of The Intimate P.D.Q. Bach stage show, featuring Professor Peter Schickele and the Semi-Pro Musica Antiqua" and was released on Vanguard Records in 1974.

## Timeline of computational mathematics

This is a timeline of key developments in computational mathematics.

## Titanic prime

Titanic prime is a term coined by Samuel Yates in the 1980s, denoting a prime number of at least 1000 decimal digits.

## Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right.

## Trygve Nagell

Trygve Nagell (July 13, 1895 in Oslo &ndash; January 24, 1988 in Uppsala) was a Norwegian mathematician, known for his works on the Diophantine equations within number theory.

## Unique prime

In recreational number theory, a unique prime or unique period prime is a certain kind of prime number.

## Universal hashing

In mathematics and computing universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below).

## University of Central Missouri

The University of Central Missouri (UCM), formerly Central Missouri State University (CMSU), is a public state university located in Warrensburg, Missouri, United States.

## Wheat and chessboard problem

The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as: The problem may be solved using simple addition.

## Wieferich prime

In number theory, a Wieferich prime is a prime number p such that p2 divides, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides.

## Wilhelm Ljunggren

Wilhelm Ljunggren (7 October 1905 – 25 January 1973) was a Norwegian mathematician, specializing in number theory.

## Woodall number

In number theory, a Woodall number (Wn) is any natural number of the form for some natural number n. The first few Woodall numbers are.

## 1,000,000,000

1,000,000,000 (one billion, short scale; one thousand million or milliard, yard, long scale) is the natural number following 999,999,999 and preceding 1,000,000,001.

## 10,000,000

10,000,000 (ten million) is the natural number following 9,999,999 and preceding 10,000,001.

## 100,000

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001.

## 107 (number)

107 (one hundred seven) is the natural number following 106 and preceding 108.

## 11 (number)

11 (eleven) is the natural number following 10 and preceding 12.

## 127 (number)

127 (one hundred twenty-seven) is the natural number following 126 and preceding 128.

## 1588 in science

The year 1588 in science and technology, Armada year, included a number of events, some of which are listed here.

## 17 (number)

17 (seventeen) is the natural number following 16 and preceding 18.

## 1772 in science

The year 1772 in science and technology involved some significant events.

## 1876 in science

The year 1876 in science and technology involved some significant events, listed below.

## 19 (number)

19 (nineteen) is the natural number following 18 and preceding 20.

## 1903 in science

The year 1903 in science and technology involved some significant events, listed below.

## 1925–26 Toronto St. Patricks season

The 1925–26 Toronto St.

## 2

2 (two) is a number, numeral, and glyph.

## 2,147,483,647

The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1.

## 2000 (number)

2000 (two thousand) is a natural number following 1999 and preceding 2001.

## 2005 in the United States

Events from the year 2005 in the United States.

## 2006 in science

The year 2006 in science and technology involved some significant events.

## 255 (number)

255 (two hundred fifty-five) is the natural number following 254 and preceding 256.

## 28 (number)

28 (twenty-eight) is the natural number following 27 and preceding 29.

## 3

3 (three) is a number, numeral, and glyph.

## 31 (number)

31 (thirty-one) is the natural number following 30 and preceding 32.

## 4000 (number)

4000 (four thousand) is the natural number following 3999 and preceding 4001.

## 496 (number)

496 (four hundred ninety-six) is the natural number following 495 and preceding 497.

## 5

5 (five) is a number, numeral, and glyph.

## 500 (number)

500 (five hundred) is the natural number following 499 and preceding 501.

## 6

6 (six) is the natural number following 5 and preceding 7.

## 61 (number)

61 (sixty-one) is the natural number following 60 and preceding 62.

## 63 (number)

63 (sixty-three) is a natural number following 62 and preceding 64.

## 64 (number)

64 (sixty-four) is the natural number following 63 and preceding 65.

## 7

7 (seven) is the natural number following 6 and preceding 8.

## 8000 (number)

8000 (eight thousand) is the natural number following 7999 and preceding 8001.

## 8128 (number)

8128 is the integer following 8127 and preceding 8129.

## 89 (number)

89 (eighty-nine) is the natural number following 88 and preceding 90.

## References

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