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

M_n = 2ⁿ − 1. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a power of two is any of the Integer powers of the number two; in other words two multiplied by itself a certain

A Mersenne prime is a Mersenne number that is a prime number. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. ^[1] 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). Distributed computing deals with Hardware and Software Systems containing more than one processing element or Storage element concurrent The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can

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. In Mathematics, the Mersenne conjectures concern the characterization of Prime numbers of a form called Mersenne primes meaning prime numbers that are a In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing 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 composite number is a positive Integer which has a positive Divisor other than one or itself In Number theory, a Prime number p is a Sophie Germain prime if 2 p  + 1 is also prime

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. A composite number is a positive Integer which has a positive Divisor other than one or itself 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. This article is about the Lucas–Lehmer test (LLT that only applies to Mersenne numbers A primality test is an Algorithm for determining whether an input number is prime. 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. Distributed computing deals with Hardware and Software Systems containing more than one processing element or Storage element concurrent

Mersenne primes are used in pseudorandom number generators such as Mersenne Twister and Park–Miller RNG. A pseudorandom number generator ( PRNG) is an Algorithm for generating a sequence of numbers that approximates the properties of random numbers The Mersenne twister is a Pseudorandom number generator developed in 1997 by and that is based on a Matrix linear recurrence over a finite binary The Park–Miller random number generator (or the Lehmer random number generator) is a variant of Linear congruential generator that operates in Multiplicative

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. A composite number is a positive Integer which has a positive Divisor other than one or itself

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. Pietro Antonio Cataldi ( April 15, 1552 - February 11, 1626) was an Italian Mathematician. 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. François Édouard Anatole Lucas ( April 4, 1842 in Amiens - October 3, 1891) was a French Mathematician. Ivan Mikheevich Pervushin ( Иван Михеевич Первушин) ( January 21, 1827 &ndash June 29, 1900) was an important Two more (M₈₉ and M₁₀₇) were found early in the 20th century, by Powers in 1911 and 1914, respectively. Details of the life of RE Powers are little-known however he was apparently the first Mathematician to demonstrate that the Mersenne number 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 1856 [1][2] and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test for Mersenne numbers. In Mathematics, a sequence is an ordered list of objects (or events François Édouard Anatole Lucas ( April 4, 1842 in Amiens - October 3, 1891) was a French Mathematician. Derrick Henry "Dick" Lehmer ( February 23 1905 &ndash May 22 1991) was an American mathematician who refined Edouard Lucas This article is about the Lucas–Lehmer test (LLT that only applies to 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, .

Graph of number of digits in largest known Mersenne prime by year - electronic era. Note that the vertical scale is logarithmic.

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. Events 1648 - Eighty Years' War: The Treaty of Münster is signed ending the conflict between the Netherlands and Spain Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. 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. The University of California Los Angeles (generally known as UCLA) is a public research university located in Westwood Los Angeles, California, United Derrick Henry "Dick" Lehmer ( February 23 1905 &ndash May 22 1991) was an American mathematician who refined Edouard Lucas R.M. Robinson. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. 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. Titanic prime is a term coined by Samuel Yates in the 1980s denoting a Prime number of at least 1000 decimal digits A gigantic prime is a Prime number with at least 10000 decimal digits A megaprime is a Prime number with at least one million decimal digits (whereas Titanic prime is a prime number with at least 1000 digits and Gigantic prime [3] All three were the first known prime of any kind of that size.

Theorems about Mersenne numbers

• 1) If n is a positive integer, by the binomial theorem we can write:

,

or

by setting c = 2^a, d = 1, and n = b

proof

= aⁿ − bⁿ

• 2) If 2ⁿ − 1 is prime, then n is prime. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says

proof

By

If n is not prime, or n = ab where 1 < a,b < n. Therefore, 2^a − 1 would divide 2ⁿ − 1, or 2ⁿ − 1 is not prime.

• 3) If p is an odd prime, then any prime q that divides 2^p − 1 must be 1 plus a multiple of 2p. This holds even when 2^p − 1 is prime. Example I: 2⁵ − 1 = 31

is prime, and 31 is 1 plus a multiple of 2*5. Example II: 2¹¹ − 1=23*89, 23=1+2*11, and 89=1+8*11, and also 23*89=1+186*11.

proof

If q divides 2^p − 1 then 2^p is congruent to 1 mod q, so p divides the order of the multiplicative group mod q, by Lagrange's Theorem. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of This group has order q−1, so q−1=kp for some k, and q=1+kp. But q must be odd, and p is odd,(except for p=2) so k is even.

• 4) If p is an odd prime, then any prime q that divides 2^p − 1 must be congruent to . Proof: 2^{p + 1} = 2(mod q), so 2^{(p + 1) / 2} is a square root of 2 modulo q. By quadratic reciprocity, any prime modulo which two has a square root is congruent to . The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability

History

Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding They are named after 17th century French scholar Marin Mersenne, who compiled a list of Mersenne primes with exponents up to 257. This article is about the country For a topic outline on this subject see List of basic France topics. Marin Mersenne, Marin Mersennus or le Père Mersenne ( September 8, 1588 &ndash September 1, 1648) was His list was only partially correct, as Mersenne mistakenly included M₆₇ and M₂₅₇ (which are composite), and omitted M₆₁, M₈₉, and M₁₀₇ (which are prime). Mersenne gave no indication how he came up with his list, and its rigorous verification was completed more than two centuries later.

List of known Mersenne primes

The table below lists all known Mersenne primes (sequence A000668 in OEIS):

^*It is not known whether any undiscovered Mersenne primes exist between the 39th (M_13,466,917) and the 44th (M_32,582,657) on this chart; the ranking is therefore provisional. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Pietro Antonio Cataldi ( April 15, 1552 - February 11, 1626) was an Italian Mathematician. Pietro Antonio Cataldi ( April 15, 1552 - February 11, 1626) was an Italian Mathematician. Ivan Mikheevich Pervushin ( Иван Михеевич Первушин) ( January 21, 1827 &ndash June 29, 1900) was an important Details of the life of RE Powers are little-known however he was apparently the first Mathematician to demonstrate that the Mersenne number M107 Details of the life of RE Powers are little-known however he was apparently the first Mathematician to demonstrate that the Mersenne number M107 François Édouard Anatole Lucas ( April 4, 1842 in Amiens - October 3, 1891) was a French Mathematician. Events 1648 - Eighty Years' War: The Treaty of Münster is signed ending the conflict between the Netherlands and Spain Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. Events 1648 - Eighty Years' War: The Treaty of Münster is signed ending the conflict between the Netherlands and Spain Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. Events 524 - Battle of Vézeronce, the Franks defeat the Burgundians Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. Events 3761 BC - The epoch (origin of the modern Hebrew calendar ( Proleptic Julian calendar) Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. Events 768 - Carloman I and Charlemagne are crowned Kings of The Franks. Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. Events 70 - Roman forces under Titus sack Jerusalem. 1264 - The Statute of Kalisz Year 1957 ( MCMLVII) was a Common year starting on Tuesday (link displays the 1957 Gregorian calendar) Hans Ivar Riesel (born 1929 in Stockholm) is a Swedish Mathematician who discovered the 18th known Mersenne prime in 1957 Events 644 - Umar ibn al-Khattab, the second Muslim Caliph, is killed by a Persian slave in Medina. Year 1961 ( MCMLXI) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar. Alexander Hurwitz is an American Mathematician born October 16 1937 who discovered the 19th and 20th known Mersenne primes in 1961. Events 644 - Umar ibn al-Khattab, the second Muslim Caliph, is killed by a Persian slave in Medina. Year 1961 ( MCMLXI) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar. Alexander Hurwitz is an American Mathematician born October 16 1937 who discovered the 19th and 20th known Mersenne primes in 1961. Events 330 - Byzantium is renamed ''Nova Roma'' during a dedication ceremony but is more popularly referred to as Constantinople Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Donald Bruce Gillies ( October 15[[ 928]] – July 17[[ 975]] was a Canadian mathematician and computer scientist known for his work in Game theory Events 1204 - Baldwin IX Count of Flanders is crowned as the first Emperor of the Latin Empire. Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Donald Bruce Gillies ( October 15[[ 928]] – July 17[[ 975]] was a Canadian mathematician and computer scientist known for his work in Game theory Events 455 - The Vandals enter Rome, and plunder the city for two weeks Year 1963 ( MCMLXIII) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Donald Bruce Gillies ( October 15[[ 928]] – July 17[[ 975]] was a Canadian mathematician and computer scientist known for his work in Game theory Events 51 - Nero, later to become Roman Emperor, is given the title Princeps iuventutis (head of the youth Year 1971 ( MCMLXXI) was a Common year starting on Friday (link will display full calendar of the 1971 Gregorian calendar. Bryant Tuckerman ( November 28, 1915 - May 19, 2002) was an American mathematician born in Lincoln Nebraska. Events 637 - Antioch surrenders to the Muslim forces under Rashidun Caliphate after the Battle of Iron bridge. Year 1978 ( MCMLXXVIII) was a Common year starting on Sunday (link displays the 1978 Gregorian calendar) Landon Curt Noll (born October 28, 1960 Walnut Creek California, United States) is the discoverer of two Mersenne primes which he Ariel T Glenn (née Laura A Nickel with Landon Curt Noll discovered on October 30 1978 that 221701 &minus 1 was the 25th Mersenne prime Events 474 - Zeno crowned as co-emperor of the Byzantine Empire. Year 1979 ( MCMLXXIX) was a Common year starting on Monday (link displays the 1979 Gregorian calendar) Landon Curt Noll (born October 28, 1960 Walnut Creek California, United States) is the discoverer of two Mersenne primes which he Events 217 - Roman Emperor Caracalla is Assassinated (and succeeded by his Praetorian Year 1979 ( MCMLXXIX) was a Common year starting on Monday (link displays the 1979 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Events 303 - On a voyage preaching the Gospel, Saint Fermin of Pamplona is beheaded in Amiens, France Year 1982 ( MCMLXXXII) was a Common year starting on Friday (link displays the 1982 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Events 1077 - Walk to Canossa: The Excommunication of Henry IV Holy Roman Emperor is lifted Year 1988 ( MCMLXXXVIII) was a Leap year starting on Friday (link displays 1988 Gregorian calendar) Luther Welsh Jr (Luke is a Mathematician and Computer scientist. Events 335 - Dalmatius is raised to the rank of Caesar by his uncle Constantine I. Year 1983 ( MCMLXXXIII) was a Common year starting on Saturday (link displays the 1983 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Events 462 - Possible start of first Byzantine indiction cycle. Year 1985 ( MCMLXXXV) was a Common year starting on Tuesday (link displays 1985 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Events 197 - Roman Emperor Septimius Severus defeats usurper Clodius Albinus in the Battle of Lugdunum Year 1992 ( MCMXCII) was a Leap year starting on Wednesday (link will display full 1992 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Paul Gage is a research computer scientist who works at Cray Supercomputers. The Cray-2 was a vector Supercomputer made by Cray Research starting in 1985. Events 46 BC - Titus Labienus defeats Julius Caesar in the Battle of Ruspina. Year 1994 ( MCMXCIV) was a Common year starting on Saturday (link will display full 1994 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Paul Gage is a research computer scientist who works at Cray Supercomputers. Events 36 BC - In the Battle of Naulochus, Marcus Vipsanius Agrippa, Admiral of Octavian, defeats Sextus Pompeius Year 1996 ( MCMXCVI) was a Leap year starting on Monday (link will display full 1996 Gregorian calendar) David Slowinski is a Mathematician involved in Prime numbers His career highlights have included the discovery of several of the largest known Mersenne primes Paul Gage is a research computer scientist who works at Cray Supercomputers. Events 1002 - English king Ethelred orders the killing of all Danes in England, known today as the St Year 1996 ( MCMXCVI) was a Leap year starting on Monday (link will display full 1996 Gregorian calendar) The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 49 BC - Julius Caesar 's General Gaius Scribonius Curio is defeated in the Second Battle of the Bagradas River Year 1997 ( MCMXCVII) was a Common year starting on Wednesday (link will display full 1997 Gregorian calendar The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 98 - Trajan becomes Roman Emperor after the death of Nerva. Year 1998 ( MCMXCVIII) was a Common year starting on Thursday (link will display full 1998 Gregorian calendar) The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 193 - Roman Emperor Didius Julianus is Assassinated 987 - Hugh Capet is elected Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 1533 - Conquistadors from Spain under the leadership of Francisco Pizarro arrive in Cajamarca, Inca Year 2001 ( MMI) was a Common year starting on Monday according to the Gregorian calendar. The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 284 - Diocletian is proclaimed emperor by his soldiers Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 1252 - Pope Innocent IV issues the Papal bull Ad exstirpanda, which authorizes but also limits the "MMIV" redirects here For the Modest Mouse album see " Baron von Bullshit Rides Again " The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 3102 BC - Epoch (origin of the Kali Yuga. 1229 - The Sixth Crusade: Frederick II Holy Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 533 - Byzantine general Belisarius defeats the Vandals, commanded by King Gelimer, at the Battle of Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Dr Curtis Cooper is a professor at the University of Central Missouri, at its Department of Mathematics and Computer Science The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Events 476 - Romulus Augustus, last emperor of the Western Roman Empire, is deposed when Odoacer proclaims himself Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can Dr Curtis Cooper is a professor at the University of Central Missouri, at its Department of Mathematics and Computer Science The Great Internet Mersenne Prime Search ( GIMPS) is a collaborative project of volunteers who use Prime95 and MPrime Computer software that can

To help visualize the size of the 44th known Mersenne prime, it would require 2,616 pages to display the number in base 10 with 75 digits per line and 50 lines per page.

Factorization of Mersenne numbers

Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorised has been a Mersenne number. The special number field sieve (SNFS is a special-purpose Integer factorization algorithm As of March 2007, 2¹⁰³⁹ − 1 is the record-holder, after a calculation taking about a year on a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland. See integer factorization records for links to more information. Numbers of a general form The first very large distributed factorisation was RSA129 a challenge number described in the Scientific American article of 1977 which first popularised

Perfect numbers

Mersenne primes are interesting to many for their connection to perfect numbers. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding In the 4th century BC, Euclid demonstrated that if M_p is a Mersenne prime then

2^p−1×(2^p−1) = M_p(M_p+1)/2

is an even perfect number. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Mathematics, the parity of an object states whether it is even or odd In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers.

Generalization

The binary representation of 2ⁿ − 1 is the digit 1 repeated n times, for example, 2⁵ − 1 = 11111₂ in the binary notation. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. The Mersenne primes are therefore the base-2 repunit primes. In Recreational mathematics, a repunit is a Number like 11, 111, or 1111 that contains only the digit 1

Notes

1. ^ The largest prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "The Largest Known Prime by Year: A Brief History", from The Prime Pages website, U of Tennessee at Martin. The University of Tennessee (also known as UT) sometimes called the University of Tennessee Knoxville ( UT Knoxville, or UTK) is the flagship

See also

External links

• Great Internet Mersenne Prime Search (GIMPS) Orlando Florida - home page of mersenne. In Recreational mathematics, a repunit is a Number like 11, 111, or 1111 that contains only the digit 1 In Mathematics, a Fermat number, named after Pierre de Fermat who first studied them is a positive integer of the form F_{n} = 2^{2^{ The Erdős–Borwein Constant is the sum of the reciprocals of the Mersenne numbers It is named after Paul Erdős and Peter Borwein In Mathematics, the Mersenne conjectures concern the characterization of Prime numbers of a form called Mersenne primes meaning prime numbers that are a Prime95 is the name of the Microsoft Windows -based Software application written by George Woltman that is used by GIMPS, a Distributed computing MPrime is the name of the Linux and BSD Software application written by George Woltman that GIMPS, a Distributed computing This article is about the Lucas–Lehmer test (LLT that only applies to Mersenne numbers In Mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2^p-1}-1 where p is a Mersenne In Number theory, a Wieferich prime is a Prime number p such that p 2 divides 2 p  &minus 1 &minus 1 A Wagstaff prime is a Prime number p of the form p= where q is another prime org
• prime Mersenne Numbers - History, Theorems and Lists Explanation
• GIMPS Mersenne Prime - status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 40-44
• M_q = (8x)² - (3qy)² Mersenne Proof (pdf)
• M_q = x² + d. y² Math Thesis (ps)
• Mersenne Prime Bibliography with hyperlinks to original publications
• dpa - reportage about prime Mersenne number - detection in detail (German)
• Mersenne prime Wiki

Math World Links

• Eric W. Weisstein, Mersenne number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Eric W. Weisstein, Mersenne prime at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• 44th Mersenne Prime Found