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

Mn = 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 (232,582,657−1) 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

## Contents

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 Mn 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 Mp, where p = 2, 3, 5, … could be prime, it may nevertheless turn out that Mp 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

$2^{ab}-1=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)$

shows that Mn can be prime only if n itself is prime—that is the primeness of n is necessary but not sufficient for Mn 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 xnyn. This states that xaya | xbyb if and only if a|b. ) The converse statement, namely that Mn 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) Mn = 2n − 1 is prime if and only if Mn divides Sn−2, where S0 = 4 and for k > 0, $S_k=S_{k-1}^2-2$.

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 M6,972,593 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.

$c^n-d^n=(c-d)\sum_{k=0}^{n-1} c^kd^{n-1-k}$,

or

$(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)=2^{ab}-1$

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

proof

$(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k}$
$=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}$
$=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n$
= anbn
• 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

$(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)=2^{ab}-1$

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

• 3) If p is an odd prime, then any prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 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 2p − 1 then 2p 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 2p − 1 must be congruent to $\pm 1 \pmod 8$. Proof: 2p + 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 $\pm 1 \pmod 8$. 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):

#nMnDigits in MnDate of discoveryDiscoverer
1231ancientancient
2371ancientancient
35312ancientancient
471273ancientancient
513819141456anonymous [4]
61713107161588Cataldi
71952428761588Cataldi
8312147483647101772Euler
9612305843009213693951191883Pervushin
1089618970019…449562111271911Powers
11107162259276…010288127331914Powers[5]
12127170141183…884105727391876Lucas
13521686479766…115057151157January 30, 1952Robinson
14607531137992…031728127183January 30, 1952Robinson
151,279104079321…168729087386June 25, 1952Robinson
162,203147597991…697771007664October 7, 1952Robinson
172,281446087557…132836351687October 9, 1952Robinson
183,217259117086…909315071969September 8, 1957Riesel
194,253190797007…3504849911,281November 3, 1961Hurwitz
204,423285542542…6085806071,332November 3, 1961Hurwitz
219,689478220278…2257541112,917May 11, 1963Gillies
229,941346088282…7894635512,993May 16, 1963Gillies
2311,213281411201…6963921913,376June 2, 1963Gillies
2419,937431542479…9680414716,002March 4, 1971Tuckerman
2521,701448679166…5118827516,533October 30, 1978Noll & Nickel
2623,209402874115…7792645116,987February 9, 1979Noll
2744,497854509824…01122867113,395April 8, 1979Nelson & Slowinski
2886,243536927995…43343820725,962September 25, 1982Slowinski
29110,503521928313…46551500733,265January 28, 1988Colquitt & Welsh
30132,049512740276…73006131139,751September 19, 1983[6]Slowinski
31216,091746093103…81552844765,050September 1, 1985[7]Slowinski
32756,839174135906…544677887227,832February 19, 1992Slowinski & Gage on Harwell Lab Cray-2 [8]
33859,433129498125…500142591258,716January 4, 1994 [9]Slowinski & Gage
341,257,787412245773…089366527378,632September 3, 1996Slowinski & Gage [10]
351,398,269814717564…451315711420,921November 13, 1996GIMPS / Joel Armengaud [11]
362,976,221623340076…729201151895,932August 24, 1997GIMPS / Gordon Spence [12]
373,021,377127411683…024694271909,526January 27, 1998GIMPS / Roland Clarkson [13]
386,972,593437075744…9241937912,098,960June 1, 1999GIMPS / Nayan Hajratwala [14]
3913,466,917924947738…2562590714,053,946November 14, 2001GIMPS / Michael Cameron [15]
40*20,996,011125976895…8556820476,320,430November 17, 2003GIMPS / Michael Shafer [16]
41*24,036,583299410429…7339694077,235,733May 15, 2004GIMPS / Josh Findley [17]
42*25,964,951122164630…5770772477,816,230February 18, 2005GIMPS / Martin Nowak [18]
43*30,402,457315416475…6529438719,152,052December 15, 2005GIMPS / Curtis Cooper & Steven Boone [19]
44*32,582,657124575026…0539678719,808,358September 4, 2006GIMPS / Curtis Cooper & Steven Boone [20]

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 Mp is a Mersenne prime then

2p−1×(2p−1) = Mp(Mp+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