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Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. 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). The first four Mersenne primes are M2 = 3, M3 = 7, M5 = 31 and M7 = 127. A basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity \begin{align}2^{ab}-1&=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)\\&=(2^b-1)\cdot \left(1+2^b+2^{2b}+2^{3b}+\cdots+2^{(a-1)b}\right). \end{align} This rules out primality for Mersenne numbers with composite exponent, such as M4 = 24 − 1 = 15 = 3×5 = (22 − 1)×(1 + 22). Though it was believed by early mathematicians that Mp is prime for all primes p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases. In fact, of the 1,881,339 prime numbers p up to 30,402,457,[4] Mp is prime for only 43 of them. The smallest counterexample is the Mersenne number M11 = 211 − 1 = 2047 = 23 × 89. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task. The 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. Mersenne primes are used in pseudorandom number generators such as the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG. Perfect numbers Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. In the 4th century BC, Euclid proved that if 2p−1 is prime, then 2p−1(2p − 1) is a perfect number. This number, also expressible as Mp(Mp+1)/2, is the Mpth triangular number and the 2p − 1th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.[5] It is unknown whether there are any odd perfect numbers. History Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. His list was largely incorrect, as Mersenne mistakenly included M67 and M257 (which are composite), and omitted M61, M89, and M107 (which are prime). Mersenne gave little indication how he came up with his list.[6] Édouard Lucas proved in 1876 that M127 is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever calculated by hand. M61 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. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M67 is actually composite. No factor was found until a famous talk by Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking.[7] He later said that the result had taken him "three years of Sundays" to find.[8] A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list. Searching for Mersenne primes Fast algorithms for finding Mersenne primes are available, and as of 2014 the ten largest known prime numbers 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. 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. Two more (M89 and M107) were found early in the 20th century, by Powers in 1911 and 1914, respectively. The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, Mp = 2p − 1 is prime if and only if Mp divides Sp−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, the number of digits, is a double logarithmic scale of the value of the prime. The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949,[9] but the first successful identification of a Mersenne prime, M521, 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. 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.[10] All three were the first known prime of any kind of that size. In September 2008, mathematicians at UCLA participating in GIMPS won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This is the eighth Mersenne prime discovered at UCLA.[11] On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. This report was apparently overlooked until June 4, 2009. The find was verified on June 12, 2009. The prime is 242,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered. On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, 257,885,161 − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.[12] This was the third Mersenne prime discovered by Dr. Cooper and his team in the past seven years. The Electronic Frontier Foundation (EFF) offers a prize of $150,000 to the first individual or group who discovers a prime number with at least 100,000,000 decimal digits[13] (the smallest Mersenne number with said amount of digits is 2332192807 − 1). Theorems about Mersenne numbers If a and p are natural numbers such that ap − 1 is prime, then a = 2 or p = 1. Proof: a ≡ 1 (mod a − 1). Then ap ≡ 1 (mod a − 1), so ap − 1 ≡ 0 (mod a − 1). Thus a − 1 | ap − 1. However, ap − 1 is prime, so a − 1 = ap − 1 or a − 1 = ±1. In the former case, a = ap, hence a = 0,1 (which is a contradiction, as neither 1 nor 0 is prime) or p = 1. In the latter case, a = 2 or a = 0. If a = 0, however, 0p − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2. If 2p − 1 is prime, then p is prime. Proof: suppose that p is composite, hence can be written p = a⋅b with a and b > 1. Then 2p − 1 = 2ab − 1 = (2a)b − 1 = (2a − 1)[(2a)b − 1 + (2a)b − 2 + … + 2a + 1] so 2p − 1 is composite contradicting our assumption that 2p − 1 is prime. 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. Examples: Example I: 25 − 1 = 31 is prime, and 31 = 1 + 3×(2×5). Example II: 211 − 1 = 23×89, where 23 = 1 + (2×11), and 89 = 1 + 4×(2×11). Proof: By Fermat's little theorem, q is a factor of 2q − 1 − 1. Since q is a factor of 2p − 1, for all positive integers c, q is also a factor of 2pc − 1. Since p is prime and q is not a factor of 21 − 1, p is also the smallest positive integer x such that q is a factor of 2x − 1. As a result, for all positive integers x, q is a factor of 2x − 1 if and only if p is a factor of x. Therefore, since q is a factor of 2q − 1 − 1, p is a factor of q − 1 so q ≡ 1 mod p. Furthermore, since q is a factor of 2p − 1, which is odd, q is odd. Therefore q ≡ 1 mod 2p. Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2p − 1 must be larger than p. If p is an odd prime, then any prime q that divides 2^p-1 must be congruent to ±1 (mod 8). Proof: 2^{p+1} = 2 \pmod q, so 2^{(p+1)/2} is a square root of 2 modulo q. By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to ±1 (mod 8). A Mersenne prime cannot be a Wieferich prime. Proof: We show if p = 2m − 1 is a Mersenne prime, then the congruence 2p − 1 ≡ 1 does not satisfy. By Fermat's Little theorem, m \mid p-1. Now write, p-1=m\lambda. If the given congruence satisfies, then p^2\mid2^{m\lambda}-1,therefore 0 ≡ (2mλ − 1)/(2m − 1) = 1 + 2m + 22m + ... + 2λ−1m ≡ −λ mod(2m − 1}. Hence 2^m-1\mid\lambda,and therefore λ ≥ 2m − 1. This leads to p − 1 ≥ m(2m − 1), which is impossible since m ≥ 2. A prime number divides at most one prime-exponent Mersenne number[14] If p and 2p + 1 are both prime (meaning that p is a Sophie Germain prime), and p is congruent to 3 (mod 4), then 2p + 1 divides 2p − 1.[15] Example: 11 and 23 are both prime, and 11 = 2×4 + 3, so 23 divides 211 − 1. All composite divisors of prime-exponent Mersenne numbers pass the Fermat primality test for the base 2. The number of digits in the decimal representation of M_n equals \lfloor n\cdot \log_{10}2\rfloor+1, where \lfloor x\rfloor denotes the floor function. List of known Mersenne primes The table below lists all known Mersenne primes (sequence A000668 in OEIS): # p Mp Mp digits Discovered Discoverer Method used 1 2 3 1 c. 430 BC Ancient Greek mathematicians[16] 2 3 7 1 c. 430 BC Ancient Greek mathematicians[16] 3 5 31 2 c. 300 BC Ancient Greek mathematicians[17] 4 7 127 3 c. 300 BC Ancient Greek mathematicians[17] 5 13 8191 4 1456 Anonymous[18][19] Trial division 6 17 131071 6 1588[20] Pietro Cataldi Trial division[21] 7 19 524287 6 1588 Pietro Cataldi Trial division[22] 8 31 2147483647 10 1772 Leonhard Euler[23][24] Enhanced trial division[25] 9 61 2305843009213693951 19 1883 November[26] I. M. Pervushin Lucas sequences 10 89 618970019642…137449562111 27 1911 June[27] Ralph Ernest Powers Lucas sequences 11 107 162259276829…578010288127 33 1914 June 1[28][29][30] Ralph Ernest Powers[31] Lucas sequences 12 127 170141183460…715884105727 39 1876 January 10[32] Édouard Lucas Lucas sequences 13 521 686479766013…291115057151 157 1952 January 30[33] Raphael M. Robinson LLT / SWAC 14 607 531137992816…219031728127 183 1952 January 30[33] Raphael M. Robinson LLT / SWAC 15 1,279 104079321946…703168729087 386 1952 June 25[34] Raphael M. Robinson LLT / SWAC 16 2,203 147597991521…686697771007 664 1952 October 7[35] Raphael M. Robinson LLT / SWAC 17 2,281 446087557183…418132836351 687 1952 October 9[35] Raphael M. Robinson LLT / SWAC 18 3,217 259117086013…362909315071 969 1957 September 8[36] Hans Riesel LLT / BESK 19 4,253 190797007524…815350484991 1,281 1961 November 3[37][38] Alexander Hurwitz LLT / IBM 7090 20 4,423 285542542228…902608580607 1,332 1961 November 3[37][38] Alexander Hurwitz LLT / IBM 7090 21 9,689 478220278805…826225754111 2,917 1963 May 11[39] Donald B. Gillies LLT / ILLIAC II 22 9,941 346088282490…883789463551 2,993 1963 May 16[39] Donald B. Gillies LLT / ILLIAC II 23 11,213 281411201369…087696392191 3,376 1963 June 2[39] Donald B. Gillies LLT / ILLIAC II 24 19,937 431542479738…030968041471 6,002 1971 March 4[40] Bryant Tuckerman LLT / IBM 360/91 25 21,701 448679166119…353511882751 6,533 1978 October 30[41] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174 26 23,209 402874115778…523779264511 6,987 1979 February 9[42] Landon Curt Noll LLT / CDC Cyber 174 27 44,497 854509824303…961011228671 13,395 1979 April 8[43][44] Harry Lewis Nelson & David Slowinski LLT / Cray 1 28 86,243 536927995502…209433438207 25,962 1982 September 25 David Slowinski LLT / Cray 1 29 110,503 521928313341…083465515007 33,265 1988 January 29[45][46] Walter Colquitt & Luke Welsh LLT / NEC SX-2[47] 30 132,049 512740276269…455730061311 39,751 1983 September 19[48] David Slowinski LLT / Cray X-MP 31 216,091 746093103064…103815528447 65,050 1985 September 1[49][50] David Slowinski LLT / Cray X-MP/24 32 756,839 174135906820…328544677887 227,832 1992 February 17 David Slowinski & Paul Gage LLT / Harwell Lab's Cray-2[51] 33 859,433 129498125604…243500142591 258,716 1994 January 4[52][53][54] David Slowinski & Paul Gage LLT / Cray C90 34 1,257,787 412245773621…976089366527 378,632 1996 September 3[55] David Slowinski & Paul Gage[56] LLT / Cray T94 35 1,398,269 814717564412…868451315711 420,921 1996 November 13 GIMPS / Joel Armengaud[57] LLT / Prime95 on 90 MHz Pentium PC 36 2,976,221 623340076248…743729201151 895,932 1997 August 24 GIMPS / Gordon Spence[58] LLT / Prime95 on 100 MHz Pentium PC 37 3,021,377 127411683030…973024694271 909,526 1998 January 27 GIMPS / Roland Clarkson[59] LLT / Prime95 on 200 MHz Pentium PC 38 6,972,593 437075744127…142924193791 2,098,960 1999 June 1 GIMPS / Nayan Hajratwala[60] LLT / Prime95 on 350 MHz Pentium II IBM Aptiva 39 13,466,917 924947738006…470256259071 4,053,946 2001 November 14 GIMPS / Michael Cameron[61] LLT / Prime95 on 800 MHz Athlon T-Bird 40 20,996,011 125976895450…762855682047 6,320,430 2003 November 17 GIMPS / Michael Shafer[62] LLT / Prime95 on 2 GHz Dell Dimension 41 24,036,583 299410429404…882733969407 7,235,733 2004 May 15 GIMPS / Josh Findley[63] LLT / Prime95 on 2.4 GHz Pentium 4 PC 42 25,964,951 122164630061…280577077247 7,816,230 2005 February 18 GIMPS / Martin Nowak[64] LLT / Prime95 on 2.4 GHz Pentium 4 PC 43 30,402,457 315416475618…411652943871 9,152,052 2005 December 15 GIMPS / Curtis Cooper & Steven Boone[65] LLT / Prime95 on 2 GHz Pentium 4 PC 44[*] 32,582,657 124575026015…154053967871 9,808,358 2006 September 4 GIMPS / Curtis Cooper & Steven Boone[66] LLT / Prime95 on 3 GHz Pentium 4 PC 45[*] 37,156,667 202254406890…022308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael Elvenich[67] LLT / Prime95 on 2.83 GHz Core 2 Duo PC 46[*] 42,643,801 169873516452…765562314751 12,837,064 2009 April 12[**] GIMPS / Odd M. Strindmo[68] LLT / Prime95 on 3 GHz Core 2 PC 47[*] 43,112,609 316470269330…166697152511 12,978,189 2008 August 23 GIMPS / Edson Smith[67] LLT / Prime95 on Dell Optiplex 745 48[*] 57,885,161 581887266232…071724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper[2] LLT / Prime95 on 3 GHz Intel Core2 Duo E8400[69] ^ * It is not verified whether any undiscovered Mersenne primes exist between the 43rd (M30,402,457) and the 48th (M57,885,161) on this chart; the ranking is therefore provisional. All Mersenne numbers below the 47th (M43,112,609) in the interval have been tested at least once but some have not been double-checked. Some Mersenne numbers above the 47th have not yet been tested.[70] Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, M43,112,609 was followed by two smaller Mersenne primes, first 2 weeks later and then 8 months later. ^ ** M42,643,801 was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad. To help visualize the size of the 48th known Mersenne prime, it would require 4,647 pages to display the number in base 10 with 75 digits per line and 50 lines per page. The largest known Mersenne prime (257,885,161 − 1) is also the largest known prime number.[2] M43,112,609 was the first discovered prime number with more than 10 million base-10 digits. In modern times, the largest known prime has almost always been a Mersenne prime.[71] Factorization of composite Mersenne numbers The factorization of a prime number is by definition the number itself. This section is about composite numbers. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of August 2012, 21,061 − 1 is the record-holder,[72] using the special number field sieve. See integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor. As of February 2014, the composite Mersenne number with largest proven prime factor is 263,703 − 1 = 42,808,417 × p, where p has 19,169 digits and was proven prime with ECPP.[73] As of February 2014, the largest factorization with probable prime factors allowed is 21,168,183 − 1 = 54,763,676,838,381,762,583 × q, where q is a 351,639-digit probable prime.[74] Mersenne numbers in nature and elsewhere In computer science, unsigned n-bit integers can be used to express numbers up to Mn. Signed (n + 1)-bit integers can express values between −(Mn + 1) and Mn, using the two's complement representation. In the mathematical problem Tower of Hanoi, solving a puzzle with an n-disc tower requires Mn steps, assuming no mistakes are made.[75] The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime (3 Juno, 7 Iris, 31 Euphrosyne and 127 Johanna having been discovered and named during the 19th century).[76] In mathematics, a Mersenne prime is a prime number of the form M_n=2^n-1. This is to say that it is a prime number which is one less than a power of two. They are named after the French monk Marin Mersenne who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31 and 127. If n is a composite number then so is 2n − 1. The definition is therefore unchanged when written M_p=2^p-1 where p is assumed prime. More generally, numbers of the form M_n=2^n-1\, without the primality requirement are called Mersenne numbers. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those pernicious numbers whose binary representation contains no zeros. The smallest composite pernicious Mersenne number is 211 − 1. As of May 2014, 48 Mersenne primes are known. The largest known prime number (257,885,161 − 1) is a Mersenne prime.[2][3] 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. |
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