Mersenne prime
Template:Infobox integer sequence In mathematics, a Mersenne prime is a prime number of the form . 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 2^{n} − 1. The definition is therefore unchanged when written where p is assumed prime.
More generally, numbers of the form 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 2^{11} − 1.
Template:As of, 48 Mersenne primes are known. The largest known prime number 2^{57,885,161} − 1 is a Mersenne prime.^{[1]}^{[2]}
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.
About Mersenne primes
Are there infinitely many Mersenne primes? |
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), for these primes p, 2p + 1 (which is also prime) will divides M_{p}, e.g., 23|M_{11}, 47|M_{23}, 167|M_{83}, 263|M_{131}, 359|M_{179}, 383|M_{191}, 479|M_{239}, and that 503|M_{251}. (sequence A002515 in OEIS)
The first four Mersenne primes are
- M_{2} = 3, M_{3} = 7, M_{5} = 31 and M_{7} = 127.
A basic theorem about Mersenne numbers states that if M_{p} is prime, then the exponent p must also be prime. This follows from the identity
This rules out primality for Mersenne numbers with composite exponent, such as M_{4} = 2^{4} − 1 = 15 = 3×5 = (2^{2} − 1)×(1 + 2^{2}).
Though the above examples might suggest that M_{p} is prime for all primes p, this is not the case, and the smallest counterexample is the Mersenne number
- M_{11} = 2^{11} − 1 = 2047 = 23 × 89.
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 M_{p} appear to grow increasingly sparse as p increases. In fact, of the 1,881,339 prime numbers p up to 30,402,457,^{[3]} M_{p} is prime for only 43 of them.
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
{{#invoke:main|main}} Mersenne primes M_{p} are also noteworthy due to their connection to perfect numbers. In the 4th century BC, Euclid proved that if 2^{p}−1 is prime, then 2^{p−1}(2^{p} − 1) is a perfect number. This number, also expressible as M_{p}(M_{p}+1)/2, is the M_{p}th triangular number and the 2^{p − 1}th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.^{[4]} This is known as the Euclid–Euler theorem. 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, as follows:
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257
His list was largely incorrect, as Mersenne mistakenly included M_{67} and M_{257} (which are composite), and omitted M_{61}, M_{89}, and M_{107} (which are prime). Mersenne gave little indication how he came up with his list.^{[5]} (sequence A109461 in OEIS)
Édouard Lucas proved in 1876 that M_{127} is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever calculated by hand. M_{61} 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 M_{67} 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.^{[6]} He later said that the result had taken him "three years of Sundays" to find.^{[7]} 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 M_{2} = 3, M_{3} = 7, M_{5} = 31 and M_{7} = 127 were known in antiquity. The fifth, M_{13} = 8191, was discovered anonymously before 1461; the next two (M_{17} and M_{19}) were found by Cataldi in 1588. After nearly two centuries, M_{31} was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was M_{127}, found by Lucas in 1876, then M_{61} by Pervushin in 1883. Two more (M_{89} and M_{107}) 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, M_{p} = 2^{p} − 1 is prime if and only if M_{p} divides S_{p}−2, where S_{0} = 4 and, for k > 0,
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,^{[8]} but the first successful identification of a Mersenne prime, M_{521}, 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, M_{607}, was found by the computer a little less than two hours later. Three more — M_{1279}, M_{2203}, M_{2281} — were found by the same program in the next several months. M_{4253} is the first Mersenne prime that is titanic, M_{44497} 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.^{[9]} 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.^{[10]}
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 2^{42,643,801} − 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, 2^{57,885,161} − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.^{[11]} 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^{[12]} (the smallest Mersenne number with said amount of digits is 2^{332192807} − 1).
Theorems about Mersenne numbers
- If a and p are natural numbers such that a^{p} − 1 is prime, then a = 2 or p = 1.
- Proof: a ≡ 1 (mod a − 1). Then a^{p} ≡ 1 (mod a − 1), so a^{p} − 1 ≡ 0 (mod a − 1). Thus a − 1 | a^{p} − 1. However, a^{p} − 1 is prime, so a − 1 = a^{p} − 1 or a − 1 = ±1. In the former case, a = a^{p}, 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, 0^{p} − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2.
- If 2^{p} − 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 2^{p} − 1 = 2^{ab} − 1 = (2^{a})^{b} − 1 = (2^{a} − 1)[(2^{a})^{b − 1} + (2^{a})^{b − 2} + … + 2^{a} + 1] so 2^{p} − 1 is composite contradicting our assumption that 2^{p} − 1 is prime.
- If p is an odd prime, then every prime q that divides 2^{p} − 1 must be 1 plus a multiple of 2p. This holds even when 2^{p} − 1 is prime.
- Examples: Example I: 2^{5} − 1 = 31 is prime, and 31 = 1 + 3×(2×5). Example II: 2^{11} − 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 2^{q − 1} − 1. Since q is a factor of 2^{p} − 1, for all positive integers c, q is also a factor of 2^{pc} − 1. Since p is prime and q is not a factor of 2^{1} − 1, p is also the smallest positive integer x such that q is a factor of 2^{x} − 1. As a result, for all positive integers x, q is a factor of 2^{x} − 1 if and only if p is a factor of x. Therefore, since q is a factor of 2^{q − 1} − 1, p is a factor of q − 1 so q ≡ 1 mod p. Furthermore, since q is a factor of 2^{p} − 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: for every odd prime p, all primes dividing 2^{p} − 1 are larger than p; thus there are always larger primes than any particular prime.
- If p is an odd prime, then every prime q that divides is congruent to ±1 (mod 8).
- Proof: , so is a square root of 2 modulo . By quadratic reciprocity, every prime modulo which the number 2 has a square root is congruent to ±1 (mod 8).
- A Mersenne prime cannot be a Wieferich prime.
- Proof: We show if p = 2^{m} − 1 is a Mersenne prime, then the congruence 2^{p} − 1 ≡ 1 does not satisfy. By Fermat's Little theorem, . Now write, . If the given congruence satisfies, then ,therefore 0 ≡ (2^{mλ} − 1)/(2^{m} − 1) = 1 + 2^{m} + 2^{2m} + ... + 2^{λ−1m} ≡ −λ mod(2^{m} − 1}. Hence ,and therefore λ ≥ 2^{m} − 1. This leads to p − 1 ≥ m(2^{m} − 1), which is impossible since m ≥ 2.
- A prime number divides at most one prime-exponent Mersenne number,^{[13]} so in other words the set of pernicious Mersenne numbers is pairwise coprime.
- 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 2^{p} − 1.^{[14]}
- Example: 11 and 23 are both prime, and 11 = 2×4 + 3, so 23 divides 2^{11} − 1.
- Proof: Let q be 2p + 1. By Fermat's Little theorem, 2^{2p} = 1 (mod q), so either 2^{p} = 1 (mod q) or 2^{p} = -1 (mod q). Supposing latter true, then 2^{p+1} = (2^{(p+1)/2})^{2} = -2 (mod q), so -2 would be a quadratic residue mod q. However, since p is congruent to 3 (mod 4), q is congruent to 7 (mod 8) and therefore 2 is a quadratic residue mod q. Also since q is congruent to 3 (mod 4), -1 is a quadratic nonresidue mod q, so -2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and 2p + 1 divides M_{p}.
- 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 equals , where denotes the floor function.
List of known Mersenne primes
The table below lists all known Mersenne primes (sequence A000043 (p) and A000668 (M_{p}) in OEIS):
# | p | M_{p} | M_{p} digits | Discovered | Discoverer | Method used |
---|---|---|---|---|---|---|
1 | 2 | 3 | 1 | c. 430 BC | Ancient Greek mathematicians^{[15]} | |
2 | 3 | 7 | 1 | c. 430 BC | Ancient Greek mathematicians^{[15]} | |
3 | 5 | 31 | 2 | c. 300 BC | Ancient Greek mathematicians^{[16]} | |
4 | 7 | 127 | 3 | c. 300 BC | Ancient Greek mathematicians^{[16]} | |
5 | 13 | 8191 | 4 | 1456 | Anonymous^{[17]}^{[18]} | Trial division |
6 | 17 | 131071 | 6 | 1588^{[19]} | Pietro Cataldi | Trial division^{[20]} |
7 | 19 | 524287 | 6 | 1588 | Pietro Cataldi | Trial division^{[21]} |
8 | 31 | 2147483647 | 10 | 1772 | Leonhard Euler^{[22]}^{[23]} | Enhanced trial division^{[24]} |
9 | 61 | 2305843009213693951 | 19 | 1883 November^{[25]} | I. M. Pervushin | Lucas sequences |
10 | 89 | 618970019642...137449562111 | 27 | 1911 June^{[26]} | Ralph Ernest Powers | Lucas sequences |
11 | 107 | 162259276829...578010288127 | 33 | 1914 June 1^{[27]}^{[28]}^{[29]} | Ralph Ernest Powers^{[30]} | Lucas sequences |
12 | 127 | 170141183460...715884105727 | 39 | 1876 January 10^{[31]} | Édouard Lucas | Lucas sequences |
13 | 521 | 686479766013...291115057151 | 157 | 1952 January 30^{[32]} | Raphael M. Robinson | LLT / SWAC |
14 | 607 | 531137992816...219031728127 | 183 | 1952 January 30^{[32]} | Raphael M. Robinson | LLT / SWAC |
15 | 1,279 | 104079321946...703168729087 | 386 | 1952 June 25^{[33]} | Raphael M. Robinson | LLT / SWAC |
16 | 2,203 | 147597991521...686697771007 | 664 | 1952 October 7^{[34]} | Raphael M. Robinson | LLT / SWAC |
17 | 2,281 | 446087557183...418132836351 | 687 | 1952 October 9^{[34]} | Raphael M. Robinson | LLT / SWAC |
18 | 3,217 | 259117086013...362909315071 | 969 | 1957 September 8^{[35]} | Hans Riesel | LLT / BESK |
19 | 4,253 | 190797007524...815350484991 | 1,281 | 1961 November 3^{[36]}^{[37]} | Alexander Hurwitz | LLT / IBM 7090 |
20 | 4,423 | 285542542228...902608580607 | 1,332 | 1961 November 3^{[36]}^{[37]} | Alexander Hurwitz | LLT / IBM 7090 |
21 | 9,689 | 478220278805...826225754111 | 2,917 | 1963 May 11^{[38]} | Donald B. Gillies | LLT / ILLIAC II |
22 | 9,941 | 346088282490...883789463551 | 2,993 | 1963 May 16^{[38]} | Donald B. Gillies | LLT / ILLIAC II |
23 | 11,213 | 281411201369...087696392191 | 3,376 | 1963 June 2^{[38]} | Donald B. Gillies | LLT / ILLIAC II |
24 | 19,937 | 431542479738...030968041471 | 6,002 | 1971 March 4^{[39]} | Bryant Tuckerman | LLT / IBM 360/91 |
25 | 21,701 | 448679166119...353511882751 | 6,533 | 1978 October 30^{[40]} | Landon Curt Noll & Laura Nickel | LLT / CDC Cyber 174 |
26 | 23,209 | 402874115778...523779264511 | 6,987 | 1979 February 9^{[41]} | Landon Curt Noll | LLT / CDC Cyber 174 |
27 | 44,497 | 854509824303...961011228671 | 13,395 | 1979 April 8^{[42]}^{[43]} | Harry Lewis Nelson & David Slowinski | LLT / Cray 1 |
28 | 86,243 | 536927995502...709433438207 | 25,962 | 1982 September 25 | David Slowinski | LLT / Cray 1 |
29 | 110,503 | 521928313341...083465515007 | 33,265 | 1988 January 29^{[44]}^{[45]} | Walter Colquitt & Luke Welsh | LLT / NEC SX-2^{[46]} |
30 | 132,049 | 512740276269...455730061311 | 39,751 | 1983 September 19^{[47]} | David Slowinski | LLT / Cray X-MP |
31 | 216,091 | 746093103064...103815528447 | 65,050 | 1985 September 1^{[48]}^{[49]} | 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^{[50]} |
33 | 859,433 | 129498125604...243500142591 | 258,716 | 1994 January 4^{[51]}^{[52]}^{[53]} | David Slowinski & Paul Gage | LLT / Cray C90 |
34 | 1,257,787 | 412245773621...976089366527 | 378,632 | 1996 September 3^{[54]} | David Slowinski & Paul Gage^{[55]} | LLT / Cray T94 |
35 | 1,398,269 | 814717564412...868451315711 | 420,921 | 1996 November 13 | GIMPS / Joel Armengaud^{[56]} | LLT / Prime95 on 90 MHz Pentium PC |
36 | 2,976,221 | 623340076248...743729201151 | 895,932 | 1997 August 24 | GIMPS / Gordon Spence^{[57]} | LLT / Prime95 on 100 MHz Pentium PC |
37 | 3,021,377 | 127411683030...973024694271 | 909,526 | 1998 January 27 | GIMPS / Roland Clarkson^{[58]} | LLT / Prime95 on 200 MHz Pentium PC |
38 | 6,972,593 | 437075744127...142924193791 | 2,098,960 | 1999 June 1 | GIMPS / Nayan Hajratwala^{[59]} | LLT / Prime95 on 350 MHz Pentium II IBM Aptiva |
39 | 13,466,917 | 924947738006...470256259071 | 4,053,946 | 2001 November 14 | GIMPS / Michael Cameron^{[60]} | LLT / Prime95 on 800 MHz Athlon T-Bird |
40 | 20,996,011 | 125976895450...762855682047 | 6,320,430 | 2003 November 17 | GIMPS / Michael Shafer^{[61]} | LLT / Prime95 on 2 GHz Dell Dimension |
41 | 24,036,583 | 299410429404...882733969407 | 7,235,733 | 2004 May 15 | GIMPS / Josh Findley^{[62]} | LLT / Prime95 on 2.4 GHz Pentium 4 PC |
42 | 25,964,951 | 122164630061...280577077247 | 7,816,230 | 2005 February 18 | GIMPS / Martin Nowak^{[63]} | 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^{[64]} | 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^{[65]} | LLT / Prime95 on 3 GHz Pentium 4 PC |
45^{[*]} | 37,156,667 | 202254406890...022308220927 | 11,185,272 | 2008 September 6 | GIMPS / Hans-Michael Elvenich^{[66]} | 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^{[67]} | LLT / Prime95 on 3 GHz Core 2 PC |
47^{[*]} | 43,112,609 | 316470269330...166697152511 | 12,978,189 | 2008 August 23 | GIMPS / Edson Smith^{[66]} | LLT / Prime95 on Dell Optiplex 745 |
48^{[*]} | 57,885,161 | 581887266232...071724285951 | 17,425,170 | 2013 January 25 | GIMPS / Curtis Cooper^{[1]} | LLT / Prime95 on 3 GHz Intel Core2 Duo E8400^{[68]} |
^{^ *} It is not verified whether any undiscovered Mersenne primes exist between the 44th (M_{32,582,657}) and the 48th (M_{57,885,161}) on this chart; the ranking is therefore provisional. All Mersenne numbers below the 47th (M_{43,112,609}) 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.^{[69]} Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, M_{43,112,609} was followed by two smaller Mersenne primes, first 2 weeks later and then 8 months later.
^{^ **} M_{42,643,801} 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 (2^{57,885,161} − 1) is also the largest known prime number.^{[1]} M_{43,112,609} was the first discovered prime number with more than 10 million decimal digits.
In modern times, the largest known prime has almost always been a Mersenne prime.^{[70]}
Factorization of composite Mersenne numbers
The factors of a prime number are by definition one, and 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. Template:As of, 2^{1,193} − 1 is the record-holder,^{[71]} using a variant on the special number field sieve allowing the factorisation of several numbers at once. 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. Template:As of, the largest factorization with probable prime factors allowed is 2^{3,464,473} − 1 = 604,874,508,299,177 × q, where q is a 1,042,896-digit probable prime.^{[72]}
(sequence A244453 in OEIS) (or Template:Oeis with both prime and composite Mersenne numbers) (for the primes p, see Template:Oeis)
# | p | Factorization of M_{p} |
---|---|---|
1 | 11 | 23 × 89 |
2 | 23 | 47 × 178481 |
3 | 29 | 233 × 1103 × 2089 |
4 | 37 | 223 × 616318177 |
5 | 41 | 13367 × 164511353 |
6 | 43 | 431 × 9719 × 2099863 |
7 | 47 | 2351 × 4513 × 13264529 |
8 | 53 | 6361 × 69431 × 20394401 |
9 | 59 | 179951 × 3203431780337 (13 digits) |
10 | 67 | 193707721 × 761838257287 |
11 | 71 | 228479 × 48544121 × 212885833 |
12 | 73 | 439 × 2298041 × 9361973132609 (13 digits) |
13 | 79 | 2687 × 202029703 × 1113491139767 (13 digits) |
14 | 83 | 167 × 57912614113275649087721 (23 digits) |
15 | 97 | 11447 × 13842607235828485645766393 (26 digits) |
16 | 101 | 7432339208719 (13 digits) × 341117531003194129 (18 digits) |
17 | 103 | 2550183799 × 3976656429941438590393 (22 digits) |
18 | 109 | 745988807 × 870035986098720987332873 (24 digits) |
19 | 113 | 3391 × 23279 × 65993 × 1868569 × 1066818132868207 (16 digits) |
20 | 131 | 263 × 10350794431055162386718619237468234569 (38 digits) |
... | ... | ... |
23 | 149 | 86656268566282183151 (20 digits) × 8235109336690846723986161 (25 digits) |
... | ... | ... |
43 | 257 | 535006138814359 (15 digits) × 1155685395246619182673033 (25 digits) × 374550598501810936581776630096313181393 (39 digits) |
... | ... | ... |
86 | 523 | 160188778313...217468039063 (69 digits) × 171417691861...101859504089 (90 digits) |
... | ... | ... |
119 | 751 | 227640245125...672549806487 (66 digits) × 649350031993...523089149897 (67 digits) × 801306808403...587821853073 (94 digits) |
... | ... | ... |
164 | 1061 | 468172263510...207943564433 (143 digits) × 527739642811...707148303247 (177 digits) |
... | ... | ... |
172 | 1109 | 30963501968569 (14 digits) × 85608965982066833903 (20 digits) × 246160192118...804809798519 (146 digits) × 106580571390...112526589967 (156 digits) |
... | ... | ... |
182 | 1193 | 121687 × 852273262013...757462472729 (104 digits) × 129706511503...433815839617 (251 digits) |
... | ... | ... |
Mersenne primitive part
The primitive part of Mersenne number M_{n} is , the n-th cyclotomic polynomial at 2, they are
- 1, 3, 7, 5, 31, 3, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 57, 524287, 205, 2359, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, ... (sequence A019320 in OEIS)
Besides, if we notice those prime factors, and delete "old prime factors", for example, 3 divides the 2nd, 6th, 18th, 54th, 162nd, ... terms of this sequence, we only allow the 2nd term divided by 3, if we do, they are
- 1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, ... (sequence A064078 in OEIS)
The numbers n which is prime are
- 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, ... (sequence A072226 in OEIS)
The numbers n which 2^{n} - 1 has an only primitive prime factor are
- 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, ... (sequence A161508 in OEIS) (Differ from last sequence, this sequence does not have the term 6, but has the terms 18, 20, 21, 54, 147, 342, 602, and 889, and it is conjectured that no others)
Mersenne numbers in nature and elsewhere
In computer science, unsigned n-bit integers can be used to express numbers up to M_{n}. Signed (n + 1)-bit integers can express values between −(M_{n} + 1) and M_{n}, using the two's complement representation.
In the mathematical problem Tower of Hanoi, solving a puzzle with an n-disc tower requires M_{n} steps, assuming no mistakes are made.^{[73]}
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).^{[74]}
Mersenne-Fermat primes
A Mersenne-Fermat number is defined as , with p prime, r natural number, and can be written as MF(p, r), when r = 1, it is a Mersenne number, and when p = 2, it is a Fermat number, the only known Mersenne-Fermat prime with r > 1 are
- MF(2, 2), MF(3, 2), MF(7, 2), MF(59, 2), MF(2, 3), MF(3, 3), MF(2, 4), and MF(2, 5).^{[75]}
In fact, MF(p, r) = , where is the cyclotomic polynomial.
Generalizations
It is natural to try to generalize primes of the form to primes of the form for (and ). However (see also theorems above), is always divisible by , so unless is a unit, the former is not a prime. There are two ways to deal with that:
Gaussian Mersenne primes
In the ring of integers, if is a unit, then is either 2 or 0. But are the usual Mersenne primes, and the formula does not lead to anything interesting. However, if we regard instead the ring of Gaussian integers, we get the case and , and can ask (WLOG) for what the number
is a Gaussian prime which will then be called a Gaussian Mersenne prime.^{[76]}
is a Gaussian prime for exponents in 2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, ... (sequence A057429 in OEIS). This sequence is in many ways similar to the list of exponents of ordinary Mersenne primes.
The norms (i.e. squares of absolute values) of these Gaussian primes are rational primes 5, 13, 41, 113, 2113, 525313, ... (sequence A182300 in OEIS).
Repunit primes
{{#invoke:main|main}} The other way to deal with the fact that is always divisible by , the integer b can be either positive or negative, but b is not a perfect power, it is to simply take out this factor and ask which n makes
to be a prime.^{[77]} If for example we take , we get values of 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence A004023 in OEIS), corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence A004022 in OEIS). These primes are called repunit primes. Another example is when we take , we get values of 2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence A057178 in OEIS). It is a conjecture that there are infinitely many values of every integer which is not a perfect power.
The least n such that is prime are (start with b = 2)
- 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ...(sequence A084740 in OEIS)
For negative base b, they are (start with b = -2)
- 3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence A084742 in OEIS), (but this OEIS sequence does not allow n = 2)
The least base b such that is prime are
- 2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence A066180 in OEIS)
For negative bases b, they are
- 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in OEIS)
Another generized Mersenne number is
with a, b any coprime integers, a > 0, -a < b < a, and that a and b are not both rth power with any natural number r > 1 (That is, if b > 0, the ath term and the bth term in Template:Oeis are coprime, if b < 0, the greatest common factor of the ath term and the |b|th term in Template:Oeis must be 1 or a power of 2). We can also ask which n makes it to be prime. If for example we take (a, b) = (11, 5), we get n values of 5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... (sequence A128347 in OEIS), and when we take (a, b) = (14, -3), we get n values of 2, 3, 7, 71, 251, 1429, 2131, 2689, 36683, 60763, ... (sequence A128072 in OEIS). It is a conjecture that there are infinitely many n values for all such values of (a, b). When b = 1, this is a generalized repunit number in base a, and when b = -1, this is a generalized repunit number in base -a.
When a = b + 1, it is
Is a difference of two perfect nth powers, and if a^{n} - b^{n} is prime, than a must be b + 1 or b - 1, because it is divisible by a - b.
The least n such that is prime are
- 2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence A058013 in OEIS)
The least b such that is prime are
- 1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, ... (sequence A222119 in OEIS)
See also
- Repunit
- Fermat prime
- Power of 2
- Erdős–Borwein constant
- Mersenne conjectures
- Mersenne twister
- Double Mersenne number
- Prime95 / MPrime
- Great Internet Mersenne Prime Search (GIMPS)
- Largest known prime number
- Titanic prime
- Gigantic prime
- Megaprime
- Wieferich prime
- Wagstaff prime
- Cullen prime
- Woodall prime
- Proth prime
- Solinas prime
- Gillies' conjecture
References
- ↑ ^{1.0} ^{1.1} ^{1.2} Template:Cite web
- ↑ Template:Cite news
- ↑ Template:Cite web
- ↑ Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists
- ↑ The Prime Pages, Mersenne's conjecture.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }} p. 228.
- ↑ Template:Cite news
- ↑ Brian Napper, The Mathematics Department and the Mark 1.
- ↑ The Prime Pages, The Prime Glossary: megaprime.
- ↑ Template:Cite news
- ↑ Template:Cite web
- ↑ https://www.eff.org/awards/coop/
- ↑ Will Edgington's Mersenne Page
- ↑ Proof of a result of Euler and Lagrange on Mersenne Divisors
- ↑ ^{15.0} ^{15.1} There is no mentioning among the ancient Egyptians of prime numbers, and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. See Prime Numbers Divide [Retrieved 2012-11-11]. "The Egyptians used ($) in the table above for the first primes r=3, 5, 7, or 11 (also for r=23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/n Table [Retrieved 2012-11-11]. In the school of Pythagoras (b. about 570 – d. about 495 BC) and the Pythagoreans, we find the first sure observations of prime numbers. Hence the first two Mersenne primes, 3 and 7, were known to and may even be said to have been discovered by them. There is no reference, though, to their special form 2^{2} − 1 and 2^{3} − 1 as such. The sources to the knowledge of prime numbers among the Pythagoreans are late. The Neoplatonic philosopher Iamblichus, AD c. 245–c. 325, states that the Greek Platonic philosopher Speusippus, c. 408 – 339/8 BC, wrote a book named On Pythagorean Numbers. According to Iamblichus this book was based on the works of the Pythagorean Philolaus, c. 470–c. 385 BC, who lived a century after Pythagoras, 570 – c. 495 BC. In his Theology of Arithmetic in the chapter On the Decad, Iamblichus writes: "Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book On Pythagorean Numbers. In the first half of the book, he elegantly expounds linear numbers [i.e. prime numbers], polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 112f. [Retrieved 2012-11-11]. Iamblichus also gives us a direct quote from Speusippus' book where Speusippus among other things writes: "Secondly, it is necessary for a perfect number [the concept "perfect number" is not used here in a modern sense] to contain an equal amount of prime and incomposite numbers, and secondary and composite numbers." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 113. [Retrieved 2012-11-11]. For the Greek original text, see Speusippus of Athens: A Critical Study with a Collection of the Related Texts and Commentary by Leonardo Tarán, 1981, p. 140 line 21–22 [Retrieved 2012-11-11] In his comments to Nicomachus of Gerasas's Introduction to Arithmetic, Iamblichus also mentions that Thymaridas, ca. 400 BC – ca. 350 BC, uses the term rectilinear for prime numbers, and that Theon of Smyrna, fl. AD 100, uses euthymetric and linear as alternative terms. Nicomachus of Gerasa, Introduction to Aritmetic, 1926, p. 127 [Retrieved 2012-11-11] It is unclear though when this said Thymaridas lived. "In a highly suspect passage in Iamblichus, Thymaridas is listed as a pupil of Pythagoras himself." Pythagoreanism [Retrieved 2012-11-11] Before Philolaus, c. 470–c. 385 BC, we have no proof of any knowledge of prime numbers.
- ↑ ^{16.0} ^{16.1} Euclid's Elements, Book IX, Proposition 36
- ↑ The Prime Pages, Mersenne Primes: History, Theorems and Lists.
- ↑ We find the oldest (undisputed) note of the result in Codex nr. 14908, which origins from Bibliotheca monasterii ord. S. Benedicti ad S. Emmeramum Ratisbonensis now in the archive of the Bayerische Staatsbibliothek, see "Halm, Karl / Laubmann, Georg von / Meyer, Wilhelm: Catalogus codicum latinorum Bibliothecae Regiae Monacensis, Bd.: 2,2, Monachii, 1876, p. 250". [retrieved on 2012-09-17] The Codex nr. 14908 consists of 10 different medieval works on mathematics and related subjects. The authors of most of these writings are known. Some authors consider the monk Fridericus Gerhart (Amman), c. 1400-d. 1465 (Frater Fridericus Gerhart monachus ordinis sancti Benedicti astrologus professus in monasterio sancti Emmerani diocesis Ratisponensis et in ciuitate eiusdem) to be the author of the part where the prime number 8191 is mentioned. Geschichte Der Mathematik [retrieved on 2012-09-17] The second manuscript of Codex nr. 14908 has the name "Regulae et exempla arithmetica, algebraica, geometrica" and the 5th perfect number and all is factors, including 8191, are mentioned on folio no. 34 a tergo (backside of p. 34). Parts of the manuscript have been published in Archiv der Mathematik und Physik, 13 (1895), pp. 388–406 [retrieved on 2012-09-23]
- ↑ "A i lettori. Nel trattato de' numeri perfetti, che giàfino dell anno 1588 composi, oltrache se era passato auáti à trouarne molti auertite molte cose, se era anco amplamente dilatatala Tauola de' numeri composti , di ciascuno de' quali si vedeano per ordine li componenti, onde preposto unnum." p. 1 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
- ↑ pp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
- ↑ pp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
- ↑ http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres 1772, pp. 35–36 EULER, Leonhard: Extrait d'une lettre à M. Bernoulli, concernant le Mémoire imprimé parmi ceux de 1771. p. 318 [intitulé: Recherches sur les diviseurs de quelques nombres très grands compris dans la somme de la progression géométrique 1 + 101 + 102 + 103 + ... + 10T = S]. Retrieved 2011-10-02.
- ↑ http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.
- ↑ Template:Cite web
- ↑ “En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre 2^{61} – 1 = 2305843009213693951 est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. http://www.biodiversitylibrary.org/item/107789#page/277/mode/1up [retrieved 2012-09-17] See also Mélanges mathématiques et astronomiques tirés du Bulletin de l’Académie impériale des sciences de St.-Pétersbourg v. 6 (1881–1888), pp. 553–554. See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48
- ↑ http://www.jstor.org/stable/2972574 The American Mathematical Monthly, Vol. 18, No. 11 (Nov., 1911), pp. 195-197. The article is signed "DENVER, COLORADO, June, 1911". Retrieved 2011-10-02.
- ↑ "M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1^{er} Juin un cablógramme à M. Bromwich [secretary of London Mathematical Society] pour M_{107}. Sur ma demande, ces deux auteurs m’ont adressé leurs remarquables résultats, et je m’empresse de les publier dans nos colonnes, avec nos felicitations." p. 103, André Gérardin, Nombres de Mersenne pp. 85, 103–108 in Sphinx-Œdipe. [Journal mensuel de la curiosité, de concours & de mathématiques.] v. 9, No. 1, 1914.
- ↑ "Power's cable announcing this same result was sent to the London Math. So. on 1 June 1914." Mersenne's Numbers, Scripta Mathematica, v. 3, 1935, pp. 112–119 http://primes.utm.edu/mersenne/LukeMirror/lit/lit_008s.htm [retrieved 2012-10-13]
- ↑ http://plms.oxfordjournals.org/content/s2-13/1/1.1.full.pdf Proceedings / London Mathematical Society (1914) s2–13 (1): 1. Result presented at a meeting with London Mathematical Society on June 11, 1914. Retrieved 2011-10-02.
- ↑ The Prime Pages, M_{107}: Fauquembergue or Powers?.
- ↑ http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3039&I=166&M=chemindefer Presented at a meeting with Académie des sciences (France) on January 10, 1876. Retrieved 2011-10-02.
- ↑ ^{32.0} ^{32.1} "Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2^{521} − 1 and 2^{607} − 1 on January 30, 1952." D. H. Lehmer, Recent Discoveries of Large Primes, Mathematics of Computation, vol. 6, No. 37 (1952), p. 61, http://www.ams.org/journals/mcom/1952-06-037/S0025-5718-52-99404-0/S0025-5718-52-99404-0.pdf [Retrieved 2012-09-18]
- ↑ "The program described in Note 131 (c) has produced the 15th Mersenne prime 2^{1279} − 1 on June 25. The SWAC tests this number in 13 minutes and 25 seconds." D. H. Lehmer, A New Mersenne Prime, Mathematics of Computation, vol. 6, No. 39 (1952), p. 205, http://www.ams.org/journals/mcom/1952-06-039/S0025-5718-52-99387-3/S0025-5718-52-99387-3.pdf [Retrieved 2012-09-18]
- ↑ ^{34.0} ^{34.1} "Two more Mersenne primes, 2^{2203} − 1 and 2^{2281} − 1, were discovered by the SWAC on October 7 and 9, 1952." D. H. Lehmer, Two New Mersenne Primes, Mathematics of Computation, vol. 7, No. 41 (1952), p. 72, http://www.ams.org/journals/mcom/1953-07-041/S0025-5718-53-99371-5/S0025-5718-53-99371-5.pdf [Retrieved 2012-09-18]
- ↑ "On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number M_{3217} = 2^{3217} − 1 is a prime." Hans Riesel, A New Mersenne Prime, Mathematics of Computation, vol. 12 (1958), p. 60, http://www.ams.org/journals/mcom/1958-12-061/S0025-5718-1958-0099752-6/S0025-5718-1958-0099752-6.pdf [Retrieved 2012-09-18]
- ↑ ^{36.0} ^{36.1} A. Hurwitz and J. L. Selfridge, Fermat numbers and perfect numbers, Notices of the American Mathematical Society, v. 8, 1961, p. 601, abstract 587-104.
- ↑ ^{37.0} ^{37.1} "If p is prime, M_{p} = 2^{}p − 1 is called a Mersenne number. The primes M_{4253} and M_{4423} were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18]
- ↑ ^{38.0} ^{38.1} ^{38.2} "The primes M_{9689}, M_{9941}, and M_{11213} which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois." Donald B. Gillies, Three New Mersenne Primes and a Statistical Theory, Mathematics of Computation, vol. 18, No. 85 (1964), pp. 93–97, http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0159774-6/S0025-5718-1964-0159774-6.pdf [Retrieved 2012-09-18]
- ↑ "On the evening of March 4, 1971, a zero Lucas-Lehmer residue for p = p_{24} = 19937 was found. Hence, M_{19937} is the 24th Mersenne prime." Bryant Tuckerman, The 24th Mersenne Prime, Proceedings of the National Academy of Sciences of the United States of America, vol. 68:10 (1971), pp. 2319–2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]
- ↑ "On October 30, 1978 at 9:40 pm, we found M_{21701} to be prime. The CPU time required for this test was 7:40:20. Tuckerman and Lehmer later provided confirmation of this result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
- ↑ "Of the 125 remaining M_{p} only M_{23209} was found to be prime. The test was completed on February 9, 1979 at 4:06 after 8:39:37 of CPU time. Lehmer and McGrogan later confirmed the result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
- ↑ David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013
- ↑ "The 27th Mersenne prime. It has 13395 digits and equals 2^{44497} – 1. [...] Its primeness was determined on April 8, 1979 using the Lucas-Lehmer test. The test was programmed on a CRAY-1 computer by David Slowinski & Harry Nelson." (p. 15) "The result was that after applying the Lucas-Lehmer test to about a thousand numbers, the code determined, on Sunday, April 8th, that 2^{44497} − 1 is, in fact, the 27th Mersenne prime." (p. 17), David Slowinski, "Searching for the 27th Mersenne Prime", Cray Channels, vol. 4, no. 1, (1982), pp. 15–17.
- ↑ "An FFT containing 8192 complex elements, which was the minimum size required to test M_{110503}, ran approximately 11 minutes on the SX-2. The discovery of M_{110503} (January 29, 1988) has been confirmed." W. N. Colquitt and L. Welsh, Jr., A New Mersenne Prime, Mathematics of Computation, vol. 56, No. 194 (April 1991), pp. 867–870, http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068823-9/S0025-5718-1991-1068823-9.pdf [Retrieved 2012-09-18]
- ↑ "This week, two computer experts found the 31st Mersenne prime. But to their surprise, the newly discovered prime number falls between two previously known Mersenne primes. It occurs when p = 110,503, making it the third-largest Mersenne prime known." I. Peterson, Priming for a lucky strike Science News; 2/6/88, Vol. 133 Issue 6, pp. 85–85. http://ehis.ebscohost.com/ehost/detail?vid=3&hid=23&sid=9a9d7493-ffed-410b-9b59-b86c63a93bc4%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8824187 [Retrieved 2012-09-18]
- ↑ Template:Cite web
- ↑ "Slowinski, a software engineer for Cray Research Inc. in Chippewa Falls, discovered the number at 11:36 a.m. Monday. [i.e. 1983 September 19]" Jim Higgins, "Elusive numeral's number is up" and "Scientist finds big number" in The Milwaukee Sentinel – Sep 24, 1983, p. 1, p. 11 [retrieved 2012-10-23]
- ↑ "The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form 2^{p} – 1, where the exponent p is also a prime number. For instance, 127 is a Mersenne number for which the exponent is 7. The record prime number's exponent is 216,091." I. Peterson, Prime time for supercomputers Science News; 9/28/85, Vol. 128 Issue 13, p. 199. http://ehis.ebscohost.com/ehost/detail?vid=4&hid=22&sid=c11090a2-4670-469f-8f75-947b593a56a0%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8840537 [Retrieved 2012-09-18]
- ↑ "Slowinski's program also found the 28th in 1982, the 29th in 1983, and the 30th [known at that time] this past Labor Day weekend. [i.e. August 31-September 1, 1985]" Rad Sallee, "`Supercomputer'/Chevron calculating device finds a bigger prime number" Houston Chronicle, Friday 09/20/1985, Section 1, Page 26, 4 Star Edition [retrieved 2012-10-23]
- ↑ The Prime Pages, The finding of the 32nd Mersenne.
- ↑ Chris Caldwell, The Largest Known Primes.
- ↑ Crays press release
- ↑ Slowinskis email
- ↑ Silicon Graphics' press release http://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
- ↑ The Prime Pages, A Prime of Record Size! 2^{1257787} – 1.
- ↑ GIMPS Discovers 35th Mersenne Prime.
- ↑ GIMPS Discovers 36th Known Mersenne Prime.
- ↑ GIMPS Discovers 37th Known Mersenne Prime.
- ↑ GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
- ↑ GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
- ↑ GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 2^{24,036,583} – 1.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 2^{25,964,951} – 1.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 2^{30,402,457} – 1.
- ↑ GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 2^{32,582,657} – 1.
- ↑ ^{66.0} ^{66.1} Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
- ↑ "On April 12th [2009], the 47th known Mersenne prime, 2^{42,643,801} – 1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.", The List of Largest Known Primes Home Page, http://primes.utm.edu/primes/page.php?id=88847 [retrieved 2012-09-18]
- ↑ Template:Cite web
- ↑ GIMPS Milestones Report. Retrieved 2014-02-23
- ↑ The largest known 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, University of Tennessee at Martin.
- ↑ Thorsten Kleinjung, Joppe Bos, Arjen Lenstra "Mersenne Factorization Factory" http://eprint.iacr.org/2014/653.pdf
- ↑ Template:Cite web
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Template:Cite web
- ↑ A research of Mersenne and Fermat primes
- ↑ Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)
- ↑ Table of repunits
External links
Template:Sister Template:Wikinewspar2
- {{#invoke:citation/CS1|citation
|CitationClass=citation }}
- GIMPS home page
- GIMPS status — status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 42–47
- GIMPS, known factors of Mersenne numbers
- M_{q} = (8x)^{2} − (3qy)^{2} Mersenne proof (PDF)
- M_{q} = x^{2} + d·y^{2} math thesis (PS)
- Template:Cite web
- Mersenne prime bibliography with hyperlinks to original publications
- report about Mersenne primes — detection in detail Template:De icon
- GIMPS wiki
- Will Edgington's Mersenne Page — contains factors for small Mersenne numbers
- a file containing the smallest known factors of many tested Mersenne numbers (requires program to open)
- Decimal digits and English names of Mersenne primes
- Prime curios: 2305843009213693951
- Factorization of Mersenne numbers M_{n}, with n odd, n up to 1199
- Factorization of Mersenne numbers M_{2n}, 2n up to 2398 (n up to 1199) or 2n is on the form 8k+4 up to 4796 (n is on the form 4k+2 up to 2398)
- Factorization of completely factored Mersenne numbers
- The Cunningham project, factorization of b^{n} ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12
MathWorld links
- Weisstein, Eric W., "Mersenne number", MathWorld.
- Weisstein, Eric W., "Mersenne prime", MathWorld.
- 47th Mersenne Prime Found
Template:Prime number classes Template:Classes of natural numbers Template:Mersenne