Mersenne prime

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Mersenne prime
Named after Marin Mersenne
Publication year 1536[1]
Author of publication Regius, H.
Number of known terms 48
Conjectured number of terms Infinite
Subsequence of Mersenne numbers
First terms 3, 7, 31, 127
Largest known term 257885161 − 1 (January 2013)
OEIS index A000668

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number that can be written in the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes (sequence A000668 in OEIS) are 3, 7, 31, and 127.

If n is a composite number then so is 2n − 1. The definition is therefore unchanged when written Mp = 2p − 1 where p is assumed prime.

More generally, numbers of the form Mn = 2n − 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 = 2047 = 23 × 89.

As of January 2016, 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.

About Mersenne primes

Question dropshade.png Open problem in mathematics:
Are there infinitely many Mersenne primes?
(more open problems in mathematics)

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 divide Mp, e.g., 23|M11, 47|M23, 167|M83, 263|M131, 359|M179, 383|M191, 479|M239, and 503|M251. (sequence A002515 in OEIS)

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 the above examples might suggest that Mp is prime for all primes p, this is not the case, and the smallest counterexample is the Mersenne number

M11 = 211 − 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 Mp appear to grow increasingly sparse as p increases. In fact, of the 2,007,537 prime numbers p up to 32,582,657,[4] Mp is prime for only 44 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

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Main article: Euclid–Euler theorem

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 Mp-th triangular number and the 2p − 1-th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.[5] This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.

History

2 3 5 7 11 13 17 19
23 29 31 37 41 43 47 53
59 61 67 71 73 79 83 89
97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223
227 229 233 239 241 251 257 263
269 271 277 281 283 293 307 311
The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold.

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 completely accurate until 31, but then becomes 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] (sequence A109461 in OEIS)

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 found 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.[7] 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.[8] He later said that the result had taken him "three years of Sundays" to find.[9] 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,

Sk = Sk − 12 − 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,[10] 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.[11] 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.[12]

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.[13] This was the third Mersenne prime discovered by Dr. Cooper and his team in the past seven years.

Theorems about Mersenne numbers

  1. 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.
  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.
  3. If p is an odd prime, then every 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: for every odd prime p, all primes dividing 2p − 1 are larger than p; thus there are always larger primes than any particular prime.
  4. If p is an odd prime, then every prime q that divides 2p − 1 is congruent to ±1 (mod 8).
    • Proof: 2p + 1 ≡ 2 (mod q), so 2(p + 1) / 2 is a square root of 2 mod q. By quadratic reciprocity, every prime modulo which the number 2 has a square root is congruent to ±1 (mod 8).
  5. 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 (mod p2) does not hold. By Fermat's Little theorem, m | p − 1. Now write, p − 1 = mλ. If the given congruence is satisfied, then p2 | 2mλ − 1,therefore 0 ≡ (2mλ − 1) / (2m − 1) = 1 + 2m + 22m + ... + 2λ − 1m ≡ −λ mod(2m − 1). Hence 2m − 1 | λ,and therefore λ ≥ 2m − 1. This leads to p − 1 ≥ m(2m − 1), which is impossible since m ≥ 2.
  6. If m and n are natural numbers then m and n are coprime if and only if 2m-1 and 2n-1 are coprime. Consequently a prime number divides at most one prime-exponent Mersenne number,[14] so in other words the set of pernicious Mersenne numbers is pairwise coprime.
  7. 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.
    • Proof: Let q be 2p + 1. By Fermat's Little theorem, 22p ≡ 1 (mod q), so either 2p ≡ 1 (mod q) or 2p ≡ −1 (mod q). Supposing latter true, then 2p + 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 Mp.
  8. All composite divisors of prime-exponent Mersenne numbers pass the Fermat primality test for the base 2.
  9. The number of digits in the decimal representation of Mn equals ⌊n × log102⌋ + 1, where ⌊x⌋ denotes the floor function.

List of known Mersenne primes

The table below lists all known Mersenne primes (sequence A000043 (p) and A000668 (Mp) 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 L. 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[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 44th (M32,582,657) and the 48th (M57,885,161) on this chart; the ranking is therefore provisional. All Mersenne numbers below the 48th (M57,885,161) have been tested at least once but some have not been double-checked.[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 decimal digits.

In modern times, the largest known prime has almost always been a Mersenne prime.[71]

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. As of October 2014, 21,193 − 1 is the record-holder,[72] 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. As of January 2015, the largest factorization with probable prime factors allowed is 23,464,473 − 1 = 604,874,508,299,177 × q, where q is a 1,042,896-digit probable prime.[73]

(sequence A244453 in OEIS) (or OEIS A089162 with both prime and composite Mersenne numbers) (for the primes p, see OEIS A054723)

# p Factorization of Mp
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 (12 digits)
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 Mn is Φn (2), 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 Φn(2) 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 2n - 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 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.[74]

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).[75]

Mersenne-Fermat primes

A Mersenne-Fermat number is defined as <templatestyles src="Sfrac/styles.css" />2pr − 1/2pr − 1 − 1, 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).[76]

In fact, MF(p, r) = Φpr (2), where Φ is the cyclotomic polynomial.

Generalizations

It is natural to try to generalize primes of the form 2^n-1 to primes of the form b^n-1 for b \ne 2 (and n>1). However (see also theorems above), b^n-1 is always divisible by b-1, so unless b-1 is a unit, the former is not a prime. There are two ways to deal with that:

Complex numbers

In the ring of integers (on real numbers), if b-1 is a unit, then b is either 2 or 0. But 2^n-1 are the usual Mersenne primes, and the formula 0^n-1 does not lead to anything interesting. Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.

Gaussian Mersenne primes

If we regard the ring of Gaussian integers, we get the case b=1+i and b=1-i, and can ask (WLOG) for what n the number

(1+i)^n - 1

is a Gaussian prime which will then be called a Gaussian Mersenne prime.[77]

(1+i)^n - 1 is a Gaussian prime for exponents n:

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, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (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, 536903681, 140737471578113, ... (sequence A182300 in OEIS).

Eisenstein Mersenne primes

We can also regard the ring of Eisenstein integers, we get the case b=1+\omega and b=1-\omega, and can ask for what n the number

(1-\omega)^n - 1

is an Eisenstein prime which will then be called a Eisenstein Mersenne prime.

(1-\omega)^n - 1 is an Eisenstein prime for exponents n:

2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence A066408 in OEIS)

The norms (i.e. squares of absolute values) of these Eisenstein primes are rational primes:

7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence A066413 in OEIS)

Divide an integer

Repunit primes

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Main article: Repunit

The other way to deal with the fact that b^n-1 is always divisible by b-1, it is to simply take out this factor and ask which n makes

\frac{b^n-1}{b-1}

to be prime. (The integer b can be either positive or negative). If for example we take b=10, we get n 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 b=-12, we get n values of 2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence A057178 in OEIS), corresponding to primes -11, 19141, 57154490053, ... . It is a conjecture that for every integer b which is not a perfect power, there are infinitely many n values such that \frac{b^n-1}{b-1} is prime. (since when b is a perfect power, it can be shown that there is at most one n value such that \frac{b^n-1}{b-1} is prime)

Least n such that \frac{b^n-1}{b-1} 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) (notice this OEIS sequence does not allow n=2)

Least base b such that \frac{b^{prime(n)}-1}{b-1} 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)

Other generalized Mersenne primes

Another generized Mersenne number is

\frac{a^n-b^n}{a-b}

with a, b any coprime integers, a>1, -a<b<a. (Since a^n-b^n is always divisible by a-b, the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the Lucas number U_n(a+b, ab), since a and b are the roots of the quadratic equation x^2-(a+b)x+ab=0, and this number equals 1 when n=1) We can ask which n make this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a+b=1 and a^2+b^2 is prime. (Since \frac{a^4-b^4}{a-b}=(a+b)(a^2+b^2). Thus, in this case the pair (a,b) must be (x+1, -x) and x^2+(x+1)^2 must be prime. That is, x must in OEIS A027861.) It is a conjecture that for any pair (a,b) such that for every natural number r>1, a and b are not both perfect rth powers, and if b<0, the absolute values of a and b are not "one is a fourth power, the other is 4 times a fourth power". there are infinitely many values of n such that \frac{a^n-b^n}{a-b} is prime. (When a and b are both perfect rth powers for an r>1 or when b<0 and one of their absolute values is a fourth power, the other is 4 times a fourth power, it can be shown that there are at most two n values with this property.) However, this has not been proved for any single value of (a,b).

a b numbers n such that \frac{a^n-b^n}{a-b} is prime (some large terms are only probable primes, these n are checked up to 100000 for |b| \le 5 or |b| = a-1, 20000 for 5 < |b| < a-1) OEIS sequence
2 1 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, ..., 37156667, ..., 42643801, ..., 43112609, ..., 57885161, ... A000043
2 −1 3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... A000978
3 2 2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ... A057468
3 1 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... A028491
3 −1 2, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ... A007658
3 −2 3, 4, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ... A057469
4 3 2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ... A059801
4 1 2 (no others)
4 −1 2, 3 (no others)
4 −3 3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ... A128066
5 4 3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ... A059802
5 3 13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ... A121877
5 2 2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ... A082182
5 1 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, ... A004061
5 −1 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... A057171
5 −2 2, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ... A082387
5 −3 2, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ... A122853
5 −4 4, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ... A128335
6 5 2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ... A062572
6 1 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... A004062
6 −1 2, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... A057172
6 −5 3, 4, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ... A128336
7 6 2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ... A062573
7 5 3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ... A128344
7 4 2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ... A213073
7 3 3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ... A128024
7 2 3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ... A215487
7 1 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... A004063
7 −1 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ... A057173
7 −2 2, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ... A125955
7 −3 3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ... A128067
7 −4 2, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ... A218373
7 −5 2, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ... A128337
7 −6 3, 53, 83, 487, 743, ... A187805
8 7 7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ... A062574
8 5 2, 19, 1021, 5077, 34031, 46099, 65707, ... A128345
8 3 2, 3, 7, 19, 31, 67, 89, 9227, 43891, ... A128025
8 1 3 (no others)
8 −1 2 (no others)
8 −3 2, 5, 163, 191, 229, 271, 733, 21059, 25237, ... A128068
8 −5 2, 7, 19, 167, 173, 223, 281, 21647, ... A128338
8 −7 4, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ... A181141
9 8 2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ... A059803
9 7 3, 5, 7, 4703, 30113, ...
9 5 3, 11, 17, 173, 839, 971, 40867, 45821, ... A128346
9 4 2 (no others)
9 2 2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ... A173718
9 1 (none)
9 −1 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... A057175
9 −2 2, 3, 7, 127, 283, 883, 1523, 4001, ... A125956
9 −4 2, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ... A211409
9 −5 3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ... A128339
9 −7 2, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ...
9 −8 3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ... A187819
10 9 2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ... A062576
10 7 2, 31, 103, 617, 10253, 10691, ...
10 3 2, 3, 5, 37, 599, 38393, 51431, ... A128026
10 1 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... A004023
10 −1 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... A001562
10 −3 2, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ... A128069
10 −7 2, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
10 −9 4, 7, 67, 73, 1091, 1483, 10937, ... A217095
11 10 3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ... A062577
11 9 5, 31, 271, 929, 2789, 4153, ...
11 8 2, 7, 11, 17, 37, 521, 877, 2423, ...
11 7 5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ...
11 6 2, 3, 11, 163, 191, 269, 1381, 1493, ...
11 5 5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... A128347
11 4 3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ... A216181
11 3 3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ... A128027
11 2 2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ... A210506
11 1 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... A005808
11 −1 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... A057177
11 −2 3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ... A125957
11 −3 3, 103, 271, 523, 23087, 69833, ... A128070
11 −4 2, 7, 53, 67, 71, 443, 26497, ... A224501
11 −5 7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ... A128340
11 −6 2, 5, 7, 107, 383, 17359, 21929, ...
11 −7 7, 1163, 4007, 10159, ...
11 −8 2, 3, 13, 31, 59, 131, 223, 227, 1523, ...
11 −9 2, 3, 17, 41, 43, 59, 83, ...
11 −10 53, 421, 647, 1601, 35527, ... A185239
12 11 2, 3, 7, 89, 101, 293, 4463, 70067, ... A062578
12 7 2, 3, 7, 13, 47, 89, 139, 523, 1051, ...
12 5 2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ... A128348
12 1 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... A004064
12 −1 2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... A057178
12 −5 2, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ... A128341
12 −7 2, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
12 −11 47, 401, 509, 8609, ... A213216

(Note: if b<0 and n is even, then the numbers n are not included in the corresponding OEIS sequence)

For more information, see.[78][79][80][81][82][83][84][85]

A conjecture related to the generalized Mersenne primes:[86][87] (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many primes for all such (a,b) pairs)

For any integers a, b, which satisfy the conditions:

  1. a>1, -a<b<a.
  2. a and b are coprime.
  3. For every natural number r>1, a and b are not both perfect rth powers. (since when a and b are both perfect rth powers, it can be shown that there are at most two n value such that \frac{a^n-b^n}{a-b} is prime, and these n values are r itself or a root of r, or 2)
  4. If b<0, the absolute values of a and b are not "one is a fourth power, the other is 4 times a fourth power". (if so, then the number has aurifeuillean factorization)

has prime numbers of the form

R_p(a,b)=\frac{a^p-b^p}{a-b}

for prime p, the prime numbers will be distributed near the best fit line

Y=G \cdot log_a(log_a(R_{(a,b)}(n)))+C

where limit n\rightarrow\infty, G=\frac{1}{e^\gamma}=0.561459483566...

and there are about

(log_e(N)+m \cdot log_e(2) \cdot log_e(log_e(N))+\frac{1}{\sqrt N}-\delta) \cdot \frac{e^\gamma}{log_e(a)}

prime numbers of this form less than N.

We also have the following 3 properties:

  1. The number of prime numbers of the form \frac{a^p-b^p}{a-b} (with prime p) less than or equal to n is about e^\gamma \cdot log_a(log_a(n)).
  2. The expected number of prime numbers of the form \frac{a^p-b^p}{a-b} with prime p between n and a \cdot n is about e^\gamma.
  3. The probability that number of the form \frac{a^p-b^p}{a-b} is prime (for prime p) is about \frac{e^\gamma}{p \cdot log_e(a)}.

If this conjecture is true, then for all such (a,b) pairs, let q be the nth prime of the form \frac{a^p-b^p}{a-b}, the graph "log_a(log_a(q)) verse n" is almost linear. (See [86])

When a=b+1, it is

(b+1)^n-b^n

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.

Least n such that (b+1)^n-b^n 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)

Least b such that (b+1)^{prime(n)}-b^{prime(n)} 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

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References

  1. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  2. ↑ 2.0 2.1 2.2 Lua error in package.lua at line 80: module 'strict' not found.
  3. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  4. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  5. ↑ Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists
  6. ↑ The Prime Pages, Mersenne's conjecture.
  7. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  8. ↑ Lua error in package.lua at line 80: module 'strict' not found. p. 228.
  9. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  10. ↑ Brian Napper, The Mathematics Department and the Mark 1.
  11. ↑ The Prime Pages, The Prime Glossary: megaprime.
  12. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  13. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  14. ↑ Will Edgington's Mersenne Page
  15. ↑ Proof of a result of Euler and Lagrange on Mersenne Divisors
  16. ↑ 16.0 16.1 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 22 − 1 and 23 − 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 Arithmetic, 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.
  17. ↑ 17.0 17.1 Euclid's Elements, Book IX, Proposition 36
  18. ↑ The Prime Pages, Mersenne Primes: History, Theorems and Lists.
  19. ↑ 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]
  20. ↑ "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#
  21. ↑ pp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  22. ↑ pp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  23. ↑ 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.
  24. ↑ http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.
  25. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  26. ↑ “En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre 261 − 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
  27. ↑ 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.
  28. ↑ "M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1er Juin un cablógramme à M. Bromwich [secretary of London Mathematical Society] pour M107. 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.
  29. ↑ "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]
  30. ↑ 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.
  31. ↑ The Prime Pages, M107: Fauquembergue or Powers?.
  32. ↑ 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.
  33. ↑ 33.0 33.1 "Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2521 − 1 and 2607 − 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]
  34. ↑ "The program described in Note 131 (c) has produced the 15th Mersenne prime 21279 − 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]
  35. ↑ 35.0 35.1 "Two more Mersenne primes, 22203 − 1 and 22281 − 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]
  36. ↑ "On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number M3217 = 23217 − 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]
  37. ↑ 37.0 37.1 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.
  38. ↑ 38.0 38.1 "If p is prime, Mp = 2p − 1 is called a Mersenne number. The primes M4253 and M4423 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]
  39. ↑ 39.0 39.1 39.2 "The primes M9689, M9941, and M11213 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]
  40. ↑ "On the evening of March 4, 1971, a zero Lucas-Lehmer residue for p = p24 = 19937 was found. Hence, M19937 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]
  41. ↑ "On October 30, 1978 at 9:40 pm, we found M21701 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]
  42. ↑ "Of the 125 remaining Mp only M23209 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]
  43. ↑ David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013
  44. ↑ "The 27th Mersenne prime. It has 13395 digits and equals 244497 – 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 244497 − 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.
  45. ↑ "An FFT containing 8192 complex elements, which was the minimum size required to test M110503, ran approximately 11 minutes on the SX-2. The discovery of M110503 (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]
  46. ↑ "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]
  47. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  48. ↑ "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]
  49. ↑ "The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form 2p − 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]
  50. ↑ "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]
  51. ↑ The Prime Pages, The finding of the 32nd Mersenne.
  52. ↑ Chris Caldwell, The Largest Known Primes.
  53. ↑ Crays press release
  54. ↑ Slowinskis email
  55. ↑ Silicon Graphics' press release http://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
  56. ↑ The Prime Pages, A Prime of Record Size! 21257787 – 1.
  57. ↑ GIMPS Discovers 35th Mersenne Prime.
  58. ↑ GIMPS Discovers 36th Known Mersenne Prime.
  59. ↑ GIMPS Discovers 37th Known Mersenne Prime.
  60. ↑ GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
  61. ↑ GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
  62. ↑ GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
  63. ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583 – 1.
  64. ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951 – 1.
  65. ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457 – 1.
  66. ↑ GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657 – 1.
  67. ↑ 67.0 67.1 Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
  68. ↑ "On April 12th [2009], the 47th known Mersenne prime, 242,643,801 – 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]
  69. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  70. ↑ GIMPS Milestones Report. Retrieved 2015-10-04
  71. ↑ 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.
  72. ↑ Thorsten Kleinjung, Joppe Bos, Arjen Lenstra "Mersenne Factorization Factory" http://eprint.iacr.org/2014/653.pdf
  73. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  74. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  75. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  76. ↑ A research of Mersenne and Fermat primes
  77. ↑ Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)
  78. ↑ Ali Zalnezhad, Hossein Zalnezhad, Ghasem Shabani, Mehdi Zalnezhad "Relationships and Algorithm in order to Achieve the Largest Primes" http://arxiv.org/pdf/1503.07688.pdf
  79. ↑ (x, 1) and (x, -1) for x = 2 to 50
  80. ↑ (x, 1) for x = 2 to 152
  81. ↑ (x, -1) for x = 2 to 151
  82. ↑ (x + 1, x) for x = 1 to 150
  83. ↑ (x + 1, -x) for x = 1 to 40
  84. ↑ (x + 2, x) for x = 1 to 107
  85. ↑ (x, -1) for x = 2 to 200
  86. ↑ 86.0 86.1 Deriving the Wagstaff Mersenne Conjecture
  87. ↑ Generalized Repunit Conjecture

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