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# Mersenne prime

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### Mersenne prime

 Named after Marin Mersenne 1536 Regius, H. 48 Infinite Mersenne numbers 3, 7, 31, 127 257885161 − 1 (January 2013) A000668

In mathematics, a Mersenne prime is a prime number of the form M_n=2^n-1. This is to say that it is a prime number which is one less than a power of two. They are named after the French monk Marin Mersenne who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31, and 127.

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

More generally, numbers of the form M_n=2^n-1\, without the primality requirement are called Mersenne numbers. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those pernicious numbers whose binary representation contains no zeros. The smallest composite pernicious Mersenne number is 211 − 1.

As of October 2014, 48 Mersenne primes are known. The largest known prime number 257,885,161 − 1 is a Mersenne prime.

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.

## Contents

• Perfect numbers 2
• History 3
• Searching for Mersenne primes 4
• Theorems about Mersenne numbers 5
• List of known Mersenne primes 6
• Factorization of composite Mersenne numbers 7
• Mersenne primitive part 8
• Mersenne numbers in nature and elsewhere 9
• Mersenne-Fermat primes 10
• Generalizations 11
• Gaussian Mersenne primes 11.1
• Repunit primes 11.2
• References 13

 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 Mp, e.g., 23|M11, 47|M23, 167|M83, 263|M131, 359|M179, 383|M191, 479|M239, and that 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 1,881,339 prime numbers p up to 30,402,457, Mp 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

Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. In the 4th century BC, Euclid proved that if 2p−1 is prime, then 2p−1(2p − 1) is a perfect number. This number, also expressible as Mp(Mp+1)/2, is the Mpth triangular number and the 2p − 1th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. 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 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. (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 calculated by hand. M61 was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M67 is actually composite. No factor was found until a famous talk by Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking. He later said that the result had taken him "three years of Sundays" to find. A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.

## Searching for Mersenne primes

Fast algorithms for finding Mersenne primes are available, and as of 2014 the ten largest known prime numbers are Mersenne primes.

The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was M127, found by Lucas in 1876, then M61 by Pervushin in 1883. Two more (M89 and M107) were found early in the 20th century, by Powers in 1911 and 1914, respectively.

The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, Mp = 2p − 1 is prime if and only if Mp divides Sp−2, where S0 = 4 and, for k > 0,

S_k = S_{k-1}^2-2.\ Graph of number of digits in largest known Mersenne prime by year – electronic era. Note that the vertical scale, the number of digits, is a double logarithmic scale of the value of the prime.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949, 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. 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. 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. 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 (the smallest Mersenne number with said amount of digits is 2332192807 − 1).

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 = ab 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 2^p-1 is congruent to ±1 (mod 8).
• Proof: 2^{p+1} = 2 \pmod q, so 2^{(p+1)/2} is a square root of 2 modulo q. By quadratic reciprocity, 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 does not satisfy. By Fermat's Little theorem, m \mid p-1. Now write, p-1=m\lambda. If the given congruence satisfies, then p^2\mid2^{m\lambda}-1,therefore 0 ≡ (2mλ − 1)/(2m − 1) = 1 + 2m + 22m + ... + 2λ−1m ≡ −λ mod(2m − 1}. Hence 2^m-1\mid\lambda,and therefore λ ≥ 2m − 1. This leads to p − 1  ≥ m(2m − 1), which is impossible since m ≥ 2.
6. A prime number divides at most one prime-exponent Mersenne number, 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.
• 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 M_n equals \lfloor n\cdot \log_{10}2\rfloor+1, where \lfloor x\rfloor denotes the floor function.

## List of known Mersenne primes

The table below lists all known Mersenne primes (sequence 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
2 3 7 1 c. 430 BC Ancient Greek mathematicians
3 5 31 2 c. 300 BC Ancient Greek mathematicians
4 7 127 3 c. 300 BC Ancient Greek mathematicians
5 13 8191 4 1456 Anonymous Trial division
6 17 131071 6 1588 Pietro Cataldi Trial division
7 19 524287 6 1588 Pietro Cataldi Trial division
8 31 2147483647 10 1772 Leonhard Euler Enhanced trial division
9 61 2305843009213693951 19 1883 November I. M. Pervushin Lucas sequences
10 89 618970019642...137449562111 27 1911 June Ralph Ernest Powers Lucas sequences
11 107 162259276829...578010288127 33 1914 June 1 Ralph Ernest Powers Lucas sequences
12 127 170141183460...715884105727 39 1876 January 10 Édouard Lucas Lucas sequences
13 521 686479766013...291115057151 157 1952 January 30 Raphael M. Robinson LLT / SWAC
14 607 531137992816...219031728127 183 1952 January 30 Raphael M. Robinson LLT / SWAC
15 1,279 104079321946...703168729087 386 1952 June 25 Raphael M. Robinson LLT / SWAC
16 2,203 147597991521...686697771007 664 1952 October 7 Raphael M. Robinson LLT / SWAC
17 2,281 446087557183...418132836351 687 1952 October 9 Raphael M. Robinson LLT / SWAC
18 3,217 259117086013...362909315071 969 1957 September 8 Hans Riesel LLT / BESK
19 4,253 190797007524...815350484991 1,281 1961 November 3 Alexander Hurwitz LLT / IBM 7090
20 4,423 285542542228...902608580607 1,332 1961 November 3 Alexander Hurwitz LLT / IBM 7090
21 9,689 478220278805...826225754111 2,917 1963 May 11 Donald B. Gillies LLT / ILLIAC II
22 9,941 346088282490...883789463551 2,993 1963 May 16 Donald B. Gillies LLT / ILLIAC II
23 11,213 281411201369...087696392191 3,376 1963 June 2 Donald B. Gillies LLT / ILLIAC II
24 19,937 431542479738...030968041471 6,002 1971 March 4 Bryant Tuckerman LLT / IBM 360/91
25 21,701 448679166119...353511882751 6,533 1978 October 30 Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174
26 23,209 402874115778...523779264511 6,987 1979 February 9 Landon Curt Noll LLT / CDC Cyber 174
27 44,497 854509824303...961011228671 13,395 1979 April 8 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 Walter Colquitt & Luke Welsh LLT / NEC SX-2
30 132,049 512740276269...455730061311 39,751 1983 September 19 David Slowinski LLT / Cray X-MP
31 216,091 746093103064...103815528447 65,050 1985 September 1 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
33 859,433 129498125604...243500142591 258,716 1994 January 4 David Slowinski & Paul Gage LLT / Cray C90
34 1,257,787 412245773621...976089366527 378,632 1996 September 3 David Slowinski & Paul Gage LLT / Cray T94
35 1,398,269 814717564412...868451315711 420,921 1996 November 13 GIMPS / Joel Armengaud LLT / Prime95 on 90 MHz Pentium PC
36 2,976,221 623340076248...743729201151 895,932 1997 August 24 GIMPS / Gordon Spence LLT / Prime95 on 100 MHz Pentium PC
37 3,021,377 127411683030...973024694271 909,526 1998 January 27 GIMPS / Roland Clarkson LLT / Prime95 on 200 MHz Pentium PC
38 6,972,593 437075744127...142924193791 2,098,960 1999 June 1 GIMPS / Nayan Hajratwala LLT / Prime95 on 350 MHz Pentium II IBM Aptiva
39 13,466,917 924947738006...470256259071 4,053,946 2001 November 14 GIMPS / Michael Cameron LLT / Prime95 on 800 MHz Athlon T-Bird
40 20,996,011 125976895450...762855682047 6,320,430 2003 November 17 GIMPS / Michael Shafer LLT / Prime95 on 2 GHz Dell Dimension
41 24,036,583 299410429404...882733969407 7,235,733 2004 May 15 GIMPS / Josh Findley LLT / Prime95 on 2.4 GHz Pentium 4 PC
42 25,964,951 122164630061...280577077247 7,816,230 2005 February 18 GIMPS / Martin Nowak 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 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 LLT / Prime95 on 3 GHz Pentium 4 PC
45[*] 37,156,667 202254406890...022308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael Elvenich 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 LLT / Prime95 on 3 GHz Core 2 PC
47[*] 43,112,609 316470269330...166697152511 12,978,189 2008 August 23 GIMPS / Edson Smith LLT / Prime95 on Dell Optiplex 745
48[*] 57,885,161 581887266232...071724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper LLT / Prime95 on 3 GHz Intel Core2 Duo E8400

^ * 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 47th (M43,112,609) in the interval have been tested at least once but some have not been double-checked. Some Mersenne numbers above the 47th have not yet been tested. 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. 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.

## 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, 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 October 2014, the largest factorization with probable prime factors allowed is 22,327,417 − 1 = 23,915,387,348,002,001 × q, where q is a 700,606-digit probable prime.

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

# 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
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 \Phi_n(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 \Phi_n(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.

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).

## Mersenne-Fermat primes

A Mersenne-Fermat number is defined as \frac{2^{p^r}-1}{2^{p^{r-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).

In fact, MF(p, r) = \Phi_{p^r}(2), where \Phi 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:

### Gaussian Mersenne primes

In the ring of integers, 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. However, if we regard instead 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.

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

The other way to deal with the fact that b^n-1 is always divisible by b-1, 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

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

to be a prime. 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). It is a conjecture that there are infinitely many n values of every integer b which is not a perfect power.

The 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)

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 \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)

Another generized Mersenne number is

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

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 are coprime, if b < 0, the greatest common factor of the ath term and the |b|th term in 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

(b+1)^n-b^n

Is a difference of two perfect nth powers, and if an - bn is prime, than a must be b + 1 or b - 1, because it is divisible by a - b.

The 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)

The 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)

Interestingly, when we take b = 137, than (b+1)^n-b^n is composite when n < 196873, and it is a probable prime when n = 196873. Besides, no known prime or PRP of the form (b+1)^n-b^n when b = 139, 256, 295, 370, 373, 415, ...