Double Mersenne number

Double Mersenne primes
No. of known terms	4
Conjectured no. of terms	4
First terms	7, 127, 2147483647
Largest known term	170141183460469231731687303715884105727
OEIS index	A077586 ; a(n) = 2^(2^prime(n) − 1) − 1;

Last updated November 15, 2024

In mathematics, a double Mersenne number is a Mersenne number of the form

Examples

The first four terms of the sequence of double Mersenne numbers are^[1](sequence A077586 in the OEIS ):

M_{M_{2}}=M_{3}=7

M_{M_{3}}=M_{7}=127

M_{M_{5}}=M_{31}=2147483647

M_{M_{7}}=M_{127}=170141183460469231731687303715884105727

Double Mersenne primes

A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number M_p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number $M_{M_{p}}$ can be prime only if M_p is itself a Mersenne prime. For the first values of p for which M_p is prime, $M_{M_{p}}$ is known to be prime for p = 2, 3, 5, 7 while explicit factors of $M_{M_{p}}$ have been found for p = 13, 17, 19, and 31.

$p$	$M_{p}=2^{p}-1$	$M_{M_{p}}=2^{2^{p}-1}-1$	factorization of $M_{M_{p}}$
2	3	prime	7
3	7	prime	127
5	31	prime	2147483647
7	127	prime	170141183460469231731687303715884105727
11	not prime	not prime	47 × 131009 × 178481 × 724639 × 2529391927 × 70676429054711 × 618970019642690137449562111 × ...
13	8191	not prime	338193759479 × 210206826754181103207028761697008013415622289 × ...
17	131071	not prime	231733529 × 64296354767 × ...
19	524287	not prime	62914441 × 5746991873407 × 2106734551102073202633922471 × 824271579602877114508714150039 × 65997004087015989956123720407169 × 4565880376922810768406683467841114102689 × ...
23	not prime	not prime	2351 × 4513 × 13264529 × 76899609737 × ...
29	not prime	not prime	1399 × 2207 × 135607 × 622577 × 16673027617 × 4126110275598714647074087 × ...
31	2147483647	not prime	295257526626031 × 87054709261955177 × 242557615644693265201 × 178021379228511215367151 × ...
37	not prime	not prime
41	not prime	not prime
43	not prime	not prime
47	not prime	not prime
53	not prime	not prime
59	not prime	not prime
61	2305843009213693951	unknown

Thus, the smallest candidate for the next double Mersenne prime is $M_{M_{61}}$ , or 2^{2305843009213693951} − 1. Being approximately 1.695×10^{694127911065419641}, this number is far too large for any currently known primality test. It has no prime factor below 1 × 10³⁶.^[2] There are probably no other double Mersenne primes than the four known.^[1]^[3]

Smallest prime factor of $M_{M_{p}}$ (where p is the nth prime) are

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 10³⁶) (sequence A309130 in the OEIS )

Catalan–Mersenne number conjecture

The recursively defined sequence

c_{0}=2

c_{n+1}=2^{c_{n}}-1=M_{c_{n}}

is called the sequence of Catalan–Mersenne numbers.^[4] The first terms of the sequence (sequence A007013 in the OEIS ) are:

c_{0}=2

c_{1}=2^{2}-1=3

c_{2}=2^{3}-1=7

c_{3}=2^{7}-1=127

c_{4}=2^{127}-1=170141183460469231731687303715884105727

c_{5}=2^{170141183460469231731687303715884105727}-1\approx 5.45431\times 10^{51217599719369681875006054625051616349}\approx 10^{10^{37.70942}}

Catalan discovered this sequence after the discovery of the primality of $M_{127}=c_{4}$ by Lucas in 1876.^[1]^[5] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if $c_{5}$ is not prime, there is a chance to discover this by computing $c_{5}$ modulo some small prime $p$ (using recursive modular exponentiation). If the resulting residue is zero, $p$ represents a factor of $c_{5}$ and thus would disprove its primality. Since $c_{5}$ is a Mersenne number, such a prime factor $p$ would have to be of the form $2kc_{4}+1$ . Additionally, because $2^{n}-1$ is composite when $n$ is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

If $c_{5}$ were prime, it would also contradict the New Mersenne conjecture. It is known that ${\frac {2^{c_{4}}+1}{3}}$ is composite, with factor $886407410000361345663448535540258622490179142922169401=5209834514912200c_{4}+1$ .^[6]

In popular culture

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number $M_{M_{7}}$ is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "Martian prime".

Related Research Articles

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 of the form $M n = 2 n - 1$ for some integer $n$ . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If $n$ is a composite number then so is $2 n - 1$ . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form $M p = 2 p - 1$ for some prime $p$ .

<span class="mw-page-title-main">Prime number</span> Number divisible only by 1 or itself

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or $(41, 43)$ . In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.

In mathematics, a Fermat number, named after Pierre de Fermat (1607–1665), the first known to have studied them, is a positive integer of the form: $where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ....$

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.

In number theory, a Wieferich prime is a prime number p such that p² divides $2 p - 1 - 1$ , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides $2 p - 1 - 1$ . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

23 (twenty-three) is the natural number following 22 and preceding 24.

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

61 (sixty-one) is the natural number following 60 and preceding 62.

63 (sixty-three) is the natural number following 62 and preceding 64.

In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

127 is the natural number following 126 and preceding 128. It is also a prime number.

In number theory, a Wagstaff prime is a prime number of the form

In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi, who observed that both 2 and 5 are untouchable.

In number theory, a Pierpont prime is a prime number of the form $for some nonnegative integers u and v . That is, they are the prime numbers p for which p - 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding.$

In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named after a French mathematician, Théophile Pépin.

In mathematics, the Mersenne conjectures concern the characterization of a kind of prime numbers called Mersenne primes, meaning prime numbers that are a power of two minus one.

References

1 2 3 Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
↑ "Double Mersenne 61 factoring status". www.doublemersennes.org. Retrieved 31 March 2022.
↑ I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
↑ Weisstein, Eric W. "Catalan-Mersenne Number". MathWorld .
↑ "Questions proposées". Nouvelle correspondance mathématique. 2: 94–96. 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92:
Prouver que 2⁶¹ − 1 et 2¹²⁷ − 1 sont des nombres premiers. (É. L.) (*).
The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows:
(*) Si l'on admet ces deux propositions, et si l'on observe que 2² − 1, 2³ − 1, 2⁷ − 1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2ⁿ − 1 est un nombre premierp, 2^p − 1 est un nombre premierp', 2^p' − 1 est un nombre premier p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2ⁿ + 1 est un nombre premier. (E. C.)
↑ New Mersenne Conjecture

External links

Weisstein, Eric W. "Double Mersenne Number". MathWorld .
Tony Forbes, A search for a factor of MM61 Archived 2009-02-08 at the Wayback Machine .
Status of the factorization of double Mersenne numbers
Double Mersennes Prime Search
Operazione Doppi Mersennes

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Caldwell-1] 1 2 3 Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.

[2] "Double Mersenne 61 factoring status". www.doublemersennes.org. Retrieved 31 March 2022.

[3] I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]

[4] Weisstein, Eric W. "Catalan-Mersenne Number". MathWorld .

[5] "Questions proposées". Nouvelle correspondance mathématique. 2: 94–96. 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92:
Prouver que 2⁶¹ − 1 et 2¹²⁷ − 1 sont des nombres premiers. (É. L.) (*).
The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows:
(*) Si l'on admet ces deux propositions, et si l'on observe que 2² − 1, 2³ − 1, 2⁷ − 1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2ⁿ − 1 est un nombre premierp, 2^p − 1 est un nombre premierp', 2^p' − 1 est un nombre premier p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2ⁿ + 1 est un nombre premier. (E. C.)

[Hoegge-6] New Mersenne Conjecture

[1]

[2]

[3]

[4]

[5]

[6]