# Double Mersenne number

No. of known terms 4 4 7, 127, 2147483647 170141183460469231731687303715884105727 A077586a(n) = 2^(2^prime(n) − 1) − 1

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

$\displaystyle{ M_{M_p} = 2^{2^p-1}-1 }$

where p is prime.

## Examples

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

$\displaystyle{ M_{M_2} = M_3 = 7 }$
$\displaystyle{ M_{M_3} = M_7 = 127 }$
$\displaystyle{ M_{M_5} = M_{31} = 2147483647 }$
$\displaystyle{ M_{M_7} = M_{127} = 170141183460469231731687303715884105727 }$

## Double Mersenne primes

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

$\displaystyle{ p }$ $\displaystyle{ M_{p} = 2^p-1 }$ $\displaystyle{ M_{M_{p}} = 2^{2^p-1}-1 }$ factorization of $\displaystyle{ M_{M_{p}} }$
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 × ...
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 $\displaystyle{ M_{M_{61}} }$, or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, 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 $\displaystyle{ M_{M_{p}} }$ (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

$\displaystyle{ c_0 = 2 }$
$\displaystyle{ c_{n+1} = 2^{c_n}-1 = M_{c_n} }$

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

$\displaystyle{ c_0 = 2 }$
$\displaystyle{ c_1 = 2^2-1 = 3 }$
$\displaystyle{ c_2 = 2^3-1 = 7 }$
$\displaystyle{ c_3 = 2^7-1 = 127 }$
$\displaystyle{ c_4 = 2^{127}-1 = 170141183460469231731687303715884105727 }$
$\displaystyle{ c_5 = 2^{170141183460469231731687303715884105727}-1 \approx 5.454 \times 10^{51217599719369681875006054625051616349} \approx 10^{10^{37.7094}} }$

Catalan discovered this sequence after the discovery of the primality of $\displaystyle{ M_{127}=c_4 }$ 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 $\displaystyle{ c_5 }$ is not prime, there is a chance to discover this by computing $\displaystyle{ c_5 }$ modulo some small prime $\displaystyle{ p }$ (using recursive modular exponentiation). If the resulting residue is zero, $\displaystyle{ p }$ represents a factor of $\displaystyle{ c_5 }$ and thus would disprove its primality. Since $\displaystyle{ c_5 }$ is a Mersenne number, such a prime factor $\displaystyle{ p }$ would have to be of the form $\displaystyle{ 2kc_4 +1 }$. Additionally, because $\displaystyle{ 2^n-1 }$ is composite when $\displaystyle{ n }$ is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

## In popular culture

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number $\displaystyle{ M_{M_7} }$ is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".
The text passage provides a detailed explanation of double Mersenne numbers, their properties, and some known values. It also mentions the Catalan–Mersenne number conjecture and a reference to a double Mersenne number in popular culture.
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the passage does not contain any explicit mathematical questions or problems that require solving. it is primarily informational and does not provide answers to any specific questions.
Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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system
# Double Mersenne number

No. of known terms 4 4 7, 127, 2147483647 170141183460469231731687303715884105727 A077586a(n) = 2^(2^prime(n) − 1) − 1

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

$\displaystyle{ M_{M_p} = 2^{2^p-1}-1 }$

where p is prime.

## Examples

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

$\displaystyle{ M_{M_2} = M_3 = 7 }$
$\displaystyle{ M_{M_3} = M_7 = 127 }$
$\displaystyle{ M_{M_5} = M_{31} = 2147483647 }$
$\displaystyle{ M_{M_7} = M_{127} = 170141183460469231731687303715884105727 }$

## Double Mersenne primes

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

$\displaystyle{ p }$ $\displaystyle{ M_{p} = 2^p-1 }$ $\displaystyle{ M_{M_{p}} = 2^{2^p-1}-1 }$ factorization of $\displaystyle{ M_{M_{p}} }$
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 × ...
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 $\displaystyle{ M_{M_{61}} }$, or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, 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 $\displaystyle{ M_{M_{p}} }$ (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

$\displaystyle{ c_0 = 2 }$
$\displaystyle{ c_{n+1} = 2^{c_n}-1 = M_{c_n} }$

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

$\displaystyle{ c_0 = 2 }$
$\displaystyle{ c_1 = 2^2-1 = 3 }$
$\displaystyle{ c_2 = 2^3-1 = 7 }$
$\displaystyle{ c_3 = 2^7-1 = 127 }$
$\displaystyle{ c_4 = 2^{127}-1 = 170141183460469231731687303715884105727 }$
$\displaystyle{ c_5 = 2^{170141183460469231731687303715884105727}-1 \approx 5.454 \times 10^{51217599719369681875006054625051616349} \approx 10^{10^{37.7094}} }$

Catalan discovered this sequence after the discovery of the primality of $\displaystyle{ M_{127}=c_4 }$ 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 $\displaystyle{ c_5 }$ is not prime, there is a chance to discover this by computing $\displaystyle{ c_5 }$ modulo some small prime $\displaystyle{ p }$ (using recursive modular exponentiation). If the resulting residue is zero, $\displaystyle{ p }$ represents a factor of $\displaystyle{ c_5 }$ and thus would disprove its primality. Since $\displaystyle{ c_5 }$ is a Mersenne number, such a prime factor $\displaystyle{ p }$ would have to be of the form $\displaystyle{ 2kc_4 +1 }$. Additionally, because $\displaystyle{ 2^n-1 }$ is composite when $\displaystyle{ n }$ is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

## In popular culture

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number $\displaystyle{ M_{M_7} }$ is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".
user
The text passage provides a detailed explanation of double Mersenne numbers, their properties, and some known values. It also mentions the Catalan–Mersenne number conjecture and a reference to a double Mersenne number in popular culture.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems that require solving. It is primarily informational and does not provide answers to any specific questions.