# Double Mersenne number

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

No. of known terms 4 4 7, 127, 2147483647 170141183460469231731687303715884105727 A077586a(n) = 2^(2^prime(n) - 1) - 1
${\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}$

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 (no prime factor < 4×1033)

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 4×1033.[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 > 4×1033) (sequence A263686 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 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 came up with 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 prime factor ${\displaystyle p}$  must 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".

## References

1. ^ a b c Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
2. ^ Tony Forbes, A search for a factor of MM61. Progress: 9 October 2008. This reports a high-water mark of 204204000000×(10019 + 1)×(261 − 1), above 4×1033. Retrieved on 2008-10-22.
3. ^
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 premier p, 2p − 1 est un nombre premier p', 2p' − 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.)