Double Mersenne number

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

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

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 ${\displaystyle M_{M_{p}}}$ can be prime only if M_p is itself a Mersenne prime. For the first values of p for which M_p is prime, ${\displaystyle M_{M_{p}}}$ 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 × 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 ${\displaystyle M_{M_{61}}}$, 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 ${\displaystyle M_{M_{p}}}$ (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:

${\displaystyle c_{0}=2}$

${\displaystyle c_{1}=2^{2}-1=3}$

${\displaystyle c_{2}=2^{3}-1=7}$

${\displaystyle c_{3}=2^{7}-1=127}$

${\displaystyle c_{4}=2^{127}-1=170141183460469231731687303715884105727}$

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

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.

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

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".

See also

• Cunningham chain
• Double exponential function
• Fermat number
• Perfect number
• Wieferich prime

References

1. 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 premier p, 2^p − 1 est un nombre premier p', 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.)

6. New Mersenne Conjecture

Further reading

• Dickson, L. E. (1971) [1919], History of the Theory of Numbers, New York: Chelsea Publishing.

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