Inmathematics, adouble Mersenne number is aMersenne number of the form${\displaystyle M_{M_p}=2^{2^p-1}-1}$ where${\displaystyle p}$ isprime.

The first four terms of thesequence of double Mersenne numbers are^[1] (sequenceA077586 in theOEIS):

${\displaystyle M_{M_2}=M_3=7}$

${\displaystyle M_{M_3}=M_7=127}$

${\displaystyle M_{M_5}=M_{31}=2147483647}$

${\displaystyle M_{M_7}=M_{127}=170141183460469231731687303715884105727}$

A double Mersenne number that isprime is called adouble Mersenne prime. Since a Mersenne numberM_p can be prime only ifp is prime, (seeMersenne prime for a proof), a double Mersenne number${\displaystyle M_{M_p}}$ can be prime only ifM_p is itself a Mersenne prime. For the first values ofp for whichM_p is prime,${\displaystyle M_{M_p}}$ is known to be prime forp = 2, 3, 5, and 7 while explicit factors of${\displaystyle M_{M_p}}$ have been found forp = 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 (triple)	127
5	31	prime	2147483647
7	127	prime (quadruple)	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 × 285212639 × 76899609737 × ...
29	not prime	not prime	1399 × 2207 × 135607 × 622577 × 16673027617 × 52006801325877583 × 4126110275598714647074087 × ...
31	2147483647	not prime (triple mersenne number)	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 knownprimality 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${\displaystyle M_{M_p}}$ (wherep is thenth prime) are

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 10³⁶) (sequenceA309130 in theOEIS)

Therecursively defined sequence

${\displaystyle c_0=2}$

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

is called the sequence ofCatalan–Mersenne numbers.^[4] The first terms of the sequence (sequenceA007013 in theOEIS) 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${\displaystyle M_{127}=c_4}$ byLucas in 1876.^{[1][5][6]p. 22} Catalanconjectured 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 recursivemodular 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 aMersenne number, such a prime factor${\displaystyle p}$ would have to be of the form${\displaystyle 2k{c}_4+1}$. Additionally, because${\displaystyle 2^n-1}$ iscomposite 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 theNew Mersenne conjecture. It is known that${\displaystyle \frac{2^{c_4}+1}{3}}$ is composite, with factor${\displaystyle 886407410000361345663448535540258622490179142922169401=5209834514912200c_4+1}$.^[7]

In theFuturama movieThe Beast with a Billion Backs, the double Mersenne number${\displaystyle M_{M_7}}$ is briefly seen in "anelementary proof of theGoldbach conjecture". In the movie, this number is known as a "Martian prime".

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

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

• Weisstein, Eric W."Double Mersenne Number".MathWorld.
• Tony Forbes,A search for a factor of MM61Archived 2009-02-08 at theWayback Machine.
• Status of the factorization of double Mersenne numbers
• Double Mersennes Prime Search
• Operazione Doppi Mersennes

• Integer sequences
• Large integers
• Unsolved problems in number theory
• Mersenne primes

• Articles with short description
• Short description is different from Wikidata
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