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;

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

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):

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 × ...
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 $M_{M_{61}}$ , 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 4×10³³.^[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 > 4×10³³) (sequence A263686 in the OEIS)

Catalan–Mersenne number conjecture

Write $M(p)$ instead of $M_{p}$ . A special case of the double Mersenne numbers, namely the recursively defined sequence

2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), ... (sequence A007013 in the OEIS)

is called the Catalan–Mersenne numbers.^[4] Catalan came up with this sequence after the discovery of the primality of M(127) = M(M(M(M(2)))) by Lucas in 1876.^[1]^[5] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms (below M₁₂₇) 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 M_M₁₂₇ is not prime, there is a chance to discover this by computing M_M₁₂₇ modulo some small prime p (using recursive modular exponentiation. If the resulting residue is zero, p represents a factor of M_M₁₂₇ and thus would disprove its primality. Since M_M₁₂₇ is a Mersenne number, such prime factor p must be of the form 2·k·M₁₂₇+1.)

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".

References

1 2 3 Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
↑ 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.
↑ 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 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.)

External links

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[Caldwell-1] 1 2 3 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] 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.)

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
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