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

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

M_{M_{p}}=2^{2^{p}-1}-1

where p is a prime exponent.

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.

$p$	$M_{p}=2^{p}-1$	$M_{M_{p}}=2^{2^{p}-1}-1$
2	3	Prime
3	7	Prime
5	31	Prime
7	127	Prime
11	Not prime	—
13	8191	Not prime
17	131071	Not prime
19	524287	Not prime
23	Not prime	—
29	Not prime	—
31	2147483647	Not prime
37	Not prime	—
41	Not prime	—
43	Not prime	—
47	Not prime	—
53	Not prime	—
59	Not prime	—
61	2305843009213693951	Unknown

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]

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).^[6]

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

^ ^a ^b ^c 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.)
^ 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.

External links

[Caldwell-1] 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.)

[6] 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.

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[2]

[3]

[4]

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[6]

v t e 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 ( $x 3 - y 3)/(x - y$ ) 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
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 Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number 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
List of prime numbers