Double Mersenne number

Double Mersenne primes
Number of known terms	4
Conjectured number 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. The first values of p for which M_p is prime are p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Of these, $M_{M_p}$ is known to be prime for p = 2, 3, 5, 7. For p = 13, 17, 19, and 31, explicit factors have been found showing that the corresponding double Mersenne numbers are not prime. 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

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.)
↑ 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

Weisstein, Eric W. "Double Mersenne Number". MathWorld.

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

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 (10ⁿ − 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, 2n − 1, 2n + 1, …) 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, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 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

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