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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 Mersenne prime exponent.

Examples

The first four terms of the sequence of double Mersenne numbers are[1] (sequence A077586 in 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 Mp 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 Mp 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 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".
See also

Perfect number
Fermat number
Wieferich prime
Double exponential function

References

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, 2p − 1 est un nombre premier p', 2p' − 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 MM127 and thus would disprove its primality. Since MM127 is a Mersenne number, such prime factor p must be of the form 2·k·M127+1.

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Double Mersenne number