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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 (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 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test. It has no prime factor below 4×1033. There are probably no other double Mersenne primes than the four known.
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. Catalan came up with this sequence after the discovery of the primality of M(127) = M(M(M(M(2)))) by Lucas in 1876. 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 MM127 is not prime, there is a chance to discover this by computing MM127 modulo some small prime p (using recursive modular exponentiation).

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".

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.

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