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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.

The smallest double Mersenne numbers

The sequence of double Mersenne numbers begins [1]

\( M_{M_2} = M_3 = 7 \)
\( M_{M_3} = M_7 = 127 \)
\( M_{M_5} = M_{31} = 2147483647 \)
\( M_{M_7} = M_{127} = 170141183460469231731687303715884105727 \) (sequence A077586 in OEIS).

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 Mp 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. 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.[2] There are probably no other double Mersenne primes than the four known.[1]
The 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.[3] It is said[1] that 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 all prime.

Although the first five terms (up to M(127)) are prime, no known methods can decide if any more of these numbers are prime (in any reasonable time) simply because the numbers in question are too huge, unless the primality of M(M(127)) is disproved.
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

^ 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.
^ Weisstein, Eric W., "Catalan-Mersenne Number" from MathWorld.

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Double Mersenne number