Double Mersenne number

Examples

The first four terms of the sequence of double Mersenne numbers are (sequence A077586 in the OEIS):

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.

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. 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

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 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).

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".