Double Mersenne number

In mathematics, a double Mersenne number is a Mersenne number of the form

{\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}

where p is a prime exponent.

Examples[edit]

The first four terms of the sequence of double Mersenne numbers are^[1] (sequence A077586 in the OEIS):

{\displaystyle M_{M_{2}}=M_{3}=7}

{\displaystyle M_{M_{3}}=M_{7}=127}

{\displaystyle M_{M_{5}}=M_{31}=2147483647}

{\displaystyle M_{M_{7}}=M_{127}=170141183460469231731687303715884105727}

Double Mersenne primes[edit]

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 {\displaystyle 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, {\displaystyle M_{M_{p}}} is known to be prime for p = 2, 3, 5, 7 while explicit factors of {\displaystyle M_{M_{p}}} have been found for p = 13, 17, 19, and 31.

{\displaystyle p}	{\displaystyle M_{p}=2^{p}-1}	{\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}
2	3	prime
3	7	prime
5	31	prime
7	127	prime
11	not prime	—
13	8191	not prime
17	131071	not prime
19	524287	not prime
23	not prime	—
29	not prime	—
31	2147483647	not prime
37	not prime	—
41	not prime	—
43	not prime	—
47	not prime	—
53	not prime	—
59	not prime	—
61	2305843009213693951	unknown

Thus, the smallest candidate for the next double Mersenne prime is {\displaystyle 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[edit]

Write {\displaystyle M(p)} instead of {\displaystyle 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[edit]

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number {\displaystyle 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”.