Double Mersenne number

The first four terms of the sequence of double Mersenne numbers are : :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 Mp is itself a Mersenne prime. For the first values of p for which Mp is prime, M_{M_{p}} is known to be prime for p = 2, 3, 5, and 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 22305843009213693951 − 1. Being approximately 1.695, this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036. There are probably no other double Mersenne primes than the four known. Smallest prime factor of M_{M_{p}} (where p is the nth prime) are :7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 1036) ==Catalan–Mersenne number conjecture==

The recursively defined sequence : c_0 = 2 : c_{n+1} = 2^{c_n}-1 = M_{c_n} is called the sequence of Catalan–Mersenne numbers. The first terms of the sequence are: :c_0 = 2 :c_1 = 2^2-1 = 3 :c_2 = 2^3-1 = 7 :c_3 = 2^7-1 = 127 :c_4 = 2^{127}-1 = 170141183460469231731687303715884105727 :c_5 = 2^{170141183460469231731687303715884105727}-1 \approx 5.45431 \times 10^{51217599719369681875006054625051616349} \approx 10^{10^{37.70942}} Catalan discovered this sequence after the discovery of the primality of M_{127}=c_4 by Lucas in 1876.p. 22 Catalan conjectured that they are prime "up to a certain limit". Although the first five terms 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 c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation). If the resulting residue is zero, p represents a factor of c_5 and thus would disprove its primality. Since c_5 is a Mersenne number, such a prime factor p would have to be of the form 2kc_4 +1. Additionally, because 2^n-1 is composite when n is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence. If c_5 were prime, it would also contradict the New Mersenne conjecture. It is known that \frac{2^{c_4} + 1}{3} is composite, with factor 886407410000361345663448535540258622490179142922169401 = 5209834514912200c_4 + 1. ==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==