Double Mersenne number

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


 * $$M_{M_p} = 2^{2^p-1}-1$$

where p is prime.

Examples
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, 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&thinsp;×&thinsp;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&thinsp;×&thinsp;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. 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.

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".