In mathematics, a double Mersenne number is a Mersenne number of the form where p is prime.
The first four terms of the sequence of double Mersenne numbers are[1] (sequence A077586 in the OEIS): 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
is known to be prime for p = 2, 3, 5, 7 while explicit factors of
Thus, the smallest candidate for the next double Mersenne prime is
Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test.
(where p is the nth prime) are The recursively defined sequence is called the sequence of Catalan–Mersenne numbers.
[1][5] 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.
is not prime, there is a chance to discover this by computing
modulo some small prime
(using recursive modular exponentiation).
is a Mersenne number, such a prime factor
is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.
were prime, it would also contradict the New Mersenne conjecture.
[6] In the Futurama movie The Beast with a Billion Backs, the double Mersenne number
is briefly seen in "an elementary proof of the Goldbach conjecture".
In the movie, this number is known as a "Martian prime".
Prouver que 261 − 1 et 2127 − 1 sont des nombres premiers.
(*) Si l'on admet ces deux propositions, et si l'on observe que 22 − 1, 23 − 1, 27 − 1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2n − 1 est un nombre premier p, 2p − 1 est un nombre premier p', 2p' − 1 est un nombre premier p", etc.
Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2n + 1 est un nombre premier.