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# Double Mersenne number

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### Double Mersenne number

 Number of known terms 4 4 7, 127, 2147483647 170141183460469231731687303715884105727 A077856

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

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

where p is a Mersenne prime exponent.

## Contents

• Examples 1
• Double Mersenne primes 2
• Catalan–Mersenne number conjecture 3
• In popular culture 4
• References 6

## Examples

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

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. The first values of p for which Mp is prime are p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Of these, M_{M_p} is known to be prime for p = 2, 3, 5, 7. For p = 13, 17, 19, and 31, explicit factors have been found showing that the corresponding double Mersenne numbers are not prime. Thus, the smallest candidate for the next double Mersenne prime is M_{M_{61}}, or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, 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

2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), ... (sequence A007013 in OEIS)

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 MM127 is not prime, there is a chance to discover this by computing MM127 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".