Double Mersenne number

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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.

1 The smallest double Mersenne numbers
2 Double Mersenne primes
3 Catalan-Mersenne number
4 See also
5 References
6 External links

[edit] The smallest double Mersenne numbers

The sequence of double Mersenne numbers begins ^[1]

$M_{M_2} = M_3 = 7$

$M_{M_3} = M_7 = 127$

$M_{M_5} = M_{31} = 2147483647$

$M_{M_7} = M_{127} = 170141183460469231731687303715884105727$ (sequence A077586 in OEIS).

[edit] Double Mersenne primes

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 $M_{M_p}$ can be prime only if M_p is itself a Mersenne prime. The first values of p for which M_p is prime are p = 2, 3, 5, 7, 13, 17, 19, 31. 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. However, the smallest candidate is $M_{M_{61}}$ , or 2^{2305843009213693951}-1. At approximately 6.94×10¹⁷ decimal digits, this number is far too large for any currently known primality test.

[edit] Catalan-Mersenne number

Write $M (p)$ instead of $M p$ . A special case of the double Mersenne numbers, namely the recursively defined sequence

$2, \quad M(2), \quad M(M(2)), \quad M(M(M(2))), \quad M(M(M(M(2)))), \quad \ldots$

(sequence A007013 in OEIS) is called the Catalan-Mersenne numbers.^[2] It is said^[1] that Catalan came up with this sequence after the discovery of the primality of $M (127) = M (M (M (M (2))))$ by Lucas in 1876.

Although the first five terms (up to $M (127)$ ) are prime, no known methods can decide if any more of these numbers are prime (in any reasonable time) simply because the numbers in question are too huge.