Double Mersenne number

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

$M_{M_n} = 2^{2^n-1}-1$

where n is a positive integer.

1 The smallest double Mersenne numbers
2 Double Mersenne primes
3 See also
4 External links

[edit] The smallest double Mersenne numbers

The sequence of double Mersenne numbers (sequence A077585 in OEIS) begins

$M_{M_1} = M_1 = 1$

$M_{M_2} = M_3 = 7$

$M_{M_3} = M_7 = 127$

$M_{M_4} = M_{15} = 32767 = 7 \times 31 \times 151$

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

$M_{M_6} = M_{63} = 9223372036854775807 = 7^2 \times 73 \times 127 \times 337 \times 92737 \times 649657$

$M_{M_7} = M_{127} = 170141183460469231731687303715884105727$

[edit] Double Mersenne primes

A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number M_n can be prime only if n is prime, (see Mersenne prime for a proof of this), a double Mersenne number $M_{M_n}$ can be prime only if M_n is itself a Mersenne prime. The first values of n for which M_n is prime are n = 2, 3, 5, 7, 13, 17, 19, 31. Of these, $M_{M_n}$ is known to be prime for n = 2, 3, 5, 7; for n = 13, 17, 19, and 31, explicit factors have been found showing that the corresponding double Mersenne numbers are not prime. If another double Mersenne prime is ever found, it would almost certainly be the largest known prime number. However, the smallest candidate is $M_{M_{61}}$ , or 2^{2305843009213693951}-1. At approximately 700 thousand trillion decimal digits, this number is far, far too big for any currently known test of primality.