Double Mersenne number
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In mathematics, a double Mersenne number is a Mersenne number of the form
- <math>M_{M_n} = 2^{2^n-1}-1<math>
where n is a positive integer.
The first few double Mersenne numbers are
- <math>M_{M_1} = M_1 = 1 <math>
- <math>M_{M_2} = M_3 = 7 <math>
- <math>M_{M_3} = M_7 = 127 <math>
- <math>M_{M_4} = M_{15} = 32767 = 7 \times 31 \times 151 <math>
- <math>M_{M_5} = M_{31} = 2147483647 <math>
- <math>M_{M_6} = M_{63} = 9223372036854775807 = 7^2 \times 73 \times 127 \times 337 \times 92737 \times 649657 <math>
- <math>M_{M_7} = M_{127} = 7170141183460469231731687303715884105727 <math>
A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number Mn can be prime only if n is prime, (see Mersenne number for a proof of this), a double Mersenne number MMn can be prime only if Mn is prime. The first values of n for which Mn is prime are n = 2, 3, 5, 7, 13, 17, 19, 31. Of these, MMn is known to be prime for n = 2, 3, 5, 7; for n = 13, 17, 19, and 31, explicit factors have been found. If another double Mersenne prime is ever found, it would almost certainly be the largest known prime number.