NationMaster - Encyclopedia: Double Mersenne number

In mathematics, a double Mersenne number is a Mersenne number of the form Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ...

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

where n is a positive integer. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

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

The smallest double Mersenne numbers

The sequence of double Mersenne numbers (sequence A077585 in OEIS) begins The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

$M_{M_1} = M_1 = 1$

$M_{M_2} = M_3 = 7$

$M_{M_3} = M_7 = 127$

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

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

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

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

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 7 x 10¹⁷ decimal digits, this number is far, far too big for any currently known test of primality. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ...

External links

A project to factor the double-Mersenne number $M_{M_{61}}$
Eric W. Weisstein, Double Mersenne Number at MathWorld.

Category: Integer sequences Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

What's Special About This Number? (7255 words)
	is the number of planar partitions of 10.
	is the number of planar partitions of 11.
	is the number of planar partitions of 12.

Mersenne prime - Wikipedia, the free encyclopedia (768 words)
	Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors.
	Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number.
	The best method presently known for testing the primality of Mersenne numbers is based on the computation of a recurring sequence, as developed originally by Lucas in 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test.

More results at FactBites »

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Contents

The smallest double Mersenne numbers GA_googleFillSlot("encyclopedia_square");

Double Mersenne primes

See also

External links

The smallest double Mersenne numbers