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Double Mersenne Number


A double Mersenne number is a number of the form

 M_(M_n)=2^(2^n-1)-1,

where M_n is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).

A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime M_n can be prime only for prime n, a double Mersenne prime can be prime only for prime M_n, i.e., M_n a Mersenne prime. Double Mersenne numbers are prime for n=2, 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).

The next four M_(M_(13)), M_(M_(17)), M_(M_(19)), and M_(M_(31)) have known factors summarized in the following table. The status of all other double Mersenne numbers is unknown, with M_(M_(61)) being the smallest unresolved case. Since this number has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer test to be practical. The only possible method of determining the status of this number is therefore attempting to find small divisors (or discovery of an efficient primality test for this type of number). T. Forbes has organized a distributed search, but thus no factors have been found although about 80% of the trial divisors up to 8600·204204000000-1 have been checked. Edgington maintains a list of known factorizations of double Mersenne numbers.

nfactorsreference
13338193759479, C2455Wilfrid Keller (1976)
17231733529Raphael Robinson (1957)
1962914441Raphael Robinson (1957)
31295257526626031Guy Haworth (1983, 1987)
87054709261955177Keller (1994)
242557615644693265201Keiser and Forbes (1999)
178021379228511215367151Mayer (2005)

See also

Catalan-Mersenne Number, Mersenne Number, Mersenne Prime

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References

Edgington, W. "Will Edgington's Mersenne Page." http://www.garlic.com/~wedgingt/mersenne.html.Edgington, W. "Status of M(M(p)) where M(p) is a Mersenne Prime." http://anthony.d.forbes.googlepages.com/mm61prog.htm.Forbes, T. "MM61: A Search for a Factor of 2^(2^(61)-1)-1." http://anthony.d.forbes.googlepages.com/mm61.htm.Forbes, T. "MM61: A Search for a Factor of 2^(2^(61)-1)-1. Progress: 2 March 2004." http://www.ltkz.demon.co.uk/ar2/mm61prog.htm.Haworth, G. M. Notes on Mersenne Numbers. Privately produced manuscript, 1987.Mayer, E. W. "Fourth Known Factor of M(M31)." 21 Jun 2005. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0506&L=nmbrthry&T=0&F=&S=&P=2514.Sloane, N. J. A. Sequences A077585 and A077586 in "The On-Line Encyclopedia of Integer Sequences."

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Double Mersenne Number

Cite this as:

Weisstein, Eric W. "Double Mersenne Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DoubleMersenneNumber.html

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