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In [[mathematics]], a '''double Mersenne number''' is a [[Mersenne prime|Mersenne number]] of the form
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:<math>M_{M_p} = 2^{2^p-1}-1</math>
 
where ''p'' is a Mersenne prime exponent.
 
== The smallest double Mersenne numbers ==
 
The [[sequence]] of double Mersenne numbers begins <ref name="Caldwell">Chris Caldwell, [http://primes.utm.edu/mersenne/index.html#unknown ''Mersenne Primes: History, Theorems and Lists''] at the [[Prime Pages]].</ref>
:<math>M_{M_2} = M_3 = 7 </math>
:<math>M_{M_3} = M_7 = 127 </math>
:<math>M_{M_5} = M_{31} = 2147483647 </math>
:<math>M_{M_7} = M_{127} = 170141183460469231731687303715884105727 </math> {{OEIS|id=A077586}}.
 
== Double Mersenne primes ==
 
A double Mersenne number that is [[prime number|prime]] is called a '''double Mersenne prime'''. Since a Mersenne number ''M''<sub>''p''</sub> can be prime only if ''p'' is prime, (see [[Mersenne prime]] for a proof), a double Mersenne number <math>M_{M_p}</math> can be prime only if ''M''<sub>''p''</sub> is itself a Mersenne prime. The first values of ''p'' for which ''M''<sub>''p''</sub> is prime are ''p'' = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Of these, <math>M_{M_p}</math> 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. Thus, the smallest candidate for the next double Mersenne prime is <math>M_{M_{61}}</math>, or 2<sup>2305843009213693951</sup> − 1.
Being approximately 1.695{{e|694127911065419641}},
this number is far too large for any currently known [[primality test]]. It has no prime factor below 4×10<sup>33</sup>.<ref>Tony Forbes, [http://anthony.d.forbes.googlepages.com/mm61prog.htm A search for a factor of MM61. Progress: 9 October 2008]. This reports a high-water mark of 204204000000×(10019+1)×(2<sup>61</sup>−1), above 4×10<sup>33</sup>. Retrieved on 2008-10-22.</ref> There are probably no other double Mersenne primes than the four known.<ref name="Caldwell"/><ref>[http://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071444-6/S0025-5718-1955-0071444-6.pdf I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121] [retrieved 2012-10-19]</ref>
 
== The Catalan–Mersenne number conjecture==
 
Write <math>M(p)</math> instead of <math>M_p</math>. A special case of the double Mersenne numbers, namely the [[Recursion|recursively]] defined sequence
: ''2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), ...'' {{OEIS|id=A007013}}
is called the '''Catalan–Mersenne numbers'''.<ref>{{MathWorld|urlname=Catalan-MersenneNumber|title=Catalan-Mersenne Number}}</ref> It is said<ref name="Caldwell"/> that [[Eugène Charles Catalan|Catalan]] came up with this sequence after the discovery of the primality of M(127)=M(M(M(M(2)))) by [[Édouard Lucas|Lucas]] in 1876.<ref>''Nouvelle correspondance mathématique'' vol. 2 (1876), p. 94-96, "Questions proposées" probably collected by the editor. Almost all of the questions are signed by Édouard Lucas as is number 92: "Prouver que 2<sup>61</sup> - 1 et 2<sup>127</sup> - 1 sont des nombres premiers. (É. L.) (*)." The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows: "(*) Si l'on admet ces deux propositions, et si l'on observe que 2<sup>2</sup> - 1, 2<sup>3</sup> - 1, 2<sup>7</sup> - 1 sont aussi des nombres premiers, on a ce ''théorème empirique: Jusqu'à une certaine limite, si'' 2<sup>n</sup> - 1 ''est un nombre premiere'' p, 2<sup>p</sup> - 1 ''est une nombre premiere'' p', 2<sup>p'</sup> - 1 ''est une nombre premiere'' p", etc. Cette proposition a quelque analogie avec le théorème suviant, énoncé par Fermat, et dont Euler a montré l'inexactitude: ''Si n est une puissance de 2, 2<sup>n</sup> + 1 est une nombre premiere.'' (E. C.)" http://archive.org/stream/nouvellecorresp01mansgoog#page/n353/mode/2up [retrieved 2012-10-18]</ref> Catalan conjectured that they, up to a certain limit, are all prime.{{clarify|date=December 2012}}
 
Although the first five terms (up to <math>M(127)</math>) 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, unless the primality of M(M(127)) is disproved.
 
==In popular culture==
In the [[Futurama]] movie [[Futurama: The Beast with a Billion Backs|''The Beast with a Billion Backs'']], the double Mersenne number <math>M_{M_7}</math> is briefly seen in "an elementary proof of the [[Goldbach conjecture]]". In the movie, this number is known as a "martian prime".
 
==See also==
* [[Perfect number]]
* [[Fermat number]]
* [[Wieferich prime]]
* [[Double exponential function]]
 
== References ==
{{Reflist}}
 
==Further reading==
*{{Citation |authorlink=L. E. Dickson |last=Dickson |first=L. E. |title=[[History of the Theory of Numbers]] |origyear=1919 |publisher=Chelsea Publishing |location=New York |year=1971 |isbn= }}.
 
== External links ==
* {{MathWorld|urlname=DoubleMersenneNumber|title=Double Mersenne Number}}
* Tony Forbes, [http://anthony.d.forbes.googlepages.com/mm61.htm A search for a factor of MM61].
* [http://www.garlic.com/~wedgingt/MMPstats.txt Status of the factorization of double Mersenne numbers]
* [http://www.doublemersennes.org Double Mersennes Prime Search]
* [http://www.mersenneforum.org/forumdisplay.php?f=99 Operazione Doppi Mersennes]
 
{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}
 
[[Category:Integer sequences]]
[[Category:Large integers]]

Latest revision as of 13:34, 8 December 2014

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