# Double Mersenne number

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

where *p* is a Mersenne prime exponent.

## The smallest double Mersenne numbers

The first four terms of the sequence of double Mersenne numbers are^{[1]} (sequence A077586 in OEIS):

## Double Mersenne primes

A double Mersenne number that is prime is called a **double Mersenne prime**. Since a Mersenne number *M*_{p} can be prime only if *p* is prime, (see Mersenne prime for a proof), a double Mersenne number can be prime only if *M*_{p} is itself a Mersenne prime. The first values of *p* for which *M*_{p} is prime are *p* = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Of these, 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 , or 2^{2305843009213693951} − 1.
Being approximately 1.695Template:E,
this number is far too large for any currently known primality test. It has no prime factor below 4×10^{33}.^{[2]} There are probably no other double Mersenne primes than the four known.^{[1]}^{[3]}

## The Catalan–Mersenne number conjecture

Write instead of . A special case of the double Mersenne numbers, namely the recursively defined sequence

is called the **Catalan–Mersenne numbers**.^{[4]} It is said^{[1]} that Catalan came up with this sequence after the discovery of the primality of*M*(127) = *M*(*M*(*M*(*M*(2)))) by Lucas in 1876.^{[5]} Catalan conjectured that they are all prime and that "up to a certain limit," the sequences defined in the same way starting at any Mersenne number are composed only of primes. This limit is now known to be at most 13, because *M*_{M13} is not prime.

Although the first five terms (up to *M*_{127}) 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*_{M127} is disproved.

## In popular culture

In the Futurama movie *The Beast with a Billion Backs*, the double Mersenne number is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".

## See also

## References

- ↑
^{1.0}^{1.1}^{1.2}Chris Caldwell,*Mersenne Primes: History, Theorems and Lists*at the Prime Pages. - ↑ Tony Forbes, A search for a factor of MM61. Progress: 9 October 2008. This reports a high-water mark of 204204000000×(10019 + 1)×(2
^{61}− 1), above 4×10^{33}. Retrieved on 2008-10-22. - ↑ I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
- ↑ Weisstein, Eric W., "Catalan-Mersenne Number",
*MathWorld*. - ↑
*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^{61}− 1 et 2^{127}− 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^{2}− 1, 2^{3}− 1, 2^{7}− 1 sont aussi des nombres premiers, on a ce*théorème empirique: Jusqu'à une certaine limite, si*2^{n}− 1*est un nombre premier**p*, 2^{p}− 1*est un nombre premier**p*', 2^{p'}− 1*est un nombre premier*p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude:*Si n est une puissance de 2, 2*(E. C.)" http://archive.org/stream/nouvellecorresp01mansgoog#page/n353/mode/2up [retrieved 2012-10-18]^{n}+ 1 est un nombre premier.

## Further reading

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## External links

- Weisstein, Eric W., "Double Mersenne Number",
*MathWorld*. - Tony Forbes, A search for a factor of MM61.
- Status of the factorization of double Mersenne numbers
- Double Mersennes Prime Search
- Operazione Doppi Mersennes

Template:Prime number classes Template:Classes of natural numbers Template:Mersenne