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In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.


Examples

The first four terms of the sequence of double Mersenne numbers areChris Caldwell
''Mersenne Primes: History, Theorems and Lists''
at the
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
.
: :M_ = M_3 = 7 :M_ = M_7 = 127 :M_ = M_ = 2147483647 :M_ = M_ = 170141183460469231731687303715884105727


Double Mersenne primes

A double Mersenne number that is
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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 M_ can be prime only if ''M''''p'' is itself a Mersenne prime. For the first values of ''p'' for which ''M''''p'' is prime, M_ is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of M_ have been found for ''p'' = 13, 17, 19, and 31. Thus, the smallest candidate for the next double Mersenne prime is M_, or 22305843009213693951 − 1. Being approximately 1.695, this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036. There are probably no other double Mersenne primes than the four known. Smallest prime factor of M_ (where ''p'' is the ''n''th prime) are :7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 1036)


Catalan–Mersenne number conjecture

The
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
defined sequence : c_0 = 2 : c_ = 2^-1 = M_ is called the sequence of Catalan–Mersenne numbers. The first terms of the sequence are: :c_0 = 2 :c_1 = 2^2-1 = 3 :c_2 = 2^3-1 = 7 :c_3 = 2^7-1 = 127 :c_4 = 2^-1 = 170141183460469231731687303715884105727 :c_5 = 2^-1 \approx 5.454 \times 10^ \approx 10^
Catalan Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #1 ...
discovered this sequence after the discovery of the primality of M_=c_4 by Lucas in 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92: The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows: Catalan
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
d that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if c_5 is not prime, there is a chance to discover this by computing c_5
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
some small prime p (using recursive
modular exponentiation Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular ...
). If the resulting residue is zero, p represents a factor of c_5 and thus would disprove its primality. Since c_5 is a Mersenne number, such a prime factor p would have to be of the form 2kc_4 +1. Additionally, because 2^n-1 is
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
when n is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.


In popular culture

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


See also

*
Cunningham chain In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes. Definition A Cunningham chain of the first ki ...
* Double exponential function * Fermat number * Perfect number * Wieferich prime


References


Further reading

*.


External links

* * Tony Forbes
A search for a factor of MM61

Status of the factorization of double Mersenne numbers

Double Mersennes Prime Search

Operazione Doppi Mersennes
{{Mersenne Integer sequences Large integers Unsolved problems in number theory Mersenne primes