In
mathematics, a double Mersenne number is a
Mersenne number of the form
:
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 are
[Chris 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" ...
. :
:
:
:
:
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
can be prime only if ''M''
''p'' is itself a Mersenne prime. For the first values of ''p'' for which ''M''
''p'' is prime,
is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of
have been found for ''p'' = 13, 17, 19, and 31.
Thus, the smallest candidate for the next double Mersenne prime is
, or 2
2305843009213693951 − 1.
Being approximately 1.695,
this number is far too large for any currently known
primality test. It has no prime factor below 1 × 10
36.
There are probably no other double Mersenne primes than the four known.
Smallest prime factor of
(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 × 10
36)
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
:
:
is called the sequence of Catalan–Mersenne numbers. The first terms of the sequence are:
:
:
:
:
:
:
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
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
is not prime, there is a chance to discover this by computing
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
(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,
represents a factor of
and thus would disprove its primality. Since
is a
Mersenne number, such a prime factor
would have to be of the form
. Additionally, because
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
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
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 numbersDouble Mersennes Prime SearchOperazione Doppi Mersennes
{{Mersenne
Integer sequences
Large integers
Unsolved problems in number theory
Mersenne primes