In
mathematics, a double Mersenne number is a
Mersenne number
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
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
conjectured 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 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.
Modul ...
). If the resulting residue is zero,
represents a factor of
and thus would disprove its primality. Since
is a
Mersenne number
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
, 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 ...
*
Double exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b ...
*
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 42949672 ...
*
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...
*
Wieferich prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...
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