Double Mersenne number
Encyclopedia
In mathematics
, a double Mersenne number is a Mersenne number
of the form
where p is a Mersenne prime exponent.
of double Mersenne numbers begins
.
is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime
for a proof), a double Mersenne number
can be prime only if Mp is itself a Mersenne prime. The first values of p for which Mp is prime are p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89. 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 22305843009213693951 − 1.
Being approximately 1.695,
this number is far too large for any currently known primality test
. It has no prime factor below 4×1033.
instead of 
. A special case of the double Mersenne numbers, namely the recursively
defined sequence
is called the Catalan–Mersenne numbers. It is said that Catalan
came up with this sequence after the discovery of the primality of
by Lucas
in 1876.
Although the first five terms (up to
) 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 a factor of M(M(127)) is discovered.
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".
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a double Mersenne number is a Mersenne number
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
of the form
where p is a Mersenne prime exponent.
The smallest double Mersenne numbers
The sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of double Mersenne numbers begins
.Double Mersenne primes
A double Mersenne number that is primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
for a proof), a double Mersenne number
can be prime only if Mp is itself a Mersenne prime. The first values of p for which Mp is prime are p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89. 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 22305843009213693951 − 1.Being approximately 1.695,
this number is far too large for any currently known primality test
Primality test
A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not...
. It has no prime factor below 4×1033.
Catalan–Mersenne number
Write instead of . A special case of the double Mersenne numbers, namely the recursivelyRecursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
defined sequence
- 2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), ...
is called the Catalan–Mersenne numbers. It is said that Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...
came up with this sequence after the discovery of the primality of
by LucasEdouard Lucas
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.-Biography:...
in 1876.
Although the first five terms (up to
) 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 a factor of M(M(127)) is discovered.In popular culture
In the FuturamaFuturama
Futurama is an American animated science fiction sitcom created by Matt Groening and developed by Groening and David X. Cohen for the Fox Broadcasting Company. The series follows the adventures of a late 20th-century New York City pizza delivery boy, Philip J...
movie The Beast with a Billion Backs
Futurama: The Beast with a Billion Backs
Futurama: The Beast with a Billion Backs is an animated science-fiction comedy film, the second of the four Futurama straight-to-DVD films. The film was released in the United States and Canada on June 24, 2008, followed by a UK release on June 30, 2008 and an Australian release on August 6, 2008....
, 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
- Perfect numberPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
- Fermat number
- Wieferich primeWieferich primeIn number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1...
- Double exponential function
External links
- Tony Forbes, A search for a factor of MM61.