In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
double Mersenne number is a
Mersenne numberIn mathematics, a Mersenne number is a positive integer that is one less than a power of two:Some definitions of Mersenne numbers require that the exponent p be prime....
of the form
where
p is a Mersenne prime exponent.
The
sequenceIn 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...
of double Mersenne numbers begins .
A double Mersenne number that is
primeIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
is called a
double Mersenne prime. Since a Mersenne number
Mp can be prime only if
p is prime, (see
Mersenne primeIn mathematics, a Mersenne number is a positive integer that is one less than a power of two:Some definitions of Mersenne numbers require that the exponent p be prime....
for a proof), a double Mersenne number can be prime only if
Mp is itself a Mersenne prime.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
double Mersenne number is a
Mersenne numberIn mathematics, a Mersenne number is a positive integer that is one less than a power of two:Some definitions of Mersenne numbers require that the exponent p be prime....
of the form
where
p is a Mersenne prime exponent.
The smallest double Mersenne numbers
The
sequenceIn 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...
of double Mersenne numbers begins .
Double Mersenne primes
A double Mersenne number that is
primeIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
is called a
double Mersenne prime. Since a Mersenne number
Mp can be prime only if
p is prime, (see
Mersenne primeIn mathematics, a Mersenne number is a positive integer that is one less than a power of two:Some definitions of Mersenne numbers require that the exponent p be 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 2
2305843009213693951 − 1. At approximately 6.94 (694127911065419642) decimal digits, this number is far too large for any currently known
primality testA primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. The difference between this and integer factorization is that a primality test doesn't necessarily give prime factors , while integer factorization does...
. It has no prime factor below 4×10
33.
Catalan-Mersenne number
Write instead of . A special case of the double Mersenne numbers, namely the
recursivelyRecursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way...
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
CatalanEugène 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
LucasFrançois Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequence is named after him. He gave a formula for finding the nth term of the Fibonacci sequence.Lucas was educated at the École Normale Supérieure...
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.
In popular culture
In the
FuturamaFuturama is an animated American sci-fi sitcom created by Matt Groening and developed by Groening and David X. Cohen for the Fox network. The series follows the adventures of a late 20th-century New York City pizza delivery boy, Philip J...
movie
The Beast with a Billion BacksFuturama: The Beast with a Billion Backs is an animated science-fiction comedy, the second of the four Futurama straight-to-DVD films. The film was released in the USA and Canada on June 24, 2008, followed by a UK release on June 30, 2008 and an Australian release on August 6, 2008. It has been...
, 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 number
In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself...
- Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the formwhere n is a nonnegative integer. The first ten Fermat numbers are :...
- Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1; compare this with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1...
- Double exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows even faster than an exponential function...
External links