Cullen number
Encyclopedia
In mathematics
, a Cullen number is a natural number
of the form n · 2n + 1 (written Cn). Cullen numbers were first studied by Fr. James Cullen
in 1905. Cullen numbers are special cases of Proth numbers.
showed that the natural density
of positive integers
for which Cn is a prime is of the order o(x) for 
. In that sense, almost all
Cullen numbers are composite
. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers, and in particular also for Woodall number
s. The only known Cullen primes are those for n equal:
Still, it is conjectured that there are infinitely many Cullen primes.
, the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime
with 2,010,852 digits and was discovered by a PrimeGrid
participant from Japan.
A Cullen number Cn is divisible by p = 2n − 1 if p is a prime number
of the form 8k - 3; furthermore, it follows from Fermat's little theorem
that if p is an odd prime, then p divides Cm(k) for each m(k) = (2k − k)
(p − 1) − k (for k > 0). It has also been shown that the prime number p divides C(p + 1) / 2 when the Jacobi symbol
(2 | p) is −1, and that p divides C(3p − 1) / 2 when the Jacobi symbol (2 | p) is +1.
It is unknown whether there exists a prime number p such that Cp is also 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 Cullen number is a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
of the form n · 2n + 1 (written Cn). Cullen numbers were first studied by Fr. James Cullen
James Cullen (mathematician)
Father James Cullen, S.J. was born at Drogheda, County Louth, Ireland.He studied mathematics at the Trinity College, Dublin for a while, but eventually turned to theology and was ordained as a Jesuit on 1 July 1901....
in 1905. Cullen numbers are special cases of Proth numbers.
Properties
In 1976 Christopher HooleyChristopher Hooley
Christopher Hooley FLSW FRS is a British mathematician, emeritus professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in 1973. He was elected a Fellow of the Royal Society in 1983...
showed that the natural density
Natural density
In number theory, asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is....
of positive integers
for which Cn is a prime is of the order o(x) for . In that sense, almost allAlmost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
Cullen numbers are composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers, and in particular also for Woodall number
Woodall number
In number theory, a Woodall number is any natural number of the formfor some natural number n. The first few Woodall numbers are:Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers...
s. The only known Cullen primes are those for n equal:
- 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 .
Still, it is conjectured that there are infinitely many Cullen primes.
, the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime
Megaprime
A megaprime is a prime number with at least one million decimal digits ., 35 megaprimes are known...
with 2,010,852 digits and was discovered by a PrimeGrid
PrimeGrid
PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...
participant from Japan.
A Cullen number Cn is divisible by p = 2n − 1 if p is a prime number
Prime 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...
of the form 8k - 3; furthermore, it follows from Fermat's little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
that if p is an odd prime, then p divides Cm(k) for each m(k) = (2k − k)
(p − 1) − k (for k > 0). It has also been shown that the prime number p divides C(p + 1) / 2 when the Jacobi symbol
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
(2 | p) is −1, and that p divides C(3p − 1) / 2 when the Jacobi symbol (2 | p) is +1.
It is unknown whether there exists a prime number p such that Cp is also prime.
Generalizations
Sometimes, a generalized Cullen number is defined to be a number of the form n · bn + 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Cullen prime. Woodall numbers are sometimes called Cullen numbers of the second kind.External links
- Chris Caldwell, The Top Twenty: Cullen primes at The Prime PagesPrime pagesThe Prime Pages 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" lists for primes of various forms...
. - The Prime Glossary: Cullen number at The Prime Pages.
- Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGridPrimeGridPrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...
- Paul Leyland, Generalized Cullen and Woodall Numbers