Cullen number
From Wikipedia, the free encyclopedia
In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). Cullen numbers were first studied by Rev. James Cullen in 1905.
It has been shown that almost all Cullen numbers are composite; the only known Cullen primes are those for n = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899 and 1354828 (sequence A005849 in OEIS). Still, it is conjectured that there are infinitely many Cullen primes.
In August 2005, Mark Rodenkirch discovered the largest known Cullen prime: n = 1354828.
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.
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.
[edit] External links
- The Prime Glossary: Cullen number
- Eric W. Weisstein, Cullen number at MathWorld.
- Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGrid
[edit] References
- Cullen, James (1905). Question 15897. Educ. Times (December 1905), 534.
- Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B20.
- Wilfrid Keller, "New Cullen Primes", Mathematics of Computation, 64 (1995) 1733-1741.
