Cullen number
|
|
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, and 481899 Template:OEIS. Still, it is conjectured that there are infinitely many Cullen primes.
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.
External links
- The Prime Glossary: Cullen number (http://primes.utm.edu/glossary/page.php?sort=Cullens)
- MathWorld: Cullen number (http://mathworld.wolfram.com/CullenNumber.html)
References
- Cullen, James (1905). Question 15897. Educ. Times (December 1905), 534.de:Cullen-Zahl