In 1976 Christopher Hooley 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 numbers. 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 (sequence A005849 in 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.
For b = 1 to 12, the least n such that n · bn + 1 is prime are
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- "The Prime Database: 6679881*2^6679881+1", Chris Caldwell's The Largest Known Primes Database, retrieved December 22, 2009
- "The Prime Database: 427194 · 113^427194 + 1", Chris Caldwell's The Largest Known Primes Database, retrieved January 30, 2012
- Cullen, James (December 1905), "Question 15897", Educ. Times: 534.
- Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, Section B20, ISBN 0-387-20860-7, Zbl 1058.11001.
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- Chris Caldwell, The Top Twenty: Cullen primes at The Prime Pages.
- The Prime Glossary: Cullen number at The Prime Pages.
- Weisstein, Eric W., "Cullen number", MathWorld.
- Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGrid
- Paul Leyland, Generalized Cullen and Woodall Numbers