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.
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.
- Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.
- "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
Further reading 
- 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.
- Hooley, Christopher (1976), Applications of sieve methods, Cambridge Tracts in Mathematics 70, Cambridge University Press, pp. 115–119, ISBN 0-521-20915-3, Zbl 0327.10044.
- Keller, Wilfrid (1995), "New Cullen Primes", Mathematics of Computation 64 (212): 1733–1741,S39–S46, ISSN 0025-5718, Zbl 0851.11003.
- 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