# Cullen number

In mathematics, a Cullen number is a natural number of the form $n \cdot 2^n + 1$ (written $C_n$). Cullen numbers were first studied by Fr. James Cullen in 1905. Cullen numbers are special cases of Proth numbers.

## Properties

In 1976 Christopher Hooley showed that the natural density of positive integers $n \leq x$ for which Cn is a prime is of the order o(x) for $x\to\infty$. In that sense, almost all Cullen numbers are composite.[1] 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.

As of August 2009, 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.[2]

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.

## 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.

As of February 2012, the largest known generalized Cullen prime is 427194 × 113 427194 + 1. It has 877,069 digits and was discovered by a PrimeGrid participant from United States.[3]

## References

1. ^ 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.
2. ^ "The Prime Database: 6679881*2^6679881+1", Chris Caldwell's The Largest Known Primes Database, retrieved December 22, 2009
3. ^ "The Prime Database: 427194 · 113^427194 + 1", Chris Caldwell's The Largest Known Primes Database, retrieved January 30, 2012