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 (Sloane's A005849). Still, it is conjectured that there are infinitely many Cullen primes.

A Cullen number *C _{n}* is divisible by

It is unknown whether there exists a prime number *p* such that *C*_{p} 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**.