In mathematics, a Cullen number is a member of the integer sequence
In 1976 Christopher Hooley showed that the natural density of positive integers
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).
Cp follows the recurrence relation Sometimes, a generalized Cullen number base b 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.
It has 4,705,888 digits and was discovered by Tom Greer, a PrimeGrid participant.
[3][4] According to Fermat's little theorem, if there is a prime p such that n is divisible by p − 1 and n + 1 is divisible by p (especially, when n = p − 1) and p does not divide b, then bn must be congruent to 1 mod p (since bn is a power of bp − 1 and bp − 1 is congruent to 1 mod p).