Cullen number

In mathematics, a Cullen number is a member of the integer sequence . Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.

Properties

In 1976 Christopher Hooley showed that the natural density of positive integers for which C_n is a prime is of the order o 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·2^{n + 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 to:
Still, it is conjectured that there are infinitely many Cullen primes.
A Cullen number C_n 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 C_m for each m =
− k. It has also been shown that the prime number p divides C_/2 when the Jacobi symbol is −1, and that p divides C_/2 when the Jacobi symbol is + 1.
It is unknown whether there exists a prime number p such that C_p is also prime.
C_n follows the recurrence relation

Generalizations

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. Woodall numbers are sometimes called Cullen numbers of the second kind.
As of April 2025, the largest known generalized Cullen prime is 4052186·69^4052186 + 1. It has 7,451,366 digits and was discovered by a PrimeGrid participant.
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 and p does not divide b, then bⁿ must be congruent to 1 mod p. Thus, n·''bn'' + 1 is divisible by p, so it is not prime. For example, if some n congruent to 2 mod 6, n·''bn'' + 1 is prime, then b must be divisible by 3.
The least n such that n·''bn'' + 1 is prime are

b	Numbers n such that n × bⁿ + 1 is prime	OEIS sequence
3	2, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, 1183414,...
4	1, 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375, 1740349,...
5	1242, 18390,...
6	1, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496, 515516,..., 4582770
7	34, 1980, 9898, 474280,...
8	5, 17, 23, 1911, 20855, 35945, 42816,..., 749130,...
9	2, 12382, 27608, 31330, 117852,...
10	1, 3, 9, 21, 363, 2161, 4839, 49521, 105994, 207777,...
11	10,...
12	1, 8, 247, 3610, 4775, 19789, 187895, 345951,...
13	...
14	3, 5, 6, 9, 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545, 1198433, 1486287, 1909683,...
15	8, 14, 44, 154, 274, 694, 17426, 59430,...
16	1, 3, 55, 81, 223, 1227, 3012, 3301,...
17	19650, 236418,...
18	1, 3, 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928, 612497,...
19	6460,...
20	3, 6207, 8076, 22356, 151456, 793181, 993149,...