# Cullen number

In mathematics, a Cullen number is a member of the integer sequence $C_{n}=n\cdot 2^{n}+1$ (where $n$ is a natural number). 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 $n\leq x$ for which C$_n$ is a prime is of the order o(x) for $x\to \infty$. 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:
    1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 (sequence A005849 in the OEIS).
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$_{m(k)}$ for each m(k) = (2$^k$ − 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 C$_p$ is also prime.
C$_p$ follows the recurrence relation
    $C_{p}=4(C_{p-1}+C_{p-2})+1$.

## Generalizations

Sometimes, a generalized Cullen number base b is defined to be a number of the form n·b$^n$ + 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 October 2021, the largest known generalized Cullen prime is 2525532·73$^{2525532}$ + 1. It has 4,705,888 digits and was discovered by Tom Greer, 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 (especially, when n = p − 1) and p does not divide b, then b$^n$ must be congruent to 1 mod p (since b$^n$ is a power of b$^{p−1}$ and b$^{p−1}$ is congruent to 1 mod p). Thus, n·b$^n$ + 1 is divisible by p, so it is not prime. For example, if some n congruent to 2 mod 6 (i.e. 2, 8, 14, 20, 26, 32, ...), n·b$^n$ + 1 is prime, then b must be divisible by 3 (except b = 1).
The least n such that n·b$^n$ + 1 is prime (with question marks if this term is currently unknown) are
    1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, 2805222, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, 1806676, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, 1341174, 3, ?, 1072, 234, 1, 220, 1, 142, 1295, 8, 3, 16990, 1, 474, 129897, ?, 1, 13948, 1, ?, 3, 2, 1161, 12198, 1, 682156, 5, 350, 1, 1242, 26, 186, 3, 2, 1, 298, 14, 101670, 9, 2, 775, 202, 1, 1374, 63, 2, 1, ... (sequence A240234 in the OEIS)
| b | Numbers n such that n × b$^n$ + 1 is prime | OEIS sequence |
| 3 | 2, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, 1183414, ... | A006552 |
| 4 | 1, 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375, 1740349, ... | A007646 |
| 5 | 1242, 18390, ... | |
| 6 | 1, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496, 515516, ..., 4582770 | A242176 |
| 7 | 34, 1980, 9898, 474280, ... | A242177 |
| 8 | 5, 17, 23, 1911, 20855, 35945, 42816, ..., 749130, ... | A242178 |
| 9 | 2, 12382, 27608, 31330, 117852, ... | A265013 |
| 10 | 1, 3, 9, 21, 363, 2161, 4839, 49521, 105994, 207777, ... | A007647 |
| 11 | 10, ... | |
| 12 | 1, 8, 247, 3610, 4775, 19789, 187895, 345951, ... | A242196 |
| 13 | ... | |
| 14 | 3, 5, 6, 9, 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545, 1198433, 1486287, 1909683, ... | A242197 |
| 15 | 8, 14, 44, 154, 274, 694, 17426, 59430, ... | A242198 |
| 16 | 1, 3, 55, 81, 223, 1227, 3012, 3301, ... | A242199 |
| 17 | 19650, 236418, ... | |
| 18 | 1, 3, 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928, 612497, ... | A007648 |
| 19 | 6460, ... | |
| 20 | 3, 6207, 8076, 22356, 151456, 793181, 993149, ... | A338412 |