Cn=n ⋅ 2n+1
n
In 1976 Christopher Hooley showed that the natural density of positive integers
n\leqx
x\toinfty
1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 .
Still, it is conjectured that there are infinitely many Cullen primes.
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.
Cp follows the recurrence relation
Cp=4(Cp-1+Cp-2)+1
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.[2]
As of October 2021, the largest known generalized Cullen prime is 2525532·732525532 + 1. 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). Thus, n·bn + 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·bn + 1 is prime, then b must be divisible by 3 (except b = 1).
The least n such that n·bn + 1 is prime (with question marks if this term is currently unknown) are[5] [6]
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, ...
| 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, ... |