Cullen number
Mathematical concept
title: "Cullen number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integer-sequences", "unsolved-problems-in-number-theory", "classes-of-prime-numbers"] description: "Mathematical concept" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Cullen_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Mathematical concept ::
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·2n + 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 .
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) = (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 C**p is also prime.
Cn follows the recurrence relation :C_n=4(C_{n-1}-C_{n-2})+1.
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 4052186·694052186 + 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 (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, 4052186, 1, 13948, 1, 2525532, 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, ...
::data[format=table]
| 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, ... | |
| 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, ... | |
| :: |
References
References
- (2003). "Recurrence sequences". [[American Mathematical Society]].
- Marques, Diego. (2014). "On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers". Journal of Integer Sequences.
- (26 April 2025). "PrimeGrid Official Announcement".
- "PrimePage Primes: 4052186 · 69^4052186 + 1".
- Löh, Günter. (6 May 2017). "Generalized Cullen primes".
- Harvey, Steven. (6 May 2017). "List of generalized Cullen primes base 101 to 10000".
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::