Cullen number

 ( Integer Sequences ) 

In mathematics, a Cullen number is a member of the integer sequence $C\_n\; =\; n\; \backslash cdot\; 2^n\; +\; 1$ (where $n$ is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of .

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Properties In 1976 Christopher Hooley showed that the natural density of positive $n\; \backslash leq\; x$ for which C _n is a prime number is of the order o( x) for $x\; \backslash to\; \backslash infty$. In that sense, almost all Cullen numbers are composite number.

(2025). 9780821833872, American Mathematical Society. ISBN 9780821833872

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 . 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 that there are infinitely many Cullen primes.

A Cullen number C _n is divisibility by p = 2 n − 1 if p is a prime number of the form 8 k − 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_{(3 p − 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

$C\_p=4(C\_\{p-1\}+C\_\{p-2\})+1$.

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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. 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 (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, ...

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, ...

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Further reading

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External links

• Chris Caldwell, The Top Twenty: Cullen primes at The Prime Pages.
• The Prime Glossary: Cullen number at The Prime Pages.
• Chris Caldwell, The Top Twenty: Generalized Cullen at The Prime Pages.
• Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGrid
• Paul Leyland, (Generalized) Cullen and Woodall Numbers

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