Cullen number

In 1976 Christopher Hooley showed that the natural density of positive integers ${\displaystyle n\leq x}$ for which C_n is a prime is of the order o(x) for ${\displaystyle x\to \infty }$. In that sense, almost all Cullen numbers are composite.[1] 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

${\displaystyle C_{p}=4(C_{p-1}+C_{p-2})+1}$.

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 2525532·73^2525532 + 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 b^{p − 1} and b^{p − 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, ... (sequence A240234 in the OEIS)

• Cullen, James (December 1905), "Question 15897", Educ. Times: 534.
• Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, Section B20, ISBN 0-387-20860-7, Zbl 1058.11001.
• Hooley, Christopher (1976), Applications of sieve methods, Cambridge Tracts in Mathematics, vol. 70, Cambridge University Press, pp. 115–119, ISBN 0-521-20915-3, Zbl 0327.10044.
• Keller, Wilfrid (1995), "New Cullen Primes" (PDF), Mathematics of Computation, 64 (212): 1733–1741, S39–S46, doi:10.2307/2153382, ISSN 0025-5718, JSTOR 2153382, Zbl 0851.11003.

• 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.
• Weisstein, Eric W. "Cullen number". MathWorld.
• Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGrid
• Paul Leyland, (Generalized) Cullen and Woodall Numbers