Cullen number

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

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