Cullen number

Properties

In 1976 Christopher Hooley showed that the natural density of positive integers n ≤ x for which C_n is a prime is of the order o(x) for x → ∞ . 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 · 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:

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.

As of February 2016, the largest known Cullen prime is 6679881 × 2^6679881 + 1. It is a megaprime with 2,010,852 digits and was discovered by a PrimeGrid participant from Japan.

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.

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.

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).

Least n such that n × bⁿ + 1 is prime are (with question marks if this term is currently unknown)

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, ?, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, ?, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, ?, 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)

As of October 2016, the largest known generalized Cullen prime is 682156 × 79⁶⁸²¹⁵⁶ + 1. It has 1,294,484 digits and was discovered by a PrimeGrid participant from Austria.