Cullen number

In 1976 Christopher Hooley showed that the natural density of positive integers n \leq x for which Cn 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, sixteen in all, 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 Cn 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 Cm(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 Cp 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·b''n + 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 bn must be congruent to 1 mod p (since bn is a power of bp − 1 and bp − 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 :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, ... == References ==