Cullen number

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In mathematics, a Cullen number is a natural number of the form n · 2ⁿ + 1 (written C_n). Cullen numbers were first studied by Rev. James Cullen in 1905.

It has been shown that almost all Cullen numbers are composite; the only known Cullen primes are those for n = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899 and 1354828 (sequence A005849 in OEIS). Still, it is conjectured that there are infinitely many Cullen primes.

In August 2005, Mark Rodenkirch discovered the largest known Cullen prime: $n = 1354828$

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.

Sometimes, a generalized Cullen number 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.