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Cullen number

In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). Cullen numbers were first studied by J. 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, and 481899 (Sloane's A005849). 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.

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.

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