Properties
In 1976 Christopher Hooley showed that the natural density of positive integers for which Cn is a prime is of the order o(x) for . 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 are those for n equal:
- 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 (sequence A005849 in OEIS).
Still, it is conjectured that there are infinitely many Cullen primes.
As of August 2009, the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime with 2,010,852 digits and was discovered by a PrimeGrid participant from Japan.
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.
Read more about this topic: Cullen Number
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)