In 1905, the Reverend Cullen was interested in the numbers n.2n+1 (denoted Cn). He noticed that the first, C1=3, was prime, but with the possible exception of the 53rd, the next 99 were all composite. Very soon afterwards, Cunningham discovered that 5591 divides C53, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141. Five decades later Robinson showed C141 was a prime.
These numbers are now called the Cullen numbers. Sometimes, the name "Cullen number" is extended to also include the Woodall numbers: Wn=n.2n-1 (then these are then the "Cullen primes of the second kind").
A Cullen prime is any prime of the form Cn. The only known Cullen primes Cn are those with n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, and 6679881.
It has been shown that almost all Cullen numbers Cn are composite! Fermat's little theorem tells us if p is an odd prime, then p divides both Cp-1, Cp-2 (and more generally, Cm(k) for each m(k) = (2k-k)(p-1)-k, k > 0). It has also been shown that the prime p divides C(p+1)/2 whenever the Jacobi symbol (2|p) is -1, and p divides C(3p-1)/2 whenever the Jacobi symbol (2|p) is +1.
Still it has been conjectured that there are infinitely many Cullen primes Cn, and it is not yet known if n and Cn can be simultaneously prime.
Finally, a few authors have defined a number of the form n.bn+1 with n+2 > b, to be a generalized Cullen number, so any prime that can be written in this form could be called a generalized Cullen prime. We emphasize can be because at first glance neither of the following have the correct form:
669.2128454+1, 755.248323+1But these two primes may be written as follows:
42816.842816+1 and 6040.2566040+1 (respectively).
Related pages (outside of this work)