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 called "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)
- J. Cullen, "Question 15897," Educ. Times, (December 1905) 534. [Originated the study of Cullen numbers. See also [CW17].]
- A. J. C. Cunningham and H. J. Woodall, "Factorisation of Q=(2q ± q) and q*2q ± 1," Math. Mag., 47 (1917) 1--38. [A classic paper in the history of the study of Cullen numbers. See also [Keller95]]
- Guy94 (Section B20)
- R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, 1994. ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
- C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Math. Vol, 70, Cambridge University Press, Cambridge, 1976. pp. xiv+122, MR 53:7976
- W. Keller, "Factors of Fermat numbers and large primes of the form k· 2n +1," Math. Comp., 41 (1983) 661-673. MR 85b:11117
- W. Keller, "New Cullen primes," Math. Comp., 64 (1995) 1733-1741. Supplement S39-S46. MR 95m:11015
- Ribenboim95 (p. 360-361)
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
- R. P. Steiner, "On Cullen numbers," BIT, 19:2 (1979) 276-277. MR 80j:10009