# Cullen prime

In 1905, the Reverend Cullen was interested in the numbers *n*^{.}2^{n}+1 (denoted C_{n}). He noticed that the first, C_{1}=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 C_{53}, 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 C_{141} was a prime.

These numbers are now called the **Cullen numbers**. Sometimes, the name "Cullen number" is extended to also include the Woodall numbers: W_{n}=*n*^{.}2^{n}-1 (then these are called "Cullen primes *of the second kind*").

A **Cullen prime** is any prime of the form C_{n}. The only known Cullen primes C_{n} 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 C_{n} are composite! Fermat's little theorem tells us if *p* is an odd prime, then
*p* divides both C_{p-1},
C_{p-2} (and more generally,
C_{m(k)} for each *m*(*k*) =
(2^{k}-*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 C_{n}, and it is not yet known if *n* and C_{n} can
be simultaneously prime.

Finally, a few authors have defined a number of
the form *n*^{.}*b*^{n}+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:

669But these two primes may be written as follows:^{.}2^{128454}+1, 755^{.}2^{48323}+1

42816^{.}8^{42816}+1 and 6040^{.}256^{6040}+1 (respectively).

**See Also:** WoodallNumber, Fermats, Mersennes

**Related pages** (outside of this work)

**References:**

- Cullen05
J. Cullen, "Question 15897,"Educ. Times, (December 1905) 534. [Originated the study of Cullen numbers. See also [CW17].]- CW17
A. J. C. CunninghamandH. J. Woodall, "Factorisation ofQ=(2^{q}±q) andq*2^{q}± 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.]- Hooley76
C. Hooley,Applications of sieve methods to the theory of numbers, Cambridge Tracts in Math. Vol, 70, Cambridge University Press, 1976. Cambridge, pp. xiv+122,MR 53:7976- Keller83
W. Keller, "Factors of Fermat numbers and large primes of the formk· 2^{n}+1,"Math. Comp.,41(1983) 661-673.MR 85b:11117- Keller95
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.]- Steiner79
R. P. Steiner, "On Cullen numbers,"BIT,19:2 (1979) 276-277.MR 80j:10009