Cullen prime

In 1905, the Reverend Cullen was interested in the numbers n^.2ⁿ+1 (denoted C_n).  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: W_n=n^.2ⁿ-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^.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+1

But these two primes may be written as follows:

42816^.8⁴²⁸¹⁶+1   and   6040^.256⁶⁰⁴⁰+1   (respectively).

See Also: WoodallNumber, Fermats, Mersennes

Related pages (outside of this work)

• Cullen prime search status page
• The twenty largest Cullen primes

References:

Cullen05

J. Cullen, "Question 15897," Educ. Times, (December 1905) 534. [Originated the study of Cullen numbers. See also [CW17].]

CW17

A. J. C. Cunningham and H. J. Woodall, "Factorisation of Q=(2^q ± q) and q*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, 1994.  New York, NY, 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, Cambridge, 1976.  pp. xiv+122, MR 53:7976

Keller83

W. Keller, "Factors of Fermat numbers and large primes of the form k· 2ⁿ +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