# Cullen primes

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The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

### Definitions and Notes

A Cullen prime is any prime of the form n.2n+1 (compare these with the Woodall numbers). These numbers are named after Reverend J. Cullen who noticed [Cullen05] they were composite for all n less than 100, with the possible exception of n=53. Cunningham responded [Cunningham06] by finding that 5519 divides C53 and stating that Cn is composite for all n less than 201, with the possible exception of n=141. In 1957 Robinson showed C141 was indeed prime [Robinson58].

Now the known Cullen primes include those with n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, and 6679881. See the Cullen prime search status page for more information.

It has been shown that almost all Cullen numbers are composite [Hooley76], but it is still conjectured that there are infinitely many Cullen primes. It is also unknown if Cp can be prime for some prime p.

Keep in mind that some of these may not look like Cullens when written in canonical form. For example:

1582137.26328550+1 = 6328548.26328548+1.

### Record Primes of this Type

rankprime digitswhowhencomment
16679881 · 26679881 + 1 2010852 L917 Aug 2009 Cullen
21582137 · 26328550 + 1 1905090 L801 Apr 2009 Cullen
3338707 · 21354830 + 1 407850 L124 Aug 2005 Cullen
4481899 · 2481899 + 1 145072 gm Sep 1998 Cullen
5361275 · 2361275 + 1 108761 DS Jul 1998 Cullen
6262419 · 2262419 + 1 79002 DS Mar 1998 Cullen
790825 · 290825 + 1 27347 Y May 1997 Cullen
87457 · 259659 + 1 17964 Y May 1997 Cullen
932469 · 232469 + 1 9779 MM May 1997 Cullen
108073 · 232294 + 1 9726 MM May 1997 Cullen
11289 · 218502 + 1 5573 K Dec 1984 Cullen, generalized Fermat
126611 · 26611 + 1 1994 K Dec 1984 Cullen
135795 · 25795 + 1 1749 K Dec 1984 Cullen
144713 · 24713 + 1 1423 K Dec 1984 Cullen

### References

Cullen05
J. Cullen, "Question 15897," Educ. Times, (December 1905) 534. [Originated the study of Cullen numbers. See also [CW17].]
Cunningham06
A. Cunningham, "Solution of question 15897," Math. Quest. Educ. Times, 10 (1906) 44--47. (Annotation available)
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]]
GO2011
Grau, José Maria and Oller-Marcén, Antonio M., "An ~O(log 2(N)) time primality test for generalized Cullen numbers," Math. Comp., 80:276 (2011) 2315--2323.  (http://dx.doi.org/10.1090/S0025-5718-2011-02489-0) MR 2813363
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
Karst73
E. Karst, Prime factors of Cullen numbers n· 2n± 1.  In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, 1973.  pp. 153--163,
Karst73
E. Karst, Prime factors of Cullen numbers n· 2n± 1.  In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, 1973.  pp. 153--163,
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, 1995.  New York, NY, 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.]
Robinson58
R. M. Robinson, "A report on primes of the form k· 2n + 1 and on factors of Fermat numbers," Proc. Amer. Math. Soc., 9 (1958) 673--681.  MR 20:3097
Steiner79
R. P. Steiner, "On Cullen numbers," BIT, 19:2 (1979) 276-277.  MR 80j:10009