## Cullen primes |

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.

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.

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 C_{p} 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^{.}2^{6328550}+1 = 6328548^{.}2^{6328548}+1.

rank prime digits who when comment 1 6679881 · 2^{6679881}+ 12010852 L917 Aug 2009 Cullen 2 1582137 · 2^{6328550}+ 11905090 L801 Apr 2009 Cullen 3 338707 · 2^{1354830}+ 1407850 L124 Aug 2005 Cullen 4 481899 · 2^{481899}+ 1145072 gm Sep 1998 Cullen 5 361275 · 2^{361275}+ 1108761 DS Jul 1998 Cullen 6 262419 · 2^{262419}+ 179002 DS Mar 1998 Cullen 7 90825 · 2^{90825}+ 127347 Y May 1997 Cullen 8 7457 · 2^{59659}+ 117964 Y May 1997 Cullen 9 32469 · 2^{32469}+ 19779 MM May 1997 Cullen 10 8073 · 2^{32294}+ 19726 MM May 1997 Cullen 11 289 · 2^{18502}+ 15573 K Dec 1984 Cullen, generalized Fermat 12 6611 · 2^{6611}+ 11994 K Dec 1984 Cullen 13 5795 · 2^{5795}+ 11749 K Dec 1984 Cullen 14 4713 · 2^{4713}+ 11423 K Dec 1984 Cullen

- The Prime Glossary's: Cullen numbers
- The chronology of prime number records

- 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. 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]]- GO2011
Grau, José MariaandOller-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. In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, 1973. pp. 153--163,n· 2^{n}± 1- Karst73
E. Karst,Prime factors of Cullen numbers. In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, 1973. pp. 153--163,n· 2^{n}± 1- 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 formk· 2^{n}+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

Chris K. Caldwell
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