## 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. Comments and suggestions requested.

A Cullen prime is any prime of the form
*n*^{.}2^{n}+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 C_{53} and stating that
C_{n} is composite for all
*n* less than 201, with the possible exception of
*n*=141. In 1957 Robinson showed C_{141}
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.

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, 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.]- 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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