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.

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 C_n 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.

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.

Record Primes of this Type

rank prime digits who when comment 1 6679881 · 26679881 + 1 2010852 L917 Aug 2009 Cullen 2 1582137 · 26328550 + 1 1905090 L801 Apr 2009 Cullen 3 338707 · 21354830 + 1 407850 L124 Aug 2005 Cullen 4 481899 · 2481899 + 1 145072 gm Sep 1998 Cullen 5 361275 · 2361275 + 1 108761 DS Jul 1998 Cullen 6 262419 · 2262419 + 1 79002 DS Mar 1998 Cullen 7 90825 · 290825 + 1 27347 Y May 1997 Cullen 8 7457 · 259659 + 1 17964 Y May 1997 Cullen 9 32469 · 232469 + 1 9779 MM May 1997 Cullen 10 8073 · 232294 + 1 9726 MM May 1997 Cullen 11 289 · 218502 + 1 5573 K Dec 1984 Cullen, generalized Fermat 12 6611 · 26611 + 1 1994 K Dec 1984 Cullen 13 5795 · 25795 + 1 1749 K Dec 1984 Cullen 14 4713 · 24713 + 1 1423 K Dec 1984 Cullen

Related Pages

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

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=(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]]

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., 1973.  San Jose, CA, 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, 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 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