Cullen primes

This page : Definition(s) | Records | References | Related Pages |

The Prime Pages

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 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 and 361275 (they are composite for all smaller n). 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 C_p can be prime for some prime p.

Record Primes of this Type

rank prime digits who when comment

1 338707 · 2^1354830+1 407850 L124 Aug 2005 Cullen
2 481899 · 2⁴⁸¹⁸⁹⁹+1 145072 gm Sep 1998 Cullen
3 361275 · 2³⁶¹²⁷⁵+1 108761 DS Jul 1998 Cullen
4 262419 · 2²⁶²⁴¹⁹+1 79002 DS Mar 1998 Cullen
5 90825 · 2⁹⁰⁸²⁵+1 27347 Y May 1997 Cullen
6 7457 · 2⁵⁹⁶⁵⁹+1 17964 Y May 1997 Cullen
7 32469 · 2³²⁴⁶⁹+1 9779 MM May 1997 Cullen
8 8073 · 2³²²⁹⁴+1 9726 MM May 1997 Cullen
9 289 · 2¹⁸⁵⁰²+1 5573 K Dec 1984 Cullen, generalized Fermat
10 6611 · 2⁶⁶¹¹+1 1994 K Dec 1984 Cullen
11 5795 · 2⁵⁷⁹⁵+1 1749 K Dec 1984 Cullen
12 4713 · 2⁴⁷¹³+1 1423 K Dec 1984 Cullen

Cullen prime search status page
The Prime Glossary's: Cullen numbers
Tables of factorizations of 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]]
Hooley76
C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Math. volume 70, Cambridge University Press, Cambridge, pp. xiv+122, 1976. 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, pp. 153--163, 1973.
Karst73
E. Karst, Prime factors of Cullen numbers n· 2ⁿ± 1. In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, pp. 153--163, 1973.
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, pp. xxiv+541, ISBN 0-387-94457-5. 1995. 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