Woodall 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 Woodall prime is any prime of the form n^.2ⁿ-1 (compare with the Cullen primes). The Woodall primes with n < 30000 are those with n = 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, and 18885. It is conjectured that there are infinitely many Woodall primes.

A prime of the form n^.bⁿ-1 with b not 2, is a generalized Woodall prime.

Record Primes of this Type

rank prime digits who when comment 1 8508301 · 217016603 - 1 5122515 L4784 Mar 2018 Woodall 2 938237 · 23752950 - 1 1129757 L521 Dec 2007 Woodall 3 1183953 · 22367907 - 1 712818 L447 Sep 2007 Woodall 4 251749 · 22013995 - 1 606279 L436 Aug 2007 Woodall 5 1467763 · 21467763 - 1 441847 L381 Jun 2007 Woodall 6 1268979 · 21268979 - 1 382007 L201 Jan 2007 Woodall 7 1195203 · 21195203 - 1 359799 L124 Jul 2005 Woodall 8 667071 · 2667071 - 1 200815 g55 Sep 2000 Woodall 9 151023 · 2151023 - 1 45468 g25 May 1998 Woodall 10 71509 · 2143019 - 1 43058 g23 Apr 1998 Woodall, arithmetic progression (2, d=(143018 · 283969 - 80047) · 259049) [x12] 11 49363 · 298727 - 1 29725 Y May 1997 Woodall 12 23005 · 223005 - 1 6930 Y May 1997 Woodall 13 22971 · 222971 - 1 6920 Y May 1997 Woodall 14 18885 · 218885 - 1 5690 K Dec 1987 Woodall 15 7911 · 215823 - 1 4768 K Dec 1987 Woodall 16 12379 · 212379 - 1 3731 K Dec 1984 Woodall 17 9531 · 29531 - 1 2874 K Dec 1984 Woodall 18 7755 · 27755 - 1 2339 K Dec 1984 Woodall 19 83 · 25318 - 1 1603 K Dec 1984 Woodall

Related Pages

• The Prime Glossary's: Woodall Numbers
• The chronology of prime number records

References

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]]

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.]

Riesel69a

H. Riesel, "Lucasian criteria for the primality of N = h · 2ⁿ - 1," Math. Comp., 23:108 (1969) 869--875.  MR 41:6773