The Top Twenty--a Prime Page Collection

Generalized Woodall

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

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

In 1905, the Reverend Cullen was interested in the numbers n^.2ⁿ+1 (denoted C_n). He noticed that the first, C₁=3, was prime, but with the possible exception of the 53rd, the next 99 were all composite. Very soon afterwards, Cunningham discovered that 5591 divides C₅₃, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141. Five decades later Robinson showed C₁₄₁ was a prime.

It was natural next to seek primes of the form n^.2ⁿ-1, now called Woodall numbers, and then the Generalized Woodall primes: the primes of the form n^.bⁿ-1 with n+2 > b. The reason for the restriction on the exponent n is simple, without some restriction every prime p would be a generalized Woodall because:

p = 1^.(p+1)¹-1.

Curiously, these numbers may be hard to recognize when written in standard form. For example, they may be like

18740*3¹⁶⁸⁶⁶²-1

which could be written

168660*3¹⁶⁸⁶⁶⁰-1.

More difficult to spot are those like the following:

9750*7²⁹²⁵⁰-1 = 9750*7^3*9750-1 = 9750*343⁹⁷⁵⁰-1
8511*2³⁷⁴⁴⁸⁶-1 = (8511*2²)*2^11*8511*4-1 = 34044*2048³⁴⁰⁴⁴-1.

Record Primes of this Type

rank prime digits who when comment

1 229918 · 12²²⁹⁹¹⁸-1 248129 x37 Dec 2008 Generalized Woodall
2 9980 · 19¹⁸⁹⁶²¹-1 242483 p237 Nov 2008 Generalized Woodall
3 60205 · 2⁷⁸²⁶⁶⁵-1 235611 p243 Apr 2009 Generalized Woodall
4 133736 · 3⁴⁰¹²⁰⁹-1 191431 p120 Nov 2004 Generalized Woodall
5 162454 · 15¹⁶²⁴⁵⁴-1 191066 p242 Jan 2009 Generalized Woodall
6 146478 · 19¹⁴⁶⁴⁷⁸-1 187315 p237 Oct 2008 Generalized Woodall
7 36739 · 2⁵⁸⁷⁸²⁷-1 176959 p77 Sep 2009 Generalized Woodall
8 83660 · 72⁸³⁶⁶⁰-1 155390 g265 Sep 2003 Generalized Woodall
9 103444 · 3³¹⁰³³²-1 148072 p260 Oct 2009 Generalized Woodall
10 292340 · 3²⁹²³⁴⁰-1 139488 p120 Aug 2004 Generalized Woodall
11 61652 · 103⁶¹⁶⁵²-1 124101 p120 Mar 2004 Generalized Woodall
12 36635 · 1960³⁶⁶³⁵-1 120617 p117 Sep 2003 Generalized Woodall
13 91850 · 19⁹¹⁸⁵⁰-1 117459 p237 Oct 2008 Generalized Woodall
14 92711 · 18⁹²⁷¹¹-1 116383 p242 Jan 2009 Generalized Woodall
15 64227 · 2³⁸⁵³⁶²-1 116011 p77 Nov 2003 Generalized Woodall
16 145359 · 6¹⁴⁵³⁵⁹-1 113117 p234 Sep 2008 Generalized Woodall
17 8511 · 2³⁷⁴⁴⁸⁶-1 112736 p77 Oct 2003 Generalized Woodall
18 17883 · 2³⁵⁷⁶⁶²-1 107672 p103 Mar 2003 Generalized Woodall
19 65555 · 42⁶⁵⁵⁵⁵-1 106417 p239 Nov 2008 Generalized Woodall
20 54528 · 69⁵⁴⁵²⁸-1 100274 p120 Feb 2004 Generalized Woodall

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]]
Guy94 (section B2)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, 1994. ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references.]
Keller83
W. Keller, "Factors of Fermat numbers and large primes of the form k· 2ⁿ +1," Math. Comp., 41 (1983) 661-673. MR 85b:11117
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.]
Riesel69
H. Riesel, "Some factors of the numbers G_n = 6^2ⁿ + 1 and H_n = 10^2ⁿ + 1," Math. Comp., 23:106 (1969) 413--415. MR 39:6813

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