The Top Twenty--a Prime Page Collection

Generalized Woodall

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The Prime Pages

GIMPS has discovered a new largest known prime number: 2^82589933-1 (24,862,048 digits)

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 2740879 · 2^13704395 - 1 4125441 L4976 Oct 2019 Generalized Woodall
2 479216 · 3^8625889 - 1 4115601 L4976 Nov 2019 Generalized Woodall
3 874208 · 54^1748416 - 1 3028951 L4976 Sep 2019 Generalized Woodall
4 499238 · 10^1497714 - 1 1497720 L4976 Sep 2019 Generalized Woodall
5 583854 · 14^1167708 - 1 1338349 L4976 Sep 2019 Generalized Woodall
6 1993191 · 2^3986382 - 1 1200027 L3532 May 2015 Generalized Woodall
7 334310 · 211³³⁴³¹⁰ - 1 777037 p350 Apr 2012 Generalized Woodall
8 41676 · 7⁸⁷⁵¹⁹⁷ - 1 739632 L2777 Mar 2012 Generalized Woodall
9 404882 · 43⁴⁰⁴⁸⁸² - 1 661368 p310 Feb 2011 Generalized Woodall
10 563528 · 13⁵⁶³⁵²⁸ - 1 627745 p262 Dec 2009 Generalized Woodall
11 190088 · 5⁷⁶⁰³⁵² - 1 531469 L2841 Aug 2012 Generalized Woodall
12 30981 · 14⁴³³⁷³⁵ - 1 497121 p77 Oct 2015 Generalized Woodall
13 1035092 · 3^1035092 - 1 493871 L3544 Jun 2013 Generalized Woodall
14 321671 · 34³²¹⁶⁷¹ - 1 492638 L4780 Apr 2019 Generalized Woodall
15 216290 · 167²¹⁶²⁹⁰ - 1 480757 L2777 Oct 2012 Generalized Woodall
16 199388 · 233¹⁹⁹³⁸⁸ - 1 472028 L4780 Aug 2018 Generalized Woodall
17 341351 · 22³⁴¹³⁵¹ - 1 458243 p260 Sep 2017 Generalized Woodall
18 176660 · 18³⁵³³²⁰ - 1 443519 p325 Sep 2011 Generalized Woodall
19 182402 · 14³⁶⁴⁸⁰⁴ - 1 418118 p325 Sep 2011 Generalized Woodall
20 249798 · 47²⁴⁹⁷⁹⁸ - 1 417693 L4780 Mar 2018 Generalized Woodall

Related Pages

Generalized Woodall Primes by Steven Harvey

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, 1994. New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
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, 1995. New York, NY, 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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