## Generalized Woodall |

In 1905, the Reverend Cullen was interested in the numbers
*n*^{.}2^{n}+1 (denoted C_{n}).
He noticed that the first, C_{1}=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 C_{53}, 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 C_{141} was a prime.

It was natural next to seek primes of the form *n*^{.}2^{n}-1, now called Woodall numbers, and then the **Generalized Woodall primes**: the primes of
the form *n*^{.}*b*^{n}-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:

Curiously, these numbers may be hard to recognize when written in standard form. For example, they may be likep= 1^{.}(p+1)^{1}-1.

18740*3which could be written^{168662}-1

168660*3More difficult to spot are those like the following:^{168660}-1.

9750*7^{29250}-1 = 9750*7^{3*9750}-1 = 9750*343^{9750}-1

8511*2^{374486}-1 = (8511*2^{2})*2^{11*8511*4}-1 = 34044*2048^{34044}-1.

>rank prime digits who when comment 1 2740879 · 2^{13704395}- 14125441 L4976 Oct 2019 Generalized Woodall 2 479216 · 3^{8625889}- 14115601 L4976 Nov 2019 Generalized Woodall 3 874208 · 54^{1748416}- 13028951 L4976 Sep 2019 Generalized Woodall 4 499238 · 10^{1497714}- 11497720 L4976 Sep 2019 Generalized Woodall 5 583854 · 14^{1167708}- 11338349 L4976 Sep 2019 Generalized Woodall 6 1993191 · 2^{3986382}- 11200027 L3532 May 2015 Generalized Woodall 7 334310 · 211^{334310}- 1777037 p350 Apr 2012 Generalized Woodall 8 41676 · 7^{875197}- 1739632 L2777 Mar 2012 Generalized Woodall 9 404882 · 43^{404882}- 1661368 p310 Feb 2011 Generalized Woodall 10 563528 · 13^{563528}- 1627745 p262 Dec 2009 Generalized Woodall 11 190088 · 5^{760352}- 1531469 L2841 Aug 2012 Generalized Woodall 12 30981 · 14^{433735}- 1497121 p77 Oct 2015 Generalized Woodall 13 1035092 · 3^{1035092}- 1493871 L3544 Jun 2013 Generalized Woodall 14 321671 · 34^{321671}- 1492638 L4780 Apr 2019 Generalized Woodall 15 216290 · 167^{216290}- 1480757 L2777 Oct 2012 Generalized Woodall 16 199388 · 233^{199388}- 1472028 L4780 Aug 2018 Generalized Woodall 17 341351 · 22^{341351}- 1458243 p260 Sep 2017 Generalized Woodall 18 176660 · 18^{353320}- 1443519 p325 Sep 2011 Generalized Woodall 19 182402 · 14^{364804}- 1418118 p325 Sep 2011 Generalized Woodall 20 249798 · 47^{249798}- 1417693 L4780 Mar 2018 Generalized Woodall

- Generalized Woodall Primes by Steven Harvey

- CW17
A. J. C. CunninghamandH. J. Woodall, "Factorisation ofQ=(2^{q}±q) andq*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. See his newer editions of this text.]- Keller83
W. Keller, "Factors of Fermat numbers and large primes of the formk· 2^{n}+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 numbersG_{n}=6^{2n}+1 andH_{n}=10^{2n}+1,"Math. Comp.,23:106 (1969) 413--415.MR 39:6813

Chris K. Caldwell
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