## Generalized Woodall |

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.

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 1993191 · 2^{3986382}- 11200027 L3532 May 2015 Generalized Woodall 2 334310 · 211^{334310}- 1777037 p350 Apr 2012 Generalized Woodall 3 41676 · 7^{875197}- 1739632 L2777 Mar 2012 Generalized Woodall 4 404882 · 43^{404882}- 1661368 p310 Feb 2011 Generalized Woodall 5 563528 · 13^{563528}- 1627745 p262 Dec 2009 Generalized Woodall 6 190088 · 5^{760352}- 1531469 L2841 Aug 2012 Generalized Woodall 7 30981 · 14^{433735}- 1497121 p77 Oct 2015 Generalized Woodall 8 1035092 · 3^{1035092}- 1493871 L3544 Jun 2013 Generalized Woodall 9 321671 · 34^{321671}- 1492638 L4780 Apr 2019 Generalized Woodall 10 216290 · 167^{216290}- 1480757 L2777 Oct 2012 Generalized Woodall 11 199388 · 233^{199388}- 1472028 L4780 Aug 2018 Generalized Woodall 12 341351 · 22^{341351}- 1458243 p260 Sep 2017 Generalized Woodall 13 176660 · 18^{353320}- 1443519 p325 Sep 2011 Generalized Woodall 14 182402 · 14^{364804}- 1418118 p325 Sep 2011 Generalized Woodall 15 249798 · 47^{249798}- 1417693 L4780 Mar 2018 Generalized Woodall 16 226400 · 63^{226400}- 1407377 L4780 Sep 2018 Generalized Woodall 17 248100 · 41^{248100}- 1400138 L4779 Mar 2018 Generalized Woodall 18 272970 · 29^{272970}- 1399197 p260 Nov 2017 Generalized Woodall 19 209694 · 79^{209694}- 1397927 L4780 Apr 2018 Generalized Woodall 20 15266 · 12^{366385}- 1395401 p325 Sep 2011 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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