# Generalized Woodall

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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 Cn).  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*73*9750-1 = 9750*3439750-1
8511*2374486-1 = (8511*22)*211*8511*4-1 = 34044*204834044-1.

### Record Primes of this Type

rankprime digitswhowhencomment
11993191 · 23986382 - 1 1200027 L3532 May 2015 Generalized Woodall
2334310 · 211334310 - 1 777037 p350 Apr 2012 Generalized Woodall
341676 · 7875197 - 1 739632 L2777 Mar 2012 Generalized Woodall
4404882 · 43404882 - 1 661368 p310 Feb 2011 Generalized Woodall
5563528 · 13563528 - 1 627745 p262 Dec 2009 Generalized Woodall
6190088 · 5760352 - 1 531469 L2841 Aug 2012 Generalized Woodall
730981 · 14433735 - 1 497121 p77 Oct 2015 Generalized Woodall
81035092 · 31035092 - 1 493871 L3544 Jun 2013 Generalized Woodall
9216290 · 167216290 - 1 480757 L2777 Oct 2012 Generalized Woodall
10341351 · 22341351 - 1 458243 p260 Sep 2017 Generalized Woodall
11176660 · 18353320 - 1 443519 p325 Sep 2011 Generalized Woodall
12182402 · 14364804 - 1 418118 p325 Sep 2011 Generalized Woodall
13272970 · 29272970 - 1 399197 p260 Nov 2017 Generalized Woodall
1415266 · 12366385 - 1 395401 p325 Sep 2011 Generalized Woodall
15125132 · 6500528 - 1 389492 L2777 Jan 2012 Generalized Woodall
16163747 · 6491241 - 1 382266 L2841 Apr 2012 Generalized Woodall
17545082 · 5545082 - 1 381002 p357 Apr 2015 Generalized Woodall
1889725 · 21256151 - 1 378145 p260 Dec 2012 Generalized Woodall
19178602 · 5535806 - 1 374518 L2777 Mar 2012 Generalized Woodall
20154962 · 221154962 - 1 363297 L3269 Sep 2012 Generalized Woodall

### References

CW17
A. J. C. Cunningham and H. J. Woodall, "Factorisation of Q=(2q ± q) and q*2q ± 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 Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813