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.

(up) 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.

(up) Record Primes of this Type

rankprime digitswhowhencomment
12329989 · 216309923 - 1 4909783 A20 Feb 2024 Generalized Woodall
21419499 · 215614489 - 1 4700436 A20 Feb 2024 Generalized Woodall
34788920 · 39577840 - 1 4569798 A20 Feb 2024 Generalized Woodall
4396101 · 214259638 - 1 4292585 A20 Feb 2024 Generalized Woodall
52740879 · 213704395 - 1 4125441 L4976 Oct 2019 Generalized Woodall
6479216 · 38625889 - 1 4115601 L4976 Nov 2019 Generalized Woodall
7874208 · 541748416 - 1 3028951 L4976 Sep 2019 Generalized Woodall
8101806 · 151527091 - 1 1796004 L5765 Apr 2023 Generalized Woodall
91405486 · 121405486 - 1 1516781 L5765 Apr 2023 Generalized Woodall
10499238 · 101497714 - 1 1497720 L4976 Sep 2019 Generalized Woodall
11583854 · 141167708 - 1 1338349 L4976 Sep 2019 Generalized Woodall
121182072 · 111182072 - 1 1231008 L5765 Apr 2023 Generalized Woodall
131993191 · 23986382 - 1 1200027 L3532 May 2015 Generalized Woodall
14818764 · 32456293 - 1 1171956 L4965 Apr 2023 Generalized Woodall
15866981 · 12866981 - 1 935636 L5765 Apr 2023 Generalized Woodall
16214916 · 31934246 - 1 922876 L4965 Apr 2023 Generalized Woodall
17663703 · 20663703 - 1 863504 L5765 Apr 2023 Generalized Woodall
18334310 · 211334310 - 1 777037 p350 Apr 2012 Generalized Woodall
19215206 · 51076031 - 1 752119 L20 Sep 2023 Generalized Woodall
2041676 · 7875197 - 1 739632 L2777 Mar 2012 Generalized Woodall

(up) 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, 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 Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813
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