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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.
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
| rank | prime |
digits | who | when | comment |
|---|
| 1 | 2740879 · 213704395 - 1 |
4125441 |
L4976 |
Oct 2019 |
Generalized Woodall |
| 2 | 479216 · 38625889 - 1 |
4115601 |
L4976 |
Nov 2019 |
Generalized Woodall |
| 3 | 874208 · 541748416 - 1 |
3028951 |
L4976 |
Sep 2019 |
Generalized Woodall |
| 4 | 499238 · 101497714 - 1 |
1497720 |
L4976 |
Sep 2019 |
Generalized Woodall |
| 5 | 583854 · 141167708 - 1 |
1338349 |
L4976 |
Sep 2019 |
Generalized Woodall |
| 6 | 1993191 · 23986382 - 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 · 31035092 - 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
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