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Generalized Woodall |
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*3168662-1which 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.
>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
- 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