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Soon after the Reverend Cullen examined the numbers of
the form n.2n+1, the
numbers n.2n-1 were
looked at by Cunningham and Woodall (1917). So now these
numbers are called Cullen numbers:
and the Woodall numbers:
The Woodall numbers are sometimes called the
Cullen numbers (of the second kind).
Woodall numbers that are prime are called Woodall
primes (or Cullen primes of the second kind)
It is conjectured that there are infinitely many such
primes. The Woodall numbers Wn
are primes for
n=2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312,
7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726,
143018, 151023 and
composite for all
other exponents n less than 260,000.
Like the Cunningham numbers, the Woodall numbers have
many divisibility properties. For example,
if p is a prime, then
p divides W(p+1)/2 when the
Jacobi symbol (2|p) is 1 and
W(3p-1)/2 when the
Jacobi symbol (2|p) is -1.
Suyama appears to have shown that
almost all Woodall numbers are
Generalized Woodall primes, should we
wish to make such a definition, would be 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.
See Also: Cullens, Fermats, Mersennes
Related pages (outside of this work)
- 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, 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.]
- E. Karst, Prime factors of Cullen numbers n· 2n± 1. In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., San Jose, CA, 1973. pp. 153--163,
- W. Keller, "Factors of Fermat numbers and large primes of the form k· 2n +1," Math. Comp., 41 (1983) 661-673. MR 85b:11117
- 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.]
- H. Riesel, "Some factors of the numbers Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415. MR 39:6813
- R. M. Robinson, "A report on primes of the form k· 2n + 1 and on factors of Fermat numbers," Proc. Amer. Math. Soc., 9 (1958) 673--681. MR 20:3097