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<caldwell@utm.edu> Soon after the Reverend Cullen examined the numbers of the form n^{.}2^{n}+1, the numbers n^{.}2^{n}1 were looked at by Cunningham and Woodall (1917). So now these numbers are called Cullen numbers: C_{n}=n^{.}2^{n}+1, and the Woodall numbers: W_{n}=n^{.}2^{n}1. 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 W_{n} 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 (2p) is 1 and W_{(3p1)/2} when the Jacobi symbol (2p) is 1. Suyama appears to have shown that almost all Woodall numbers are composite [Keller95]. Generalized Woodall primes, should we wish to make such a definition, would be primes of the form n^{.}b^{n}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)
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Chris K. Caldwell © 19992014 (all rights reserved)
