# Woodall prime

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 (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
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

**References:**

- CW17
A. J. C. CunninghamandH. J. Woodall, "Factorisation ofQ=(2^{q}±q) andq*2^{q}± 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.]- Karst73
E. Karst,Prime factors of Cullen numbers. In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., 1973. San Jose, CA, pp. 153--163,n· 2^{n}± 1- Keller83
W. Keller, "Factors of Fermat numbers and large primes of the formk· 2^{n}+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 numbersG_{n}=6^{2n}+1 andH_{n}=10^{2n}+1,"Math. Comp.,23:106 (1969) 413--415.MR 39:6813- Robinson58
R. M. Robinson, "A report on primes of the formk· 2^{n}+1 and on factors of Fermat numbers,"Proc. Amer. Math. Soc.,9(1958) 673--681.MR 20:3097