Woodall number

In number theory, a Woodall number (W_n) is any natural number of the form

W_n = n × 2ⁿ − 1

for some natural number n. The first few Woodall numbers are:

1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in OEIS).

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,^{[citation needed]} inspired by James Cullen's earlier study of the similarly-defined Cullen numbers. Woodall numbers curiously arise in Goodstein's theorem.^{[citation needed]}

Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers W_n are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, … (sequence A002234 in OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence A050918 in OEIS).

In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.^{[citation needed]} Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2^n+a + b where a and b are integers,^{[citation needed]} and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.^{[citation needed]} As of December 2007, the largest known Woodall prime is 3752948 × 2^3752948 − 1. It has 1,129,757 digits and was found by Matthew J. Thompson in 2007 in the distributed computing project PrimeGrid.

Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides

W_{(p + 1) / 2} if the Jacobi symbol $\left(\frac{2}{p}\right)$ is +1 and

W_{(3p − 1) / 2} if the Jacobi symbol $\left(\frac{2}{p}\right)$ is −1.^{[citation needed]}

A generalized Woodall number is defined to be a number of the form n × bⁿ − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime.

See also

• Mersenne prime - Prime numbers of the form 2ⁿ − 1.

Further reading

• Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7 .
• Keller, Wilfrid (1995), "New Cullen Primes", Mathematics of Computation 64 (212): 1733–1741 .
• Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007 .

External links

• Chris Caldwell, The Prime Glossary: Woodall number at The Prime Pages.
• Weisstein, Eric W., "Woodall number", MathWorld.
• Steven Harvey, List of Generalized Woodall primes.
• Paul Leyland, Generalized Cullen and Woodall Numbers