Woodall number
In mathematics, a Woodall number is a natural number of the form n · 2n − 1 (written Wn). Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers. The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, ... (sequence A003261 in the OEIS). Woodall numbers curiously arise in Goodstein's theorem.
Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence A050918 in the OEIS).
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 is +1 and
- W(3p − 1) / 2 if the Jacobi symbol is −1.
It is conjectured that almost all Woodall numbers are composite; a proof has been submitted by Suyama, but it has not been verified yet. Nonetheless, it is also conjectured that there are infinitely many Woodall primes. As of December 2007, the largest known Woodall prime is 3752948 · 23752948 − 1. It has 1129757 digits and was found by Matthew J Thompson in 2007.
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.
References
- Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B20.
- Wilfrid Keller, "New Cullen Primes", Mathematics of Computation, 64 (1995) 1733-1741.
- Chris Caldwell, The Top Twenty: Woodall Primes at The Prime Pages.
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.