Woodall number

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In mathematics, a Woodall number is a natural number of the form n · 2ⁿ − 1 (written W_n). 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 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 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).

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 H. Suyama, but it has not been verified yet. Nonetheless, it is also conjectured that there are infinitely many Woodall primes. As of December 2007^[update], the largest known Woodall prime is 3752948 · 2^3752948 − 1. It has 1129757 digits and was found by Matthew J Thompson in 2007 in the distributed computing project PrimeGrid.

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.

[edit] See also

• Mersenne numbers and primes - Numbers of the form 2ⁿ − 1.

[edit] 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. Retrieved December 29, 2007.

[edit] External links

• Chris Caldwell, The Prime Glossary: Woodall number at The Prime Pages.
• Weistein, Eric W., "Woodall number" from MathWorld.
• Steven Harvey, List of Generalized Woodall primes.