- Woodall number
In
mathematics , a Woodall number is anatural number of the form "n" · 2"n" − 1 (written "W""n"). Woodall numbers were first studied by Allan J. C. Cunningham andH. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-definedCullen number s. The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, ... OEIS|id=A003261.Woodall numbers curiously arise inGoodstein's theorem .Woodall numbers that are also
prime number s 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, ... OEIS|id=A002234; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... OEIS|id=A050918.Like
Cullen number s, 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"(3"p" − 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 , the largest known Woodall prime is 3752948 · 23752948 − 1. It has 1129757 digits and was found by Matthew J Thompson in 2007 in thedistributed computing projectPrimeGrid .A generalized Woodall number is defined to be a number of the form "n" · "b""n" − 1, where "n" + 2 > "b"; if a prime can be written in this form, it is then called a generalized Woodall prime.
ee also
* Mersenne numbers and primes - Numbers of the form 2"n" − 1.
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, [http://primes.utm.edu/top20/page.php?id=7 "The Top Twenty: Woodall Primes"] at ThePrime Pages . RetrievedDecember 29 2007 .External links
* Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=WoodallNumber The Prime Glossary: Woodall number] at The
Prime Pages .
*
* Steven Harvey, [http://www.geocities.com/harvey563/GeneralizedWoodallPrimes.txt List of Generalized Woodall primes] .
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