Woodall number

Woodall number

In mathematics, a Woodall number is a natural number of the form "n" · 2"n" − 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, ... OEIS|id=A003261.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, ... OEIS|id=A002234; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... OEIS|id=A050918.

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} ight) is +1 and

:"W"(3"p" − 1) / 2 if the Jacobi symbol left(frac{2}{p} ight) 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 the distributed computing project PrimeGrid.

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 The Prime Pages. Retrieved December 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] .


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • 10000 (number) — Number number = 10000 prev = 9999 next = 100000 range = 10000 100000 cardinal = 10000 ordinal = th ordinal text = ten thousandth numeral = decamillesimal factorization = 2^4 cdot 5^4 prime = divisor = roman = overline|X unicode = overline|X, ↂ… …   Wikipedia

  • 20000 (number) — Number number = 20000 range = 10000 100000 cardinal = 20000 ordinal = th ordinal text = twenty thousandth factorization = 2^5 cdot 5^4 bin = 100111000100000 oct = 47040 hex = 4E2020,000 (twenty thousand) is the number that comes after 19,999 and… …   Wikipedia

  • 40000 (number) — Number number = 40000 range = 10000 100000 cardinal = 40000 ordinal = th ordinal text = thirty thousandth factorization = 2^6 cdot 5^4 bin = 1001110001000000 oct = 116100 hex = 9C4040,000 (forty thousand) is the number that comes after 39,999 and …   Wikipedia

  • Nombre De Woodall — En mathématiques, un nombre de Woodall ou nombre de Riesel est un entier naturel de la forme (écrit ). Les nombres de Woodall ont été étudiés en premier par A. J. C. Cunningham et H. J. Woodall en 1917, inspirés par l étude précédente de James… …   Wikipédia en Français

  • Nombre de woodall — En mathématiques, un nombre de Woodall ou nombre de Riesel est un entier naturel de la forme (écrit ). Les nombres de Woodall ont été étudiés en premier par A. J. C. Cunningham et H. J. Woodall en 1917, inspirés par l étude précédente de James… …   Wikipédia en Français

  • Nombre de Woodall — En mathématiques, un nombre de Woodall ou nombre de Riesel est un entier naturel de la forme (écrit ). Les nombres de Woodall ont été étudiés en premier par A. J. C. Cunningham et H. J. Woodall en 1917, inspirés par l étude précédente de James… …   Wikipédia en Français

  • H. J. Woodall — Herbert J. Woodall was a British mathematician.In 1925 Lt. Col. Allan J.C. Cunningham and Woodall gathered together all that was known about the primality and factorization of such numbers and published a small book of tables. These tables… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • 800 (number) — This article is about the number 800. For the Common Era Year 800, see 800. For other uses, see 800 (disambiguation) 800 (eight hundred) is the natural number following 799 and preceding 801. List of numbers Integers ← 0 100 200 300 400 500 600… …   Wikipedia

  • 2000 (number) — 2000 (two thousand) is the natural number following 1999 and preceding 2001.See also: millennium, year 2000 AD, Y2K, 2000 (disambiguation).Two thousand is the highest number expressible using only two unmodified characters in roman numerals… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”