Coding theory - Revision history

en>John of Reading: /* Properties */Typo fixing, replaced: vice-versa → vice versa using AWB

2015-01-02T15:40:26Z

Properties: Typo fixing, replaced: vice-versa → vice versa using AWB

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Salix alba: /* Linear block codes */fix tags, replaced: ^{''r'' → ^''r'' (2) using AWB}

2014-02-28T22:02:27Z

Linear block codes: fix tags, replaced: ''r'' → ''r'' (2) using AWB

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en>Monkbot: /* Neural coding */Fix CS1 deprecated date parameter errors

2014-02-04T00:28:47Z

Neural coding: Fix CS1 deprecated date parameter errors

New page

In [[number theory]], a '''Woodall number''' (Wn) is any [[natural number]] of the form

:Wn = ''n'' × 2''n'' − 1

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

:1, 7, 23, 63, 159, 383, 895, … {{OEIS|id=A003261}}.

Woodall numbers were first studied by [[Allan J. C. Cunningham]] and [[H. J. Woodall]] in 1917,{{Citation needed|date=December 2011}} inspired by [[James Cullen (mathematician)|James Cullen]]'s earlier study of the similarly-defined [[Cullen number]]s. Woodall numbers curiously arise in [[Goodstein's theorem]].{{Citation needed|date=October 2009}}

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}}.

In 1976 [[Christopher Hooley]] showed that [[almost all]] Cullen numbers are [[composite number|composite]].{{Citation needed|date=December 2011}} 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|date=December 2011}} and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.{{Citation needed|date=December 2011}} {{As of|2007|12}}, the largest known Woodall prime is 3752948 × 23752948 − 1.<ref>{{Citation |url=http://primes.utm.edu/primes/page.php?id=83407 |title=The Prime Database: 938237*2^3752950-1 |work=Chris Caldwell's The Largest Known Primes Database |accessdate=December 22, 2009 }}</ref> 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]] <math>\left(\frac{2}{p}\right)</math> is +1 and

:''W''(3''p'' − 1) / 2 if the Jacobi symbol <math>\left(\frac{2}{p}\right)</math> is −1.{{Citation needed|date=December 2011}}

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'''.

==See also==
* [[Mersenne prime]] - Prime numbers of the form 2''n'' − 1.

==References==
{{Reflist}}

==Further reading==
* {{Citation |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |location=New York |year=2004 |isbn=0-387-20860-7 |pages=section B20 }}.
* {{Citation |first=Wilfrid |last=Keller |title=New Cullen Primes |journal=[[Mathematics of Computation]] |volume=64 |issue=212 |year=1995 |pages=1733–1741 |url=http://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf }}.
* {{Citation |first=Chris |last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=7 |title=The Top Twenty: Woodall Primes |work=The [[Prime Pages]] |accessdate=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]].
* {{MathWorld|urlname=WoodallNumber|title=Woodall number}}
* Steven Harvey, [http://harvey563.tripod.com/GeneralizedWoodallPrimes.txt List of Generalized Woodall primes].
* Paul Leyland, [http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.html Generalized Cullen and Woodall Numbers]

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}
__NOTOC__

{{DEFAULTSORT:Woodall Number}}
[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]

Coding theory - Revision history

en>John of Reading: /* Properties */Typo fixing, replaced: vice-versa → vice versa using AWB

Salix alba: /* Linear block codes */fix tags, replaced: ''r'' → ''r'' (2) using AWB

en>Monkbot: /* Neural coding */Fix CS1 deprecated date parameter errors

Salix alba: /* Linear block codes */fix tags, replaced: ^{''r'' → ^''r'' (2) using AWB}