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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The mathematician Allan Joseph Champneys Cunningham (Delhi 1842 - London 1928) started a military career with the East India Companys Bengal (later Royal) Engineers. ... Herbert J. Woodall was a British mathematician. ... 1917 (MCMXVII) was a common year starting on Monday of the Gregorian calendar (see link for calendar) or a common year starting on Tuesday of the 13-day slower Julian calendar (see: 1917 Julian calendar). ... Father James Cullen, S. J. (April 19, 1867 – December 7, 1933) was born at Drogheda, Co Meath, Ireland. ... In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). ... Look up one in Wiktionary, the free dictionary. ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... 23 (twenty-three) is the natural number following 22 and preceding 24. ... Sixty-three is a natural number following 62 and preceding 64. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... In mathematical logic, Goodsteins theorem is a statement about the natural numbers that is undecidable in Peano arithmetic but can be proven to be true using the stronger axiom system of set theory, in particular using the axiom of infinity. ...

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). In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). ...

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 August 2007, the largest known Woodall prime is 2013992 · 2^2013992 − 1. It has 606279 digits and was found by Lasse Mejling Andersen of Denmark in the distributed computing project PrimeGrid. The Jacobi symbol generalises the Legendre symbol. ... In mathematics, the phrase almost all has a number of specialised uses. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... August 2007 is the eighth month of that year. ... Distributed computing is a method of computer processing in which different parts of a program run simultaneously on two or more computers that are communicating with each other over a network. ... PrimeGrid is a distributed computing project for factoring the number in the RSA Factoring Challenge, and also to test PerlBOINC. It makes use of the Berkeley Open Infrastructure for Network Computing (BOINC) platform. ...

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, from The Prime Pages.

Richard Kenneth Guy (born 1916) is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ... Unsolved Problems in Number Theory may refer to: Unsolved problems in mathematics in the field of Number theory. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... Mathematics of Computation is a scientific journal run by American Mathematical Society focused on computational mathematics. ... The Prime pages is a website about prime numbers maintained by Prof. ...

External links

• The Prime Glossary: Woodall number
• MathWorld: Woodall number
• List of Generalized Woodall primes