In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Woodall number (''W''
''n'') is any
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
of the form
:
for some natural number ''n''. The first few Woodall numbers are:
:1, 7, 23, 63, 159, 383, 895, … .
History
Woodall numbers were first studied by
Allan J. C. Cunningham
Allan Joseph Champneys Cunningham (1842–1928) was a British-Indian mathematician.A. E. Western, ''J. London Math. Soc.'' 317–318 (1928)
Biography
Born in Delhi, Cunningham was the son of Sir Alexander Cunningham, archaeologist and the founde ...
and
H. J. Woodall
Herbert J. Woodall was a British mathematician, known as the namesake for the Woodall numbers.
In an 1889 publication, Woodall listed his affiliation as the Normal School of Science (now part of the Royal College of Science) in South Kensington. H ...
in 1917, inspired by
James Cullen's earlier study of the similarly defined
Cullen number
In mathematics, a Cullen number is a member of the integer sequence C_n = n \cdot 2^n + 1 (where n is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.
Properties
In 1 ...
s.
Woodall primes
Woodall numbers that are also
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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, ... ; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... .
In 1976
Christopher Hooley
Christopher Hooley (7 August 1928 – 13 December 2018) was a British mathematician, professor of mathematics at Cardiff University.
He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in ...
showed that
almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
Cullen numbers are
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
.
In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to
factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from
Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers , where ''a'' and ''b'' are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, and in particular, that almost all Woodall numbers are composite. It is an
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...
whether there are infinitely many Woodall primes. , the largest known Woodall prime is 17016602 × 2
17016602 − 1. It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the
distributed computing
A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed com ...
project
PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...
.
Restrictions
Starting with ''W''
4 = 63 and ''W''
5 = 159, every sixth Woodall number is
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by 3; thus, in order for ''W''
''n'' to be prime, the index ''n'' cannot be
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to 4 or 5 (modulo 6). Also, for a positive integer ''m'', the Woodall number ''W''
2''m'' may be prime only if 2
''m'' + ''m'' is prime. As of January 2019, the only known primes that are both Woodall primes and
Mersenne primes
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
are ''W''
2 = ''M''
3 = 7, and ''W''
512 = ''M''
521.
Divisibility properties
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
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
is +1 and
:''W''
(3''p'' − 1) / 2 if the Jacobi symbol
is −1.
Generalization
A generalized Woodall number base ''b'' 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.
The smallest value of ''n'' such that ''n'' × ''b''
''n'' − 1 is prime for ''b'' = 1, 2, 3, ... are
:3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ...
, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 32
2740879 − 1.
See also
*
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
- Prime numbers of the form 2
''n'' − 1.
References
Further reading
* .
* .
* .
External links
* Chris Caldwell
The Prime Glossary: Woodall number an
The Top Twenty: Woodall an
The Top Twenty: Generalized Woodall at The
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.
The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
.
*
* Steven Harvey
List of Generalized Woodall primes
* Paul Leyland
__NOTOC__
{{DEFAULTSORT:Woodall Number
Integer sequences
Unsolved problems in number theory
Classes of prime numbers