In
mathematics, specifically 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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, a Cunningham number is a certain kind of
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 ...
named after English mathematician
A. J. C. Cunningham.
Definition
Cunningham numbers are a simple type of
binomial number
In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number.
Definition
A binomial number is a ...
– they are of the form
:
where ''b'' and ''n'' are integers and ''b'' is not a
perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n' ...
. They are denoted ''C''
±(''b'', ''n'').
Primality
Establishing whether or not a given Cunningham number is
prime
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 way ...
has been the main focus of research around this type of number. Two particularly famous families of Cunningham numbers in this respect are the
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 4294967 ...
s, which are those of the form ''C''
+(2, 2
''m''), and the
Mersenne numbers
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 1 ...
, which are of the form ''C''
−(2, ''n'').
Cunningham worked on gathering together all known data on which of these numbers were prime. In 1925 he published tables which summarised his findings with
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 ...
, and much computation has been done in the intervening time to fill these tables.
[R. P. Brent and H. J. J. te Riele, ''Factorizations of an±1, 13≤a<100'' Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.]
See also
*
Cunningham project
References
{{Reflist
External links
Cunningham Number at MathWorld
Number theory