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 an integer obtained by evaluating a homogeneous polynomial containing two terms, also called a binomial. The form of this binomial is , with and . However, since is always divisible by , when studying the numbers generated from the version with the negative sign, they are usually divided by first. Binomial numbers formed this way form Lucas sequences. Specifically: and Binomial numbers are a generalization of a Cunningham numbers, and it will be seen that the Cunningham numbers are binomial numbers where . Other subsets of the binomial numbers are the Mersenne numbers and the repunits. Factorization The main reason for studying these numbers is to obtain their factorizations. Aside from algebraic factors, which are obtained by factoring the underlying polynomial (binomial) that was used to define the number, there are other prime factors (called primitive prime factors, because for a given they do not factorize with ) which occur seemingly at random, and it is these which the number theorist is looking for. Some binomial numbers' underlying binomials have Aurifeuillian factorizations, which can assist in finding prime factors. Cyclotomic polynomials are also helpful in finding factorizations. The amount of work required in searching for a factor is considerably reduced by applying Legendre's theorem. This theorem states that all factors of a binomial number are of the form if is even or if it is odd. Observation Some people write "binomial number" when they mean binomial coefficient, but this usage is not standard and is deprecated. See also Cunningham project References External links Binomial Number at MathWorld Number theory