Binomial Number - Factorization

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.