Aurifeuillean factorization


In number theory, an aurifeuillean factorization, or aurifeuillian factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.

Examples

Before the discovery of Aurifeuillean factorizations,, through a tremendous manual effort, obtained the following factorization into primes:
Then in 1871, Aurifeuille discovered the nature of this factorization; the number for, with the formula from the previous section, factors as:
Of course, Landry's full factorization follows from this. The general form of the factorization was later discovered by Lucas.
536903681 is an example of a Gaussian Mersenne norm.