Aurifeuillean factorization

In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless, certain families of integers coming from cyclotomic polynomials have factorizations given by formulas applying to the whole family, as in the examples below. Numbers of the form have the following factorization (Sophie Germain's identity): Setting and , one obtains the following aurifeuillean factorization of , where is the fourth cyclotomic polynomial: Numbers of the form have the following factorization, where the first factor () is the algebraic factorization of sum of two cubes: Setting and , one obtains the following factorization of : Here, the first of the three terms in the factorization is and the remaining two terms provide an aurifeuillean factorization of , where . Numbers of the form or their factors , where with square-free , have aurifeuillean factorization if and only if one of the following conditions holds: and and Thus, when with square-free , and is congruent to modulo , then if is congruent to 1 mod 4, have aurifeuillean factorization, otherwise, have aurifeuillean factorization. When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers.