## Tuesday, April 19, 2011

### Aurifeuillian Factorizations

n^4 + 4 is composite (i.e., not prime) for all n > 1

Well, n^4 + 4 = (n^2 - 2n + 2) (n^2 + 2n + 2) is a proper factorization, since the smaller of the two factors is greater than 1 when n > 1.

This type of factorization is often called Aurifeuillian (sometimes also spelled Aurifeuillean) in honor of the French mathematician [Léon François] Antoine Aurifeuille.

For example, the following Aurifeuillian (preliminary) factorization often pops up when x is a power of 2:

4 x^4 + 1 = (2x^2 - 2x + 1)( 2x^2 + 2x + 1)

If x = 2y^3, then:

(2 x^2 + 2x + 1) = ( 2 y^2 - 2y + 1)(4 y^4 + 4 y^3 + 2 y^2 + 2y + 1)
(2 x^2 - 2x + 1) = ( 2 y^2 + 2y + 1)(4 y^4 - 4 y^3 + 2 y^2 - 2y + 1)

In either case, the first factor has a similar factorization when y = 2z^3 etc.

Thus, we have two preliminary factors of 2^n+1 when n is congruent to 2 modulo 4, we've 6 of them when n is congruent to 6 modulo 12, and so forth...

4 a^4 + b^4 = (2 a^2 - 2ab + b2 )(2 a^2 + 2ab + b^2 )
27 a^6 + b^6 = (3 a^2 - 3ab + b2 )(3 a^2 + b^2)(3 a^2 + 3ab + b^2)