Definition:Aurifeuillian Factorization/Examples

Examples of Aurifeuillian Factorizations

Factorization of $2^{4 n + 2} + 1$

$2^{4 n + 2} + 1 = \paren {2^{2 n + 1} - 2^{n + 1} + 1} \paren {2^{2 n + 1} + 2^{n + 1} + 1}$

Factorization of $3^{6 n + 3} + 1$

$3^{6 n + 3} + 1 = 3^{2 n + 1} \paren {3^{2 n + 1} - 3^{n + 1} + 1} \paren {3^{2 n + 1} + 3^{n + 1} + 1}$

Sophie Germain's Identity

$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$