Definition:Aurifeuillian Factorization

Definition

An Aurifeuillian factorization is an operation to find the prime factors of integers of various forms based upon the identity:

$a^2 + b^2 = \paren {a - \sqrt {2 a b} + b} \paren {a + \sqrt {2 a b} + b}$

$a^3 + b^3 = \paren {a + b} \paren {a - \sqrt {3 a b} + b} \paren {a + \sqrt {3 a b} + b}$

Examples

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

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

Also known as

The term Aurifeuillian can also be seen as Aurifeuillean.

Source of Name

This entry was named for Léon-François-Antoine Aurifeuille.

Sources

• Aurifeuillian factor
• Aurifeuillian Factorizations
• The Cunningham Project
• Online Factor Collection
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{58} + 1$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{58} + 1$