Definition:Aurifeuillian Factorization
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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): $2^{58} + 1$