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# Aurifeuillean factorization

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 Title: Aurifeuillean factorization Author: World Heritage Encyclopedia Language: English Subject: Cyclotomic polynomial Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Aurifeuillean factorization

In number theory, an aurifeuillean factorization is a special type of algebraic factorizations that comes from non-trivial factorizations of cyclotomic polynomials.

## Examples

• Numbers of the form $2^\left\{4n+2\right\}+1$ have the following aurifeuillean factorization:[1]
$2^\left\{4n+2\right\}+1 = \left(2^\left\{2n+1\right\}-2^\left\{n+1\right\}+1\right)\cdot \left(2^\left\{2n+1\right\}+2^\left\{n+1\right\}+1\right).$
• Numbers of the form $b^n - 1$, where $b = s^2\cdot k$ with square-free $k$, have aurifeuillean factorization if one of the following conditions holds:
(i) $k\equiv 1 \pmod 4$ and $n\equiv k \pmod\left\{2k\right\};$
(ii) $k\equiv 2, 3 \pmod 4$ and $n\equiv 2k \pmod\left\{4k\right\}.$
• Numbers of the form $a^4 + 4b^4$ have the following aurifeuillean factorization:
$a^4 + 4b^4 = \left(a^2 - 2ab + 2b^2\right)\cdot \left(a^2 + 2ab + 2b^2\right).$

## History

In 1871, Aurifeuille discovered the factorization of $2^\left\{4n+2\right\}+1$ for n = 14 as the following:[1][2]

$2^\left\{58\right\}+1 = 536838145 \cdot 536903681. \,\!$

The second factor is prime, and the factorization of the first factor is $5 \cdot 107367629.$[2] The general form of the factorization was later discovered by Lucas.[1]