Aurifeuillean factorization

(Redirected from Aurifeuillian 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 $2^{4n+2}+1$ have the following aurifeuillean factorization:[1]
$2^{4n+2}+1 = (2^{2n+1}-2^{n+1}+1)\cdot (2^{2n+1}+2^{n+1}+1).$
• 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 $n\equiv k \pmod{2k};$
(ii) $k\equiv 2, 3 \pmod 4$ and $n\equiv 2k \pmod{4k}.$
• Numbers of the form $a^4 + 4b^4$ have the following aurifeuillean factorization:
$a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2).$

History

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

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

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]