# Non-autonomous mechanics

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

## Examples

${\displaystyle 2^{4n+2}+1=(2^{2n+1}-2^{n+1}+1)\cdot (2^{2n+1}+2^{n+1}+1).}$
${\displaystyle a^{4}+4b^{4}=(a^{2}-2ab+2b^{2})\cdot (a^{2}+2ab+2b^{2}).}$

## History

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

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

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

## References

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