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

• Numbers of the form $2^{4n+2}+1$ have the following aurifeuillean factorization:

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

• 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) $k\equiv 1(\mathrm{mod}4)$ and $n\equiv k(\mathrm{mod}2k);$

  (ii) $k\equiv 2,3(\mathrm{mod}4)$ and $n\equiv 2k(\mathrm{mod}4k).$

• 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:

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

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

References

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External links

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• Aurifeuillian Factorisation, Colin Barker