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Aurifeuillean factorization

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Title: Aurifeuillean factorization  
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Subject: Cyclotomic polynomial
Publisher: World Heritage Encyclopedia

Aurifeuillean factorization

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


  • 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).


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]


External links

  • Aurifeuillian Factorisation, Colin Barker
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