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Computing Aurifeuillian Factors

  • Richard P. Brent
Part of the Mathematics and Its Applications book series (MAIA, volume 325)

Abstract

For odd square-free n > 1 the cyclotomic polynomial Φ n (x) satisfies an identity Φ n (x) = C n (x)2 ± nx D n (x)2 of Aurifeuille, Le Lasseur and Lucas. Here C n (x) and D n (x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require O(n 2) arithmetic operations over the integers. Also, there are explicit formulas and generating functions for C n (x) and D n (x). This paper is a preliminary report which states the results for the case n ≡ 1 mod 4, and gives some numerical examples. The proofs, generalisations to other square-free n, and similar results for the identities of Gauss and Dirichlet, appear elsewhere[1].

Keywords

Arithmetic Operation Monic Polynomial Quadratic Field Integer Coefficient Euclidean Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Richard P. Brent
    • 1
  1. 1.Computer Sciences LabAustralian National UniversityCanberraAustralia

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