## Computing Aurifeuillian factors (1995)

Venue: | In Computational Algebra and Number Theory, Mathematics and its Applications Vol. 325 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Brent95computingaurifeuillian,

author = {Richard P. Brent},

title = {Computing Aurifeuillian factors},

booktitle = {In Computational Algebra and Number Theory, Mathematics and its Applications Vol. 325},

year = {1995},

pages = {201--212}

}

### OpenURL

### Abstract

Abstract. For odd square-free n> 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) 2 ± nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(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 Cn(x) and Dn(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, will appear elsewhere. 1.

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Citation Context .... P. Brent. rpb127 typeset using AMS-L ATEX. (3)s2 R. P. BRENT Φn(x) satisfies an identity Φn(x) = Cn(x) 2 − nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas 1 . For a proof, see Lucas [15] or Schinzel =-=[17]-=-. Here Cn(x) and Dn(x) are symmetric, monic polynomials with integer coefficients. For example, if n = 5, we have Φ5(x) = x 4 + x 3 + x 2 + x + 1 = (x 2 + 3x + 1) 2 − 5x(x + 1) 2 , so C5(x) = x 2 + 3x... |

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Citation Context ...yright c○ 1992, R. P. Brent. rpb127 typeset using AMS-L ATEX. (3)s2 R. P. BRENT Φn(x) satisfies an identity Φn(x) = Cn(x) 2 − nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas 1 . For a proof, see Lucas =-=[15]-=- or Schinzel [17]. Here Cn(x) and Dn(x) are symmetric, monic polynomials with integer coefficients. For example, if n = 5, we have Φ5(x) = x 4 + x 3 + x 2 + x + 1 = (x 2 + 3x + 1) 2 − 5x(x + 1) 2 , so... |

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