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On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) ..."
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For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.
Computing Aurifeuillian factors
 In Computational Algebra and Number Theory, Mathematics and its Applications Vol. 325
, 1995
"... Abstract. For odd squarefree 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 ..."
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Abstract. For odd squarefree 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 squarefree n, and similar results for the identities of Gauss and Dirichlet, will appear elsewhere. 1.