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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 ..."
Abstract

Cited by 13 (5 self)
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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.