## Computing Aurifeuillian factors

127. R. P. Brent,
Computing Aurifeuillian factors,
in * Computational Algebra and Number Theory*
(edited by W. Bosma and A. van der Poorten),
Mathematics and its Applications, vol. 325,
Kluwer Academic Publishers, Boston, 1995, 201-212.
MR 96m:11111, 96c:00019.
Abstract:
dvi (3K),
pdf (70K),
ps (26K).

Paper:
dvi (20K),
pdf (174K),
ps (73K).

## Abstract

For odd square-free *n* > 1,
the cyclotomic polynomial *Phi*_{n}(*x*)
satisfies an identity
*Phi*_{n}(*x*)
=
*C*_{n}^{2}
__+__
*nxD*_{n}^{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, will appear in
[135].

## Comments

For a more comprehensive (but more difficult) paper on the same topic,
see [135].
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