For odd square-free
n
>
1
n > 1
the cyclotomic polynomial
Φ
n
(
x
)
{\Phi _n}(x)
satisfies the identity of Gauss,
\[
4
Φ
n
(
x
)
=
A
n
2
−
(
−
1
)
(
n
−
1
)
/
2
n
B
n
2
.
4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2.
\]
A similar identity of Aurifeuille, Le Lasseur, and Lucas is
\[
Φ
n
(
(
−
1
)
(
n
−
1
)
/
2
x
)
=
C
n
2
−
n
x
D
n
2
{\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2
\]
or, in the case that n is even and square-free,
\[
±
Φ
n
/
2
(
−
x
2
)
=
C
n
2
−
n
x
D
n
2
.
\pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2.
\]
Here,
A
n
(
x
)
,
…
,
D
n
(
x
)
{A_n}(x), \ldots ,{D_n}(x)
are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require
O
(
n
2
)
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
)
{A_n}(x), \ldots ,{D_n}(x)
, and illustrate the application to integer factorization with some numerical examples.