In number theory, an aurifeuillean factorization, or aurifeuillian factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.
Numbers of the form
2
4
k
+
2
+
1
have the following aurifeuillean factorization:
Numbers of the form
b
n
−
1
or
Φ
n
(
b
)
, where
b
=
s
2
⋅
t
with square-free
t
, have aurifeuillean factorization if and only if one of the following conditions holds:
t
≡
1
(
mod
4
)
and
n
≡
t
(
mod
2
t
)
t
≡
2
,
3
(
mod
4
)
and
n
≡
2
t
(
mod
4
t
)
Thus, when
b
=
s
2
⋅
t
with square-free
t
, and
n
is congruent to
t
mod
2
t
, then if
t
is congruent to 1 mod 4,
b
n
−
1
have aurifeuillean factorization, otherwise,
b
n
+
1
have aurifeuillean factorization.
When the number is of a particular form (the exact expression varies with the base), Aurifeuillian factorization may be used, which gives a product of two or three numbers. The following equations give Aurifeuillian factors for the Cunningham project bases as a product of
F,
L and
M:
If we let
L =
C −
D,
M =
C +
D, the Aurifeuillian factorizations for
bn ± 1 with the bases 2 ≤
b ≤ 24 (perfect powers excluded, since a power of
bn can also be regarded as a power of
b) are: (for the coefficients of the polynomials for all square-free bases up to 199, see )
(Number =
F * (
C −
D) * (
C +
D) =
F *
L *
M)
(See for more information (square-free bases up to 199))
Numbers of the form
a
4
+
4
b
4
have the following aurifeuillean factorization:
Lucas numbers
L
10
k
+
5
have the following aurifeuillean factorization:
where
L
n
is the
n
th Lucas number,
F
n
is the
n
th Fibonacci number.
In 1871, Aurifeuille discovered the factorization of
2
4
k
+
2
+
1
for k = 14 as the following:
2
58
+
1
=
536838145
⋅
536903681.
The second factor is prime, and the factorization of the first factor is
5
⋅
107367629
. The general form of the factorization was later discovered by Lucas.
