In mathematics, a Lehmer number is a generalization of a Lucas sequence.
If a and b are complex numbers with
a
+
b
=
R
a
b
=
Q
under the following conditions:
Q and R are relatively prime nonzero integers
a
/
b
is not a root of unity.
Then, the corresponding Lehmer numbers are:
U
n
(
R
,
Q
)
=
a
n
−
b
n
a
−
b
for n odd, and
U
n
(
R
,
Q
)
=
a
n
−
b
n
a
2
−
b
2
for n even.
Their companion numbers are:
V
n
(
R
,
Q
)
=
a
n
+
b
n
a
+
b
for n odd and
V
n
(
R
,
Q
)
=
a
n
+
b
n
for n even.
Lehmer numbers form a linear recurrence relation with
U
n
=
(
R
−
2
Q
)
U
n
−
2
−
Q
2
U
n
−
4
=
(
a
2
+
b
2
)
U
n
−
2
−
a
2
b
2
U
n
−
4
with initial values
U
0
=
0
,
U
1
=
1
,
U
2
=
1
,
U
3
=
R
−
Q
=
a
2
+
a
b
+
b
2
. Similarly the companions sequence satisfies
V
n
=
(
R
−
2
Q
)
V
n
−
2
−
Q
2
V
n
−
4
=
(
a
2
+
b
2
)
V
n
−
2
−
a
2
b
2
V
n
−
4
with initial values
V
0
=
2
,
V
1
=
1
,
V
2
=
R
−
2
Q
=
a
2
+
b
2
,
V
3
=
R
−
3
Q
=
a
2
−
a
b
+
b
2
.
