Lehmer number

In mathematics, a Lehmer number is a generalization of a Lucas sequence.

Algebraic relations

If a and b are complex numbers with

$a + b = \sqrt{R}$

$ab = Q$

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:

$U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a-b}$

for n odd, and

$U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a^2-b^2}$

for n even.

Their companion numbers are:

$V_n(\sqrt{R},Q) = \frac{a^n+b^n}{a+b}$

for n odd and

$V_n(\sqrt{R},Q) = a^n+b^n$

for n even.

Recurrence

Lehmer numbers form a linear recurrence relation with

$U_n=(R-2Q)U_{n-2}-Q^2U_{n-4}=(a^2+b^2)U_{n-2}-a^2b^2U_{n-4}$

with initial values $U_0=0,U_1=1,U_2=1,U_3=R-Q=a^2+ab+b^2$. Similarly the companions sequence satisfies

$V_n=(R-2Q)V_{n-2}-Q^2V_{n-4}=(a^2+b^2)V_{n-2}-a^2b^2V_{n-4}$

with initial values $V_0=2,V_1=1,V_2=R-2Q=a^2+b^2,V_3=R-3Q=a^2-ab+b^2$.