Lehmer number

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

Algebraic relations

If a and b are complex numbers with

$a %2B b = \sqrt{R}$

$ab = Q$

under the following conditions:

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:

$V_n(\sqrt{R},Q) = \frac{a^n%2Bb^n}{a%2Bb}$

for n odd and

$V_n(\sqrt{R},Q) = a^n%2Bb^n$

for n even.