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

* $frac\{a\}\{b\}$ 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.