Lehmer Number

A number generated by a generalization of a Lucas Sequence. Let $\alpha$ and $\beta$ be Complex Numbers with

$\begin{displaymath} \alpha+\beta=\sqrt{R} \end{displaymath}$

(1)

$\begin{displaymath} \alpha\beta=Q, \end{displaymath}$

(2)

where

and

are Relatively Prime Nonzero Integers and $\alpha/\beta$ is a Root of Unity. Then the Lehmer numbers are

$\begin{displaymath} U_n(\sqrt{R}, Q)={\alpha^n-\beta^n\over\alpha-\beta}, \end{displaymath}$

(3)

and the companion numbers

$\begin{displaymath} V_n(\sqrt{R}, Q)=\cases{ {\alpha^n+\beta^n\over \alpha+\beta} & for $n$\ odd\cr \alpha^n+\beta^n & for $n$\ even.\cr} \end{displaymath}$

(4)

References

Lehmer, D. H. ``An Extended Theory of Lucas' Functions.'' Ann. Math. 31, 419-448, 1930.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 61 and 70, 1989.

Williams, H. C. ``The Primality of .'' Canad. Math. Bull. 15, 585-589, 1972.