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Lehmer Number

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

\end{displaymath} (1)

\end{displaymath} (2)

where $Q$ and $R$ are Relatively Prime Nonzero Integers and $\alpha/\beta$ is a Root of Unity. Then the Lehmer numbers are
U_n(\sqrt{R}, Q)={\alpha^n-\beta^n\over\alpha-\beta},
\end{displaymath} (3)

and the companion numbers
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)


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 $N=2A3^n-1$.'' Canad. Math. Bull. 15, 585-589, 1972.

© 1996-9 Eric W. Weisstein