Lehmer Number
A Lehmer number is a number generated by a generalization of a Lucas sequence. Let 
and 
be complex numbers with
where 
and 
are relatively prime nonzero integers and 
is not a root of unity. Then the corresponding Lehmer numbers are
![U_n(sqrt(R),Q)=<span style=]() {(alpha^n-beta^n)/(alpha-beta) for n odd,; (alpha^n-beta^n)/(alpha^2-beta^2) for n even, " src="https://mathworld.wolfram.com/images/equations/LehmerNumber/NumberedEquation1.gif" style="height:90px; width:223px" /> |
(3)
|
and the companion numbers
![V_n(sqrt(R),Q)=<span style=]() {(alpha^n+beta^n)/(alpha+beta) for n odd; alpha^n+beta^n for n even. " src="https://mathworld.wolfram.com/images/equations/LehmerNumber/NumberedEquation2.gif" style="height:64px; width:221px" /> |
(4)
|
REFERENCES:
Lehmer, D. H. "An Extended Theory of Lucas' Functions." Ann. Math. 31, 419-448, 1930.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 61 and 70, 1989.
Shorey, T. N. and Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, 2." J. London Math. Soc. 23, 17-23, 1981.
Stewart, C. L. "On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers." Proc. London Math. Soc. 35, 425-447, 1977.
Williams, H. C. "The Primality of 
." Canad. Math. Bull. 15, 585-589, 1972.
