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Lehmer Number

A Lehmer number is a number generated by a generalization of a Lucas sequence. Let alpha and beta be complex numbers with

alpha+beta=sqrt(R)
(1)
alphabeta=Q,
(2)

where Q and R are relatively prime nonzero integers and alpha/beta is not a root of unity. Then the corresponding Lehmer numbers are

U_n(sqrt(R),Q)={(alpha^n-beta^n)/(alpha-beta) for n odd,; (alpha^n-beta^n)/(alpha^2-beta^2) for n even,
(3)

and the companion numbers

V_n(sqrt(R),Q)={(alpha^n+beta^n)/(alpha+beta) for n odd; alpha^n+beta^n for n even.
(4)

See also

Lehmer's Mahler Measure Problem

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

Referenced on Wolfram|Alpha

Lehmer Number

Cite this as:

Weisstein, Eric W. "Lehmer Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LehmerNumber.html

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