Sylvester Cyclotomic Number
Given a Lucas sequence with parameters P and Q, discriminant D!=0, and roots a and b, the Sylvester cyclotomic numbers are
where
| zeta=e^(2pii/n) |
(2)
|
is a primitive root of unity and the product is over all exponents r relatively prime to n such that r in [1,n).
For small n, the first few values are
Q_0 = 1
(3)
Q_1 = 1
(4)
Q_2 = P
(5)
Q_3 = P^2-Q
(6)
Q_4 = P^2-2Q
(7)
Q_5 = P^4-3QP^2+Q^2
(8)
Q_6 = P^2-3Q.
(9)
These numbers satisfy
| [画像: U_n=product_(d|n)Q_d, ] |
(10)
|
where as usual U_n=(a^n-b^n)/(a-b).
Ward (1954) gave a primality test involving these numbers.
See also
Lucas SequenceExplore with Wolfram|Alpha
WolframAlpha
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References
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 82, 1989.Ward, M. "Prime Divisors of Second Order Recurring Sequences." Duke Math. J. 21, 607-614, 1954.Referenced on Wolfram|Alpha
Sylvester Cyclotomic NumberCite this as:
Weisstein, Eric W. "Sylvester Cyclotomic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SylvesterCyclotomicNumber.html