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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

 Q_n=product_(r)(a-zeta^rb),
(1)

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

 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 Sequence

Explore with Wolfram|Alpha

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 Number

Cite this as:

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

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