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Lucas's Theorem

Lucas's theorem states that if n>=3 be a squarefree integer and Phi_n(z) a cyclotomic polynomial, then

Phi_n(z)=U_n^2(z)-(-1)^((n-1)/2)nzV_n^2(z),
(1)

where U_n(z) and V_n(z) are integer polynomials of degree phi(n)/2 and phi(n)/2-1, respectively. This identity can be expressed as

{Phi_n((-1)^((n-1)/2)z)=C_n^2(z)-nzD_n^2(z) for n odd; Phi_(n/2)(-z^2)=C_n^2(z)-nzD_n^2(z) n=4k+2; -Phi_1(-z^2)=C_2^2(z)-2zD_2^2(z) for n=2,
(2)

with C_n(z) and D_n(z) symmetric polynomials. The following table gives the first few C_n(z) and D_n(z)s (Riesel 1994, pp. 443-456).

nC_n(z)D_n(z)
2z+11
3z+11
5z^2+3z+1z+1
6z^2+3z+1z+1
7z^3+3z^2+3z+1z^2+z+1
10z^4+5z^3+7z^2+5z+1z^3+2z^2+2z+1

See also

Cyclotomic Polynomial, Gauss's Cyclotomic Formula

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References

Brent, R. P. "On Computing Factors of Cyclotomic Polynomials." Math. Comput. 61, 131-149, 1993.Kraitchik, M. Recherches sue la théorie des nombres, tome I. Paris: Gauthier-Villars, pp. 126-128, 1924.Riesel, H. "Lucas's Formula for Cyclotomic Polynomials." In tables at end of Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 443-456, 1994.

Referenced on Wolfram|Alpha

Lucas's Theorem

Cite this as:

Weisstein, Eric W. "Lucas's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LucassTheorem.html

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