Lucas's Theorem

Lucas's theorem states that if  be a squarefree integer and  a cyclotomic polynomial, then

(1)

where  and  are integer polynomials of degree  and , respectively. This identity can be expressed as

(2)

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

2		1
3		1
5
6
7
10

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.