CYCLOTOMIC POLYNOMIALS AND FACTORIZATION THEOREMS

For example, when f(x) =9x 2+x+ 1 = 3(x) and a =2, we find from Theorem 1 that a necessary condition for 4r + 2r + 1 to be prime is that r be a power of 3. A similar result, which has been applied by Liu, Reed, and Truong [1] based on f(x) = x2x+1= 16(x), and a = 2, is that a necessary condition for 4r 2r + 1 to be prime is that r > 1 be a multiple of 4 with no prime factor greater than three, i.e., that r = 2a3ft with a > 2, ,B > 0. When r is of this form, it can be shown (Theorem 4) that every prime q which divides 4r 2r + 1 is of the form 6rk + 1. Thus, in attempting to factor 412-212+1 =16,773,121, it suffices to look only for prime factors of the form q=72k+ 1. (In fact, the factors are q1 = 433 = 72 x 6 + 1 and q2= 38,737 = 72 x 538 + 1, where 433 is only the second prime number in the sequence 72k + 1.) The techniques introduced in this paper involve the factorization of Dn(x r) over the rational field, the factorization of tDn(a) over the integers, and, for comparison and completeness, the factorization of tDn(x) and Dn (Xr) over the integers modulo q.