Recursive sequences and polynomial congruences

We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer m. We show that the period of such a sequence with characteristic polynomial f can be expressed in terms of the order of ω = x + f as a unit in the quotient ring ‫ޚ‬ m [ω] = ‫ޚ‬ m [x]/ f. When m = p is prime, this order can be described in terms of the factorization of f in the polynomial ring ‫ޚ‬ p [x]. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree k ≤ 5 in ‫ޚ‬ p [x].