Recursive sequences and polynomial congruences

Abstract

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!D xCh fi as a unit in the quotient ring ZmT!UDZmTxU=h fi. When mD p is prime, this order can be described in terms of the factorization of f in the polynomial ring Z pTxU. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree k 5 in Z pTxU. This article grew out of an undergraduate research project, performed by the second author under the direction of the first, to determine if results about the periodicity of second-order linear homogeneous recurrence relations modulo positive integers could be extended to higher orders. We arrived, somewhat unexpectedly, at algorithms to determine the degrees of the irreducible factors of quintic and smaller degree polynomials modulo prime numbers. The algebraic properties of certain finite rings, particularly automorphisms of those rings, provided the connection between these two topics. To illustrate some of the ideas in this article, we begin with the famous example of the Fibonacci sequence, defined by FnD Fn 1C Fn 2 with F0D 0 and F1D 1. If, for some positive integer m, we replace each Fn by its remainder on division by m, we obtain a new sequence of integers. For example, the Fibonacci sequence modulo mD 10 begins 0; 1; 1; 2; 3; 5; 8; 3; 1; 4; 5; 9; 4; 3; 7; 0; 7; 7; 4; 1; 5; 6; 1; 7; 8; 5;:::; with the n-th term simply the last digit of Fn. We can also view such a sequence as having terms in ZmDZ=hmi, the ring of integers modulo m. This has the advantage