**Linear Difference Equations and Periodic Sequences over Finite Fields **

* Dang Vu Giang *

**Abstract**

First, we study linear equations over finite fields in general. An explicit formula for a common period is found for every solution of a linear difference equation over a finite field. It will help to estimate the p-adic modulus of polynomial roots. Second, we focus our attention on periodic sequences over finite fields and Hamiltonian cycles in de Bruijn directed graph.