$G$-Sets and Linear Recurrences Modulo Primes

Abstract

Let $p$ be a prime number with $p\neq 2$. We consider second order linear recurrence relations of the form $S_n=aS_{n-1}+bS_{n-2}$ over the finite field $Z_p$ (we assume $b\neq 0$). Results regarding the period and distribution of elements in the sequence $\{ S_0, S_1, \ldots \}$ are well-known (see works by Kuipers, Niederreiter, Wall, and Webb). We examine these second order recurrences using matrices, groups, and $G$-sets.