Generalized Fibonacci Sequences modulo Powers of a Prime

Let us begin by defining a generalized Fibonacci sequence (gn) with all gn in some abelian group as a sequence that satisfies the recurrence gn = gn−1 + gn−2 as n ranges over Z. The Fibonacci sequence (Fn) is the generalized Fibonacci sequence with integer values defined by F0 = 0 and F1 = 1. Recall also the Binet formula: for any integer n, Fn = (α − β)/ √ 5, where α = (1 + √ 5)/2 and β = (1− √ 5)/2 are the roots of the equation f(x) = x − x− 1. Let us define (gn) as the generalized Fibonacci sequence (gn) evaluated modulo m. For example, the sequence (Fn) = (. . . , 0, 1, 1, 2, 0, 2, 2, 1, 0, . . . ) is the Fibonacci sequence evaluated modulo 3. It may be shown that any generalized Fibonacci sequence with all elements in Z modulo any integer m > 1 is a periodic generalized Fibonacci sequence with elements in Z/m. In this paper we will look at the period of such sequences modulo powers of prime numbers.