Upper Bounds for Frequencies of Elements in Second-Order Recurrences over A Finite Field

Abstract

Let F _q be a finite field with q elements and characteristic p. Let w(a,b) = (w) be a second-order linear recurrence over F _q satisfying the relation

$$w_{n + 2} = aw_{n + 1} - bw_n$$

(1)

with initial terms w ₀, w ₁ Given the consecutive terms w _n ,w _{n +1}, the preceding term

$$w_{n - 1} = (w_{n + 1} - aw_n )/( - b)$$

(2)

is uniquely determined. Thus, we will treat $\{ wn\} _n^\infty = - \infty$ as a doubly infinite sequence. It is known (see [3, pp. 344–45]) that if b ≠ 0, then w(a, b) is purely periodic. Throughout this paper we will assume that 6 khac 0. For the recurrence w(a,b), let g = ord(b), where ord(b) denotes the multiplicative order of b in F _q . Let H denote the length of a shortest period of (w). Then H is the least positive integer r such that w _n+r = w _n for all n.