Distribution of Residues of Certain Second-Order Linear Recurrences Modulo P

Abstract

Let (w) = w(a, b) be a second-order linear recurrence defined by the relation

$${w_{n + 2}} = a{w_{n + 1}} + b{w_n},$$

((1))

where the parameters a and b and the initial terms w₀, w₁ are all integers. Let D = a² + 4b be the discriminant of w(a, b). Let

$${x^2} - ax - b$$

be the characteristic polynomial associated with w(a, b) and let r₁ and r₂ be its characteristic roots. Throughout this paper, p will denote an odd prime unless specified otherwise. Further, d will always denote a residue modulo p. We say that the recurrence (w) is defective modulo p if (w) has an incomplete system of residues modulo p.