On Sums of the Reciprocals of Prime Divisors of Terms of a Linear Recurrence

Abstract

Let $R = \{ {R_n}\} \begin{array}{*{20}{c}} \infty \\ {n = 0} \\ \end{array}$ be a linear recurrent sequence of rational integers defined by the recursion

$${R_n} - A{R_n}_{ - 1} + B{R_{n - 2}}{\rm{ }}(n > 1)$$

with coprime integers A,B and initial terms R ₀ = 0, R ₁ = 1. We suppose that the sequence R is non-degenerate, i.e. α/β is not a root of unity, where α and β are the roots of the characteristic equation ${x^2} - Ax - B = 0$.