Primitive Divisors of Lucas Numbers

Abstract

Let $R = \{ {R_n}\} _{n = 1}^\infty$ be a Lucas sequence defined by fixed rational integers A and B and by the recursion relation

$$ {R_n} = A \cdot {R_{n - 1}} + B \cdot {R_{n - 2}} $$

for n > 2, where the initial values are R₁ = 1 and R₂ = A. The terms of R are called Lucas numbers. We shall denote the roots of the characteristic polynomial

$$ f(x) = {x^2} - Ax - B $$

by α and β. We may assume that |α| ≥ |β| and the sequence is not degenerate, that is, AB ≠ 0, A² + 4B ≠ 0 and α/ß is not a root of unity. In this case, as it is wellknown, the terms of the sequence R can be expressed as

$$ {R_n} = \frac{{{\alpha ^n} - {\beta ^n}}}{{\alpha - \beta }}\quad (n = 1,2,...) $$

.