Primitive Divisors of Lucas Numbers

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