Divisibility of Terms in Lucas Sequences by Their Subscripts

Abstract

Let (U) = U(P,Q) be a Lucas sequence of the first kind (LSFK) satisfying the second-order relation

$$U_{n + 2} = PU_{n + 1} - QU_n$$

(1)

and having initial terms U ₀ = 0, U ₁ = 1, where P and Q are integers. Let D = P ² − 4Q be the discriminant of U(P,Q). Associated with U(P,Q) is the characteristic polynomial

$$f(x) = x^2 - Px + Q$$

(2)

with characteristic roots α and β. By the Binet formula

$$U_n = (\alpha ^n - \beta ^n )/(\alpha - \beta )$$

(3)

if D ≠ 0. It is also known that

$$U_n = n\alpha ^{n - 1}$$

(4)

if D = 0.