Generalized Fibonacci and Lucas Numbers of the form $5x^{2}$

TL;DR

This paper investigates the solutions of generalized Fibonacci and Lucas numbers of the form 5 times a perfect square, characterizing when such numbers occur and proving certain equations have no solutions.

Contribution

It determines all indices where generalized Fibonacci numbers are multiples of 5 times a perfect square and shows specific equations involving generalized Lucas numbers have no solutions.

Findings

Identifies all n with U_n = 5 * square

Shows V_n = 5 * square only when n=1 for odd P

Proves V_n = 5 * V_m * square has no solutions

Abstract

Let $(U_{n}(P,Q) $ and $(V_{n}(P,Q) $ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_{n}=5\square $ and $U_{n}=5U_{m}\square $ under some assumptions on $P.$ We show that the equation $V_{n}=5\square $ has the solution only if $n=1$ for the case when $% P$ is odd. Moreover, we show that the equation $V_{n}=5V_{m}\square $ has no solutions.