On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers

TL;DR

This paper investigates the characterization of balancing and Lucas-balancing numbers that can be expressed as the product of two $k$-generalized Fibonacci numbers, extending understanding of these special numbers within generalized Fibonacci sequences.

Contribution

It provides a complete characterization of balancing and Lucas-balancing numbers that are products of two $k$-Fibonacci numbers, a novel extension of classical number theory results.

Findings

Identifies all balancing numbers as products of two $k$-Fibonacci numbers.

Determines conditions under which Lucas-balancing numbers are such products.

Extends classical results to generalized Fibonacci sequences.

Abstract

A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a balancing number, then $8n^{2}+1$ is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer $k \geq 2$, let $\{F_{n}^{(k)} \}_{n \geq -(k-2)}$ denote $k$-generalized Fibonacci sequence which starts with $0, \dots ,1$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as the product of two $k$-generalized Fibonacci numbers.