On Concatenations of Two $ k $-Generalized Fibonacci Numbers

TL;DR

This paper investigates when concatenations of two $k$-generalized Fibonacci numbers result in another such number, providing a complete solution for all $k \\geq 3$, extending previous work on standard Fibonacci numbers.

Contribution

It generalizes the problem of concatenations of Fibonacci numbers to $k$-generalized Fibonacci sequences and completely characterizes solutions for all $k \\geq 3$.

Findings

Identifies all concatenations of two $k$-generalized Fibonacci numbers that are also $k$-generalized Fibonacci numbers for $k \\geq 3$.

Extends previous results from classical Fibonacci numbers to the generalized case.

Provides a complete classification of such concatenations for all $k \\geq 3$.

Abstract

Let $ k \geq 2 $ be an integer. The $ k- $generalized Fibonacci sequence is a sequence defined by the recurrence relation $ F_{n}^{(k)}=F_{n-1}^{(k)} + \cdots + F_{n-k}^{(k)}$ for all $ n \geq 2$ with the initial values $ F_{i}^{(k)}=0 $ for $ i=2-k, \ldots, 0 $ and $ F_{1}^{(k)}=1.$ In 2020, Banks and Luca, among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem by taking into account $ k-$generalized Fibonacci numbers as concatenations of two terms of the same sequence. We completely solve this problem for all $ k \geq 3.