$k$--Fibonacci numbers with two blocks of repdigits

TL;DR

This paper characterizes all $k$--Fibonacci numbers that are formed by concatenating two repdigits, extending previous results from Fibonacci and Tribonacci numbers to the general $k$--Fibonacci sequence.

Contribution

It provides a complete classification of $k$--Fibonacci numbers that are concatenations of two repdigits, generalizing earlier specific cases.

Findings

Identifies all $k$--Fibonacci numbers as concatenations of two repdigits.

Extends previous results from Fibonacci and Tribonacci sequences.

Provides a comprehensive solution for all $k \

Abstract

A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we find all $k$--Fibonacci numbers that are concatenations of two repdigits. This generalizes prior results which dealt with the above problem for the particular cases of Fibonacci and Tribonacci numbers.