A Characterization of k−Generalized Fibonacci Numbers as Concatenations and Differences of repdigits

Abstract

The k-Generalized Fibonacci sequence is a generalization of the classic Fibonacci sequence with some fixed integer k ≥ 2. In this paper, we identify all k-Generalized Fibonacci numbers which can be represented as concatenation of three repdigits. This work builds upon and extends the previous research by Erduvan and Keskin, who identified all the Fibonacci numbers with this property. Using the former result we explore all the k−Generalized Fibonacci numbers, which can be expressed as the difference of two repdigits. A modified version of the Baker-Davenport reduction method (due to Dujella and Peth˝o) and lower bounds for linear forms in logarithms of algebraic numbers are used for the proof of our main theorem. The calculations were performed using Mathematica.