On Concatenations of Two k-Generalized Fibonacci Numbers

Let k ≥ 2 be an integer. The $ k- $generalized Fibonacci sequence is a sequence defined by the recurrence relation Fₙ⁽ᵏ⁾=Fₙ₋₁⁽ᵏ⁾ + ⋯ + Fₙ₋ₖ⁽ᵏ⁾ for all n ≥ 2 with the initial values Fᵢ⁽ᵏ⁾=0 for i=2-k, …, 0 and F₁⁽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 ≥ 3.