Repdigits as Product of Terms of k-Bonacci Sequences

Abstract

For any integer k >= 2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F-(k-2)(k)=...=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa horizontal ellipsis a, with a is an element of[1,9]) in the sequence (F(n)((k))F(n)((k+m)))n, for m is an element of[1,9]. This result generalizes a recent work of Bednarik and Trojovska (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).