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…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (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).

Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences

“…Now, we follow the same calculations as we did for (16) to show that the bound n(2) < 1216 is valid. Finally, we write a short computer programme to check that the variables n, m, k, l, a and d are satisfying (1) by using the bounds n(k) for 2 ≤ k ≤ 650 together with (19) and (8). As a result, we find that there is no new solution of (1) except for those that were given in Theorem 1.…”

“…In recent years, many authors have worked on problems involving relations between terms of some binary recurrence sequences and repdigits [9,10,13,17,22]. Some authors extended these problems to the case involving order k generalization of these binary recurrence sequences [1,3,4,6,8,18,24,23]. In fact, in [16], Luca found all repdigits in Lucas sequence whereas in [7], the authors extend this result to the k−Lucas sequences.…”

Curious Generalized Fibonacci Numbers

“…See [13] for a problem involving repdigits in generalized pell sequences. In addition, Bravo-Luca [14] found all repdigits, which are sums of two k−Fibonacci numbers (see [15] for a product version). Problems concerning powers of two and coincidences in generalized Fibonacci numbers can be found in [16,17].…”