Repdigits as Product of Fibonacci and Tribonacci Numbers

Abstract

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö.

Padovan numbers which are palindromic concatenations of two distinct repdigits

“…[1], the second author considered all Padovan numbers that can be written as a concatenation of two distinct repdigits and showed that the largest such number is P 21 = 200. More specifically, it was shown that if P n is a solution of the Diophantine equation Other related interesting results in this research direction include: the result of Bednařík and Trojovská [3], the result of Boussayoud, et al [4], the result of Bravo and Luca [5], the result of the second author [7], the result of Erduvan and Keskin [11], the result of Rayaguru and Panda [16], the results of Trojovský [17,18], and the result of Qu and Zeng [15]. A natural continuation of the result in Ref.…”

“…With the notation of Lemma 3, we let r := 3, L := n, and H := 5 • 10 58 and notice that this data meets the conditions of the lemma. Applying the lemma, we have that n < 2 3 • 5 • 10 58 (log(5 • 10 58 )) 3 .…”

Perrin numbers that are concatenations of two distinct repdigits

“…Consider the Lucas number sequence {L n } n≥0 , which starts with L 0 = 2, L 1 = 1, and follows the pattern L n+2 = L n+1 + L n for all n ≥ 0. The initial numbers in this sequence are 2, 1, 3,4,7,11,18,29,47,76,123,199, . .…”

On Repdigits as Sums of Fibonacci and Tribonacci Numbers

“…In the last years, many authors have worked on Diophantine problems related to repdigits (e.g., their sums, concatenations) and linear recurrences (e.g., their product, sums). For more about this subject, we refer the reader to [14][15][16][17][18][19][20][21][22][23][24] and references therein.…”