On Repdigits as Sums of Fibonacci and Tribonacci Numbers

Pavel Trojovský

doi:10.3390/sym12111774

Abstract

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form Fn+Tn, for some n≥1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively.

Highlights

A palindromic number is a number that has the same form when written forwards or backwards, i.e., of the form c1 c2 c3 . . . c3 c2 c1
An old open problem consists in proving the existence of infinitely many prime repunit numbers, where theth repunit is defined as
Many authors have worked on Diophantine problems related to repdigits and linear recurrences

Summary

Introduction

A palindromic number is a number that has the same form when written forwards or backwards, i.e., of the form c1 c2 c3 . . . c3 c2 c1 ( it can be said that they are “symmetrical” with respect to an axis of symmetry). 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. The first one is the omnipresent sequence ( Fn )n These numbers are defined by the second order linear recurrence. The sequence of Tribonacci numbers ( Tn )n (generalizes the Fibonacci sequence) is defined by the third-order recurrence. In positive integers (n, `, a), with a ∈ [1, 9], are (n, `, a) ∈ {(1, 1, 2), (2, 1, 2), (3, 1, 4), (4, 1, 7)}

Auxiliary Results

Finding an Upper Bound for n and

Reducing the Bound

Conclusions