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Sums of three Fibonacci numbers as concatenations of three repdigits in base $b$

Passimzouwé Dagou, Pagdame Tiebekabe, Kouèssi Norbert Adédji, Kokou Tchariè

Abstract

In this paper, we investigate sums of three Fibonacci numbers that can be expressed as concatenations of three repdigits in base $b$, where $b\ge 2$ is an integer. We prove that for bases $2\le b\le 10$, only finitely many such sums exist, and we determine all of them explicitly. Among these solutions, the largest occurs for $b=4$ and is given by $$ F_{42}+F_{29}+F_{20}=268435290=\overline{33333333331122}_4. $$

Sums of three Fibonacci numbers as concatenations of three repdigits in base $b$

Abstract

In this paper, we investigate sums of three Fibonacci numbers that can be expressed as concatenations of three repdigits in base , where is an integer. We prove that for bases , only finitely many such sums exist, and we determine all of them explicitly. Among these solutions, the largest occurs for and is given by
Paper Structure (6 sections, 6 theorems, 224 equations, 11 tables)

This paper contains 6 sections, 6 theorems, 224 equations, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear form in logarithms
  4. Reduction method
  5. Proof the Theorem \ref{['thrm1']}
  6. Proof of theorem \ref{['thrm2']}

Key Result

Theorem 1

Let $(n_1, n_2, n_3, \ell_1, \ell_2, \ell_3, d_1, d_2, d_3)$ be a solution to equation eq1 satisfying Then we have the explicit upper bound

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • proof
  • Remark 4.1