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Balancing and Lucas-balancing numbers as difference of two repdigits

Monalisa Mohapatra, Pritam Kumar Bhoi, Gopal Krishna Panda

TL;DR

This work determines which Balancing numbers $B_k$ and Lucas-balancing numbers $C_k$ can be written as the difference of two repdigits. It reduces the problem to solving $B_k=d_1\frac{10^n-1}{9}-d_2\frac{10^m-1}{9}$ and $C_k=d_1\frac{10^n-1}{9}-d_2\frac{10^m-1}{9}$ by leveraging the recurrences and Binet formulas with $\alpha=3+2\sqrt{2}$ and $\beta=3-2\sqrt{2}$, and applying Matveev's lower bounds on linear forms in logarithms together with the Baker–Davenport reduction. The authors obtain that only small indices $k$ (up to 25) yield solutions, with explicit small-$k$ solutions listed, and no solutions exist for larger $k$. This extends prior results on repdigits in second-order recurrences and demonstrates the effectiveness of Baker-type methods for exponential Diophantine problems within recurrence sequences.

Abstract

Positive integers with all digits equal are called repdigits. In this paper, we find all balancing and Lucas-balancing numbers, which can be expressed as the difference of two repdigits. The method of proof involves the application of Baker's theory for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure.

Balancing and Lucas-balancing numbers as difference of two repdigits

TL;DR

This work determines which Balancing numbers and Lucas-balancing numbers can be written as the difference of two repdigits. It reduces the problem to solving and by leveraging the recurrences and Binet formulas with and , and applying Matveev's lower bounds on linear forms in logarithms together with the Baker–Davenport reduction. The authors obtain that only small indices (up to 25) yield solutions, with explicit small- solutions listed, and no solutions exist for larger . This extends prior results on repdigits in second-order recurrences and demonstrates the effectiveness of Baker-type methods for exponential Diophantine problems within recurrence sequences.

Abstract

Positive integers with all digits equal are called repdigits. In this paper, we find all balancing and Lucas-balancing numbers, which can be expressed as the difference of two repdigits. The method of proof involves the application of Baker's theory for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure.
Paper Structure (4 sections, 3 theorems, 68 equations)

This paper contains 4 sections, 3 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Results
  3. Balancing numbers as difference of two repdigits
  4. Lucas-balancing numbers as difference of two repdigits

Key Result

Theorem 2.1

m2000. Let $\mathbb{L}$ be an algebraic number field of degree $d_{\mathbb{L}}$. Let $\gamma_1,\ldots,\gamma_l \in \mathbb{L}$ be positive real numbers and $b_1, \ldots, b_l$ be nonzero integers. If $\Gamma = \prod\limits_{i=1}^{l} \gamma_{i}^{b_i} -1$ is not zero, then where $D\geq$ max$\{|b_1|,\ldots,|b_l|\}$ and $A_1, \cdots, A_l$ are positive integers such that $A_j \geq h'(\gamma_j)$ = $max\

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof