Repdigits as difference of two balancing or Lucas-balancing numbers

TL;DR

This paper investigates when repdigits can be expressed as the difference of two balancing or Lucas-balancing numbers, using advanced number theory techniques and computational tools.

Contribution

It provides new results on the representation of repdigits as differences of balancing or Lucas-balancing numbers, applying Baker's theory and computational methods.

Findings

Identifies all repdigits that can be expressed as such differences.

Develops bounds and conditions for these representations.

Uses computational verification to confirm theoretical results.

Abstract

Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we study the problem of writing repdigits as the difference of two balancing or Lucas-balancing numbers. 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. Computations are done with the help of a simple computer program in {\it Mathematica}.