Balancing and Lucas-balancing numbers as difference of two repdigits

TL;DR

This paper characterizes all balancing and Lucas-balancing numbers that can be expressed as the difference of two repdigits, using advanced number theory techniques.

Contribution

It provides a complete classification of such numbers, applying Baker's theory and reduction methods to solve the problem.

Findings

Identified all balancing numbers as differences of two repdigits.

Identified all Lucas-balancing numbers as differences of two repdigits.

Applied Baker's theory and reduction procedures to derive results.

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.