Thabit and Williams Numbers Base $b$ as a Sum or Difference of Two $g$-Repdigits

TL;DR

This paper explores conditions under which Thabit and Williams numbers in various bases can be written as sums or differences of two repdigits, providing parametric solutions, bounds, and complete solutions for specific cases.

Contribution

It introduces new parametric solutions and bounds for expressing Thabit and Williams numbers as sums or differences of two repdigits across different bases.

Findings

Infinite solutions for certain base and digit combinations.

Upper bounds established for parameters in finite solutions.

Complete solutions provided for specific equations.

Abstract

We investigate cases where Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$-repdigits. For specific values of $b$ and $g$, we describe parametric solutions yielding infinitely many solutions for some equations and establish upper bounds for the parameters of the remaining finitely many solutions. As an illustration, we also provide a complete solution for some equations.