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

TL;DR

This work investigates when generalized Thabit numbers base $b$ and Williams numbers base $b$ can be expressed as sums or differences of two $g$-repdigits, i.e., representations of the form $N = d_1 (g^l-1)/(g-1) \pm d_2 (g^m-1)/(g-1)$ with $1\le d_i \le g-1$. The authors combine linear forms in logarithms with the Bravo–Gómez–Luca reduction framework to prove finiteness of non-infinite cases and to construct infinite parametric families in special parameter choices. They derive explicit upper bounds for the indices, e.g. $l,m < 2.71 \cdot 10^{29} \log^2 g \log^3 b \log^2(\max\{b,g\})$ and $n < 2.08 \cdot 10^{29} \log^3 g \log^2 b \log^2(\max\{b,g\})$, and perform detailed computations for $g=2$ and $g=10$ with $2 \le b \le 12$, yielding complete enumerations in those regimes (e.g., 164 finite sum-solution cases and 160 finite difference-solution cases) with supporting tables and examples. These results advance the understanding of repdigit representations within generalized Thabit and Williams sequences and illustrate the effectiveness of the reduction method in Diophantine problems involving base-$b$ representations.

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.

Theorems & Definitions (31)

• Definition 1
• Proposition 1
• Proposition 2
• Theorem 3
• Theorem 4
• Lemma 5: Theorem 9.4 of BMS:2006
• Lemma 6: Lemma 7 of SGL
• Lemma 7
• proof
• Lemma 8