On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function

TL;DR

This work investigates the distribution of integer solutions to the Ratat-Goormaghtigh type equation for fixed $N>1$, focusing on how many pairs $(x,m)$ satisfy $\frac{x^m-1}{x-1}=N$ and how small the sum of reciprocals $\sum 1/x$ can be. The authors reformulate the problem as counting lattice points near the graph of a smooth function $f_N(x)=\frac{\log N+\log(x-1)}{\log x}$ and combine explicit near-graph bounds with Matveev's bounds on linear forms in logarithms to control the second solution $x_2$ and the overall solution set. They obtain explicit bounds: $\sum_{i\ge2} 1/x_i<5.9037$, with the sum vanishing as $N$ grows, and a tighter bound for prime $x$ of $\sum_{i\ge2} 1/q_i<0.73194$ plus a product bound $\prod_{i\ge2} q_i/(q_i-1)<2.07913$. These results advance quantitative understanding of the distribution of Ratat-Goormaghtigh solutions and demonstrate the effectiveness of combining lattice-point counts with lower bounds for linear forms in logarithms.

Abstract

We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3