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

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 $0.73194$.

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ArXiv.org
Published
2025-10-13
Primary Topic
Analytic Number Theory Research
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article

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

ArXiv.org
Analytic Number Theory Research
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On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function

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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 $0.73194$.

ArXiv.org
Osaka Nishi Clinic (JP)
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Analytic Number Theory Research
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On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function · ArXiv.org (2025) | TGRS Research Map | TGRS