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

We prove that the sum of reciprocals $1\mathrm{/}x$ of integer solutions of $(x^m-1)\mathrm{/}(x-1)=N$ with $x,m\ge 2$ 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 $0.73194$.