This conjecture is due to Stolarsky.

A negative answer was proved by Kovač and Tao [KoTa24], who proved even more: there exists a strictly increasing sequence of positive integers $a_n$ such that\[\sum \frac{1}{a_n+t}\]converges to a rational number for every $t\in \mathbb{Q}$ (with $t\neq -a_n$ for all $n$).

View the LaTeX source

This page was last edited 28 September 2025.