DISPROVED
This has been solved in the negative.
Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t\geq 1$ such that\[\sum \frac{1}{a_n+t}\]is irrational.
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 $n$).
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This page was last edited 28 September 2025.
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