Erdős Problem #264

The problem

Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum

$$\sum \frac{1}{a_n+b_n}$$

is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?

Statement quoted from erdosproblems.com/264, maintained by Thomas Bloom. Original source: [ErGr80,p.63][Er88c,p.105]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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