OPEN
This is open, and cannot be resolved with a finite computation.
Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum\[\sum\frac{1}{b_n}\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?
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One possible definition of an 'irrationality sequence' (see also
[262] and
[264]). A folklore result states that $\sum \frac{1}{a_n}$ is irrational whenever $\lim a_n^{1/2^n}=\infty$.
Kovač and Tao
[KoTa24] have proved that any strictly increasing sequence such that $\sum \frac{1}{a_n}$ converges and $\lim a_{n+1}/a_n^2=0$ is not such an irrationality sequence. On the other hand, if\[\liminf \frac{a_{n+1}}{a_n^{2+\epsilon}}>0\]for some $\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.
This problem was originally wrongly stated without the assumption that the sequence be increasing; DeepMind has given a counterexample without this assumption (see the comments).
Koizumi
[Ko25c] has proved that $a_n=\lfloor \alpha^{2^n}\rfloor$ is an irrationality sequence of this type for all but countably many $\alpha>1$.
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This page was last edited 02 April 2026. View history
Additional thanks to: Swarat Chaudhuri and Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #263, https://www.erdosproblems.com/263, accessed 2026-08-10
3 claimed proofs for this problem