OPEN
This is open, and cannot be resolved with a finite computation.
Let $a_n$ be a sequence of 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?
A possible definition of an 'irrationality sequence' (see also
[262] and
[263]). One example is $a_n=2^{2^n}$. In
[ErGr80] they also ask whether such a sequence can have polynomial growth, but Erdős later retracted this in
[Er88c], claiming 'It is not hard to show that it cannot increase slower than exponentially'.
Kovač and Tao
[KoTa24] have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\sum\frac{1}{a_n}$ converges and\[\liminf \left(a_n^2\sum_{k>n}\frac{1}{a_k^2}\right) >0 \]is not such an irrationality sequence. In particular, any strictly increasing sequence with $\limsup a_{n+1}/a_n <\infty$ is not such an irrationality sequence.
On the other hand, Kovač and Tao do prove that for any function $F$ with $\lim F(n+1)/F(n)=\infty$ there exists such an irrationality sequence with $a_n\sim F(n)$.
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This page was last edited 28 September 2025.
Additional thanks to: 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 #264, https://www.erdosproblems.com/264, accessed 2025-11-16
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