Erdős Problem #1051

The problem

Is it true that if $1\leq a_1<a_2<\cdots$ is a sequence of integers with

$$\liminf a_n^{1/2^n}>1$$

then

$$\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}$$

is irrational?

Statement quoted from erdosproblems.com/1051, maintained by Thomas Bloom. Original source: [ErGr80,p.64][Er88c,p.106]. Current status there: proved (Lean).

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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