OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that if $a_1<a_2<\cdots$ 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?
In
[Er88c] Erdős notes this is true if $a_n\to \infty$ 'rapidly'.
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This page was last edited 04 January 2026.
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