Problem Statement
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?
Categories:
Irrationality
Progress
In [Er88c] Erdős notes this is true if $a_n\to \infty$ 'rapidly'.Source: erdosproblems.com/1051 | Last verified: January 19, 2026