PROVED (LEAN) This has been solved in the affirmative and the proof verified in Lean.
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?
In [Er88c] Erdős notes this is true if $a_n\to \infty$ 'rapidly'. In [ErGr80] they further ask 'what the strongest theorem of this type' would be.
This was solved in the affirmative by Aletheia [Fe26]. This was extended by Barreto, Kang, Kim, Kovač, and Zhang [BKKKZ26], who essentially give a complete answer: if $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio and $1\leq a_1<a_2<\cdots$ is a monotonically increasing sequence of integers such that\[\limsup a_n^{1/\phi^{n}}=\infty\]then\[\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}\]is irrational. Conversely, for any $1<C<\infty$ there exists a sequence of integers $1\leq a_1<\cdots$ such that\[\lim a_n^{1/\phi^{n}}=C\]where this infinite sum is a rational number. (Further, more general, results are available in [BKKKZ26].)
This page was last edited 01 February 2026. View history
External data from the database - you can help update this
Formalised statement?
Yes
Additional thanks to: Superhuman Reasoning team at Google DeepMind
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 #1051, https://www.erdosproblems.com/1051, accessed 2026-03-17
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This preprint, by myself and collaborators, gives a positive answer to the stated question. The solution was found completely autonomously by an AI agent powered by Gemini Deep Think, but I will report on that in more detail in a few days, when the methodology is officially released by a Google DeepMind team. Also, I have formalised the solution to the original problem variant in Lean, which can be viewed here.
We are hoping to improve our results further, but this will take some time to explore.
(The site has been updated to address this comment.)
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A positive answer to this question is given by K. Barreto, J. Kang, S. Kim, V. Kovac and S. Zhang in [BKKKZ26].
(The site has been updated to address this comment.)
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