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arXiv · 2601.21442

Irrationality of rapidly converging series: a problem of Erd\H{o}s and Graham

Kevin Barreto·Jiwon Kang·Sang-hyun Kim·Vjekoslav Kovač·Shengtong Zhang
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Abstract

Answering a question of Erd\H{o}s and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/\phi^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $\phi$ denotes the golden ratio. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

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Kevin Barreto, Jiwon Kang, Sang-hyun Kim, Vjekoslav Kovač, Shengtong Zhang. 2026-01-29. Irrationality of rapidly converging series: a problem of Erd\H{o}s and Graham. https://arxiv.org/abs/2601.21442

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