Irrationality of rapidly converging series: a problem of Erdős and Graham

Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/ϕ^n}=\infty$ 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 $ϕ$ 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.

💡 Research Summary

The paper addresses a classic problem posed by Paul Erdős and Ronald Graham concerning the irrationality of the series
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