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Title: Irrationality of rapidly converging series: A problem of Erdős and Graham

Abstract: Answering a question of Erd & odblac;s and Graham, we show that the double-exponential growth condition for a strictly increasing sequence of positive integers is sufficient for the series to have an irrational sum; here denotes the golden ratio. We also provide a positive generalization to , and a negative result showing that some of these instances are essentially optimal. The original problem was autonomously solved by the AI agent Aletheia, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

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