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

Abstract Answering a question of Erdős 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.