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 ${\mathrm{limsup}}_{n\to \infty }{a}_n^{1/{\varphi }^n}=\infty$ 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 $\varphi$ 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.

Minor modifications to the exposition. To appear in the Bulletin of the London Mathematical Society