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 ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}{a}_n^{1\mathrm{/}{\phi }^n}=\mathrm{\infty }$ for a strictly increasing sequence of positive integers $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$ is sufficient for the series ${\sum }_{n=1}^{\mathrm{\infty }}1\mathrm{/}(a_n{a}_{n+1})$ to have an irrational sum; here $\phi$ denotes the golden ratio. We also provide a positive generalization to ${\sum }_{n=1}^{\mathrm{\infty }}1\mathrm{/}(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, 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.