Irrationality of rapidly converging series: a problem of Erd\H{o}s and Graham

TL;DR

This paper proves that certain rapidly growing integer sequences produce irrational sums in related series, answering a question by Erdős and Graham, with some results being optimal and others generalized.

Contribution

It establishes sufficient growth conditions for series to have irrational sums and provides a generalization and optimality results, combining human and AI research efforts.

Findings

Rapid exponential growth ensures irrational series sums.

Generalization to multi-term weighted series.

Some cases are proven to be essentially optimal.

Abstract

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