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

TL;DR

The paper resolves a question of Erdős and Graham by showing that, for a strictly increasing sequence $\{a_n\}$ and appropriate weighted products, the Cantor-series-like sum $\sum_{n=1}^\infty \frac{b_n}{a_n^{w_0} a_{n+1}^{w_1}\cdots a_{n+d-1}^{w_{d-1}}}$ is irrational under a fast-growth condition $\lim_{n\to\infty} a_n^{1/c_{\boldsymbol{w}}^n}=\infty$, where $c_{\boldsymbol{w}}$ is the unique positive root of $P_{\boldsymbol{w}}(x)=(x-1)\sum_{j=0}^{d-1} w_j x^j - W x^{d-1}$ with $W=\max w_j$. The authors also provide a positive generalization for arbitrary $b_n$ and show sharpness: there exist sequences with $\limsup_{n\to\infty} a_n^{1/c_{\boldsymbol{w}}^n}$ finite that yield a rational sum, and a complementary negative result using the root $\tilde{c}_{\boldsymbol{w}}$ clarifies optimality in many cases. The methodology combines a classical Mahler-type irrationality criterion with delicate tail-sum analysis and a local-peak argument anchored by the growth parameter $c_{\boldsymbol{w}}$, illustrating a productive human-AI collaboration in mathematical discovery. The work extends earlier Erdős–Graham themes and clarifies the boundary between irrationality and rationality for a broad class of series.

Abstract

Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/φ^n}=\infty$ for a monotonically increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$, together with the bound $a_n a_{n+1}\geq n^{1+τ}$, is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $φ$ denotes the golden ratio $(1+\sqrt{5})/2$ and $τ>0$. 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.

Theorems & Definitions (21)

• Theorem 2
• Theorem 3
• Remark 4
• Theorem 5
• Remark 6
• Example 7
• Lemma 8
• proof
• Lemma 9: cf. Erdos1975
• proof