Irrationality of rapidly converging series: a problem of Erdős and Graham
Kevin Barreto; Jiwon Kang; Sang-hyun Kim; Vjekoslav Kovač; Shengtong Zhang · 2026 · arXiv
WASTE classifies this as Negative / Null Result Report · AI classification, approximate
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Abstract (excerpt)
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 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 $φ$ 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{Ale
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Metadata source: arXiv