Irrationality of rapidly converging series: a problem of Erdős and Graham
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918320377167872 |
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| author | Barreto, Kevin Kang, Jiwon Kim, Sang-hyun Kovač, Vjekoslav Zhang, Shengtong |
| author_facet | Barreto, Kevin Kang, Jiwon Kim, Sang-hyun Kovač, Vjekoslav Zhang, Shengtong |
| contents | 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{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21442 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Irrationality of rapidly converging series: a problem of Erdős and Graham Barreto, Kevin Kang, Jiwon Kim, Sang-hyun Kovač, Vjekoslav Zhang, Shengtong Number Theory Classical Analysis and ODEs 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{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions. |
| title | Irrationality of rapidly converging series: a problem of Erdős and Graham |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2601.21442 |