Irrationality of the reciprocal sum of doubly exponential sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909571633643520 |
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| author | Koizumi, Junnosuke |
| author_facet | Koizumi, Junnosuke |
| contents | We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only sequence with $a_n^2/a_{n+1}\in [2/3,4/3]$ whose reciprocal sum is equal to $1$. This result has applications to irrationality problems. We prove that for almost every real number $α> 1$, sequences asymptotic to $α^{2^n}$ have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erdős and Graham. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irrationality of the reciprocal sum of doubly exponential sequences Koizumi, Junnosuke Number Theory 11J72, 11D68 We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only sequence with $a_n^2/a_{n+1}\in [2/3,4/3]$ whose reciprocal sum is equal to $1$. This result has applications to irrationality problems. We prove that for almost every real number $α> 1$, sequences asymptotic to $α^{2^n}$ have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erdős and Graham. |
| title | Irrationality of the reciprocal sum of doubly exponential sequences |
| topic | Number Theory 11J72, 11D68 |
| url | https://arxiv.org/abs/2504.05933 |