On a conjecture of Erdős and Graham about the Sylvester's sequence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917964380372992 |
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| author | Li, Zheng Tang, Quanyu |
| author_facet | Li, Zheng Tang, Quanyu |
| contents | Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \cdots $ be any other positive integer sequence satisfying $ \sum_{i=1}^\infty \frac{1}{a_i} = 1 $. In this paper, we solve a conjecture of Erdős and Graham, which asks whether $$ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots. $$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erdős and Graham that "all rationals have eventually greedy best Egyptian underapproximations" holds, we establish a generalization of this conjecture using a non-constructive approach. [This paper solves Problem 315 on Bloom's website "Erdős problems".] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Erdős and Graham about the Sylvester's sequence Li, Zheng Tang, Quanyu Number Theory Classical Analysis and ODEs Primary 11D68, Secondary 11D75, 11P99 Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \cdots $ be any other positive integer sequence satisfying $ \sum_{i=1}^\infty \frac{1}{a_i} = 1 $. In this paper, we solve a conjecture of Erdős and Graham, which asks whether $$ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots. $$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erdős and Graham that "all rationals have eventually greedy best Egyptian underapproximations" holds, we establish a generalization of this conjecture using a non-constructive approach. [This paper solves Problem 315 on Bloom's website "Erdős problems".] |
| title | On a conjecture of Erdős and Graham about the Sylvester's sequence |
| topic | Number Theory Classical Analysis and ODEs Primary 11D68, Secondary 11D75, 11P99 |
| url | https://arxiv.org/abs/2503.12277 |