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arXiv · 2503.12277

On a conjecture of Erd\H{o}s and Graham about the Sylvester's sequence

Zheng Li·Quanyu Tang
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Abstract

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 Erd\H{o}s 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 Erd\H{o}s 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 "Erd\H{o}s problems".]

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Zheng Li, Quanyu Tang. 2025-03-15. On a conjecture of Erd\H{o}s and Graham about the Sylvester's sequence. https://doi.org/10.1007/s10474-025-01566-8

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