PROVED
This has been solved in the affirmative.
For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that\[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erdős
[BlEr76] have shown that\[D(b)\ll b(\log b)^2.\]If $b=p$ is a prime then\[D(p) \gg p\log p.\]This was solved by Yokota
[Yo88], who proved that\[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\]This was improved by Liu and Sawhney
[LiSa24] to\[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
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This page was last edited 18 November 2025.
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #305, https://www.erdosproblems.com/305, accessed 2026-01-16
