PROVED
This has been solved in the affirmative.
Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]with $t=n_1<\cdots <n_k\leq N$. Estimate $t(N)$.
Erdős and Graham
[ErGr80] could show\[t(N)\ll\frac{N}{\log N},\]but had no idea of the true value of $t(N)$.
Solved by Liu and Sawhney
[LiSa24] (up to $(\log\log N)^{O(1)}$), who proved that\[\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N) \ll \frac{N}{\log N}.\]
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This page was last edited 18 November 2025.
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T. F. Bloom, Erdős Problem #294, https://www.erdosproblems.com/294, accessed 2026-01-16
