PROVED
This has been solved in the affirmative.
Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<\cdots <n_k$ with\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Is it true that\[f(k)=(1+o(1))\frac{e}{e-1}k?\]
It is trivial that $f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any $u\geq 1$\[\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),\]and so if $eu\approx f(k)$ then $k\leq \frac{e-1}{e}f(k)$. Proved by Martin
[Ma00].
This problem has been
formalised in Lean as part of the
Google DeepMind Formal Conjectures project.
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This page was last edited 01 October 2025.
Additional thanks to: Zach Hunter
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 #285, https://www.erdosproblems.com/285, accessed 2026-01-14
