Problem Statement
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\[f(k)=(1+o(1))\frac{e}{e-1}k?\]
Categories:
Number Theory Unit Fractions
Progress
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.
Source: erdosproblems.com/285 | Last verified: January 14, 2026