Problem Statement
For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?
Categories:
Number Theory Unit Fractions
Progress
Erdős [Er50c] proved that\[\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.\]The upper bound was improved by Vose [Vo85] to\[N(b) \ll \sqrt{\log b}.\]One can also investigate the average of $N(a,b)$ for fixed $b$, and it is known that\[\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.\]Related to [18]. There is also a close connection to [293] (particularly with $N(b-1,b)$), as elucidated by van Doorn and Tang [vDTa25b].This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/304 | Last verified: January 14, 2026