Problem 304

Erdős Problem #304: declared status 'open'. Formalized: yes. 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 $N(b) \ll \log\log b$? Current best: Erd\H{o}s [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]. Prize: no. OEIS: A097847, A097849. Tags: number theory, unit fractions.