Problem Statement
Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Is it true that\[\lim_{N\to \infty} k(N)-(e-1)N=\infty?\]
Categories:
Number Theory Unit Fractions
Progress
Erdős and Straus [ErSt71b] have proved the existence of some constant $c>0$ such that\[-c < k(N)-(e-1)N \ll \frac{N}{\log N}.\]Source: erdosproblems.com/295 | Last verified: January 14, 2026