Problem 287

Erdős Problem #287 [status: falsifiable; formalized: no]. Let $k\geq 2$. Is it true that, for any distinct integers $1<n_1<\cdots <n_k$ such that$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$we must have $\max(n_{i+1}-n_i)\geq 3$? Current best: The lower bound of $\geq 2$ is equivalent to saying that $1$ is not the sum of reciprocals of consecutive integers, proved by Erd\H{o}s [Er32]. Prize: no. Tags: number theory, unit fractions.