Problem Statement
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 $\max(n_{i+1}-n_i)\geq 3$?
Categories:
Number Theory Unit Fractions
Progress
The example $1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$ shows that $3$ would be best possible here. The lower bound of $\geq 2$ is equivalent to saying that $1$ is not the sum of reciprocals of consecutive integers, proved by Erdős [Er32].This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large $N$, there exists a prime $p\in [N,2N]$ such that $\frac{p+1}{2}$ is also prime.
Source: erdosproblems.com/287 | Last verified: January 14, 2026