Problem Statement
Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that\[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]
Categories:
Number Theory Unit Fractions
Progress
Erdős and Graham posed this in [ErGr80] without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely [ErGr80] just forgot to mention this natural restriction.As an example representing $2$ rather than $1$, Hickerson and Montgomery, in the solution to AMS Monthly problem E2689 proposed by Hahn, found\[2=\sum_{i=1}^5 \sum_{n\in I_i}\frac{1}{n}\]where $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$.
Source: erdosproblems.com/289 | Last verified: January 14, 2026