Problem #298: Does every set $A\subseteq \mathbb{N}$ of positive density...

Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?

SOLVED

Problem Statement

Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?

Categories: Number Theory Unit Fractions

The answer is yes, proved by Bloom [Bl21].

See also [46] and [47].

This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.

Source: erdosproblems.com/298 | Last verified: January 14, 2026