Erdős Problem #321

Erdős problem #321 · erdosproblems.com/321

What is the largest $A\subseteq\{1,\dots,N\}$ such that all subset sums $\sum_{n\in S}1/n$ (over $S\subseteq A$ ) are distinct?

Result: Proved
Field: Number Theory, Unit Fractions
Posed by: Paul Erdős, Ronald Graham
Year posed: 1980
Years open: 46y
Solved: 2026-07
Model: GPT-5.6 Sol
Vendor: OpenAI
Collaborators: Young, Zhu, Luo
Verification: Site-confirmed
Notability: No dedicated article

What the AI did

GPT-5.6 Sol (prompted by Young, Zhu, and Luo) proved the matching upper bound $R(N)\asymp \frac{N}{\log N}\prod_{j\ge 3}\log_j N$ (companion to #320).

Verification

Marked solved on erdosproblems.com via a proof claim; follows from the resolution of #320.

Source

erdosproblems.com