Erdős Problem #321 — What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?

Erdős Problem $321$ asks for the growth of $R(N)$, the largest subset of $\{1,\ldots,N\}$ whose reciprocal subset sums are all distinct. The formal Lean statement remains unfinished: its target is recorded as R N = answer(sorry) with a sorry proof.

Known results

• Bleicher and Erdős ($1975$, $1976$): iterated-logarithmic lower and upper bounds for $R(N)$.
• Bettin, Grenié, Molteni, and Sanna ($2025$): the currently recorded lower-bound scale.
• Certified computation reaches $N=54$, with $R(54)=37$.

July 2026 order-of-magnitude claim

The Erdős Problems page attributes a matching upper bound to GPT 5.6 Sol, prompted by Young, Zhu, and Luo, yielding the claimed estimate $R(N)\asymp \frac{N}{\log N}\prod_{j=3}^{k}\log_j N$. No independently checkable proof was found, and the formalization still treats the exact result as open.

Current status (as of September 2026): classical bounds and computation are established, but the claimed matching asymptotic remains unverified and the exact problem is open.