Erdős Problem #320 — Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$.

Erdős Problem #320 asks how many different totals can be formed by choosing any collection of the first $N$ unit fractions. The established bounds had matching broad shape but did not settle the estimate precisely.

Known results

• Bleicher and Erdős proved lower and upper bounds of iterated-logarithmic scale for $\log S(N)$.
• Bettin, Grenié, Molteni, and Sanna (2025) strengthened the lower bound to

$$\log S(N)\geq \frac{N}{\log N}\left(2\log 2\left(1-\frac{3/2}{\log_kN}\right)\prod_{i=3}^{k}\log_iN\right).$$

August 2026 claimed proof

The Erdős Problems entry reports that GPT 5.6 Sol, prompted by Young, Zhu, and Luo, obtained an upper bound matching the known lower-bound order. A related repository claims a more precise asymptotic and a Lean 4 formalization, but its four axioms encode imported results and computational certificates; the claimed resolution remains unverified.

Current status (as of August 2026): The classical order-of-magnitude bounds are established, while the claimed matching proof and sharper asymptotic remain unconfirmed.