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 $S\subseteq A$ ?

Source: erdosproblems.com/321

Number theory
Unit fractions

Wiki page

Status

SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved, in the site's label, which the site glosses as a resolution other than a proof or disproof, for the order of magnitude

R(N)\ \asymp\ \log S(N)\ \asymp\ \frac{N}{\log N}\prod_{j=3}^{k}\log_jN,\qquad \log_kN=O(1).

The lower bound is refereed and is written out below: Bleicher and Erdős's set of products of rapidly growing primes (1975) has distinct subset reciprocal sums and gives the site's displayed lower bound, and the set $\mathcal U(N)$ in Bettin, Grenié, Molteni and Sanna's proof (Math. Comp., online 22 January 2026) has the same property and gives $R(N)\ge2(1-\frac{3/2}{\log_kN})\frac{N}{\log N}\prod_{j=3}^k\log_jN$ , a bound the paper does not state and the site calls implicit in it. The refereed upper bound, $R(N)\le\log S(N)/\log2$ with Bleicher and Erdős's 1976 Theorem 3, carries the extra factor $\log_rN$ ; the upper bound of the same order as the lower is the bound for $\log S(N)$ obtained with the AI system GPT 5.6 Sol Pro and accepted by the site on Problem 320 (claim of 15 July 2026, unrefereed), restated for this problem in the accepted claim on this page's tab. So the matching of the two bounds behind the label rests on that accepted claim, whose page for this problem is the order of magnitude of R(N). The three refereed bounds have their own partial claim pages, linked under Progress and known results, and the standing derives from the four pages.

BlEr75: Bleicher, M. N. and Erdős, P., The number of distinct subsums of $\sum_{i=1}^N 1/i$ . Math. Comp. 29 (1975), no. 129, 29--42, DOI 10.1090/S0025-5718-1975-0366795-4. The theorem on $Q_k(N)$ , p. 30; the theorem $S(N)\ge2^{Q(N)}$ and its Lemma, pp. 39--42. Library home: Bleicher and Erdős 1975.
BlEr76b: Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions. II. Illinois J. Math. 20 (1976), 598--613. Theorem 3, p. 610.
BGMS25: Bettin, S., Grenié, L., Molteni, G. and Sanna, C., A lower bound for the number of Egyptian fractions. arXiv:2509.10030v1 (12 September 2025); Math. Comp., DOI 10.1090/mcom/4190, published online 22 January 2026. Theorem 1 and Section 2 (the set $\mathcal U$ ). Library home: Bettin et al. 2025.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 43. Library home: Erdős and Graham 1980.
OEIS-a: Lu, C., Sequence A391592, The On-Line Encyclopedia of Integer Sequences (10 January 2026; created 18 January 2026): $R(N)$ for $N\le54$ ; per the entry, terms $1$ -- $20$ by the account epistemologist, $21$ -- $36$ by Stijn Cambie, $37$ -- $54$ by Cong Lu; accessed 2026-09-18.
OEIS-b: Xie, Y., Sequence A384927 (7 September 2025; last modified 21 January 2026): the largest $S\subseteq\{1,\ldots,n\}$ with $t+u\nmid tu$ for distinct $t,u\in S$ , the sequence of Problem 327, which agrees with $R(N)$ for $N\le20$ and differs at $N=21$ (thread); accessed 2026-09-18.
YZL26: Young, R., Zhu, K. and Luo, Y., the manuscript of the accepted claim, the same one as on Problem 320 (an Overleaf read link, which served no document to a request on 2026-09-18); Lean repository Zarathustra23/erdos-320-harmonic-subset-sums at its commit of 11 July 2026, pinned on the claim page.

Problem 321

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