Problem 47

If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$ , and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that $\sum_{n\in S}\frac{1}{n}=1$ ?

Source: erdosproblems.com/47

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. Bloom's Theorem 3 (J. Eur. Math. Soc. 27 (2025)) gives a unit subsum from a reciprocal sum of order $\log N\log\log\log N/\log\log N$ , which is below $\delta\log N$ for large $N$ ; Liu and Sawhney (Int. Math. Res. Not. 2026) lowered the threshold to $(\log N)^{4/5+o(1)}$ . The site records "PROVED (LEAN)"; the Lean suffix is a catalog label qualified under Existing formalization below, and no local kernel credit is claimed. The claim pages Bloom 2021 and Liu and Sawhney 2024 record the two results, their postings and the acceptance evidence from which the standing above derives.

Bl21: Bloom, T. F., On a density conjecture about unit fractions, with an appendix co-written by Bloom and B. Mehta. arXiv:2112.03726 (2021), v2 (12 October 2023); J. Eur. Math. Soc. 27 (2025), no. 11, 4563--4589, DOI 10.4171/JEMS/1456, published online 11 July 2024. Theorem 3 (Theorem 1.3 in the published version). Library home: Bloom 2021.
LiSa24: Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024); Int. Math. Res. Not. 2026, no. 2, rnaf382, DOI 10.1093/imrn/rnaf382, published online 14 January 2026. Theorem 1.1. Library home: Liu and Sawhney 2024.
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). Library home: Erdős and Graham 1980; the site gives no page; the reciprocal-sum growth sentence is on printed p. 36, quoted on the card, with the book's guess of a threshold near $\log\log n$ .
Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; p. 105 as cited by the site. Library home: Erdos 1980; the finite form with the $o((\log\log n)^\alpha)$ or $\varepsilon\log n$ guess is quoted on the card from printed p. 105.
Er92c: Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34--50. Library home: Erdős 1992; printed p. 46 is quoted on the card: it states the finite coloring form and the positive-lower-density question of Problem 298, not the $\delta\log N$ threshold.
Er95: Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas 2 (1995), 165--186. Library home: Erdős 1995; the passage is item 8 of Part I, p. 6.
Er96b: Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333--335. Not held; no library home.
Er97c: Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; the reciprocal-sum thresholds after display (4.4), printed pp. 63--64: "we do believe that if $\sum_{a_i<n}\frac1{a_i}>c\log n$ then (4.4) has a solution in the $a_i$ 's", the statement's hypothesis, after the $(\log\log n)^2$ speculations. Library home: Erdős 1997; paged at Erdős 1997, Display (4.4).

Problem 47

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