Problem 47

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claims/: The 2 claim pages of Problem 47, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\sum_{n\in S}\frac{1}{n}=1$?

Formulation. The site's wording on 2026-09-17 (page last edited 7 April 2026). For each fixed $\delta>0$ the conclusion is required for every $N\ge N_0(\delta)$ and every $A\subseteq\{1,\ldots,N\}$ whose reciprocal sum exceeds $\delta\log N$; $S$ is a subset of $A$, hence a set of distinct integers. Write $R(A)=\sum_{a\in A}1/a$ for the reciprocal sum.

Status. 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.

Source. erdosproblems.com/47, accessed 2026-09-17: the problem page (PROVED (LEAN); a prize; last edited 7 April 2026), its empty discussion thread and its empty proof-claim tab. The site cites [Er80, p. 105], [ErGr80], [Er92c], [Er95], [Er96b] and [Er97c] as the problem's sources and [Bl21] and [LiSa24] in its commentary, and links Problems 46 and 298. Cite as: T. F. Bloom, Erdős Problem #47, https://www.erdosproblems.com/47, accessed 2026-09-17.

References.

• [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_density_conjecture_about_unit_fractions.
• [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_2024_further_questions_regarding_unit_fractions.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory; 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_survey_problems_combinatorial_number_theory; 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: erdos_1992_my_forgotten_problems_number_theory; 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: erdos_1995_my_favourite_problems_number_theory_combinatorics; 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: erdos_1997_some_my_favorite_problems_results; paged at display_4_4.

Formalization. Statement in ErdosProblems/47.lean of formal-conjectures, fetched at the linked revision, with an external proof tag; the tagged file rests on the Lean 4 port of the Bloom–Mehta formalization of Theorem 3. This corpus has built neither. See Existing formalization.

Current assessment

The question. On 2026-09-17 the site asks whether, for fixed $\delta>0$ and large $N$, $R(A)>\delta\log N$ forces a subset of $A$ with reciprocal sum one, shows PROVED (LEAN) with a prize, and says in its commentary that Bloom solved it with the threshold $R(A)\gg\log N\log\log\log N/\log\log N$, that Liu and Sawhney improved this to $R(A)\gg(\log N)^{4/5+o(1)}$, and that Erdős speculated that even $\gg(\log\log N)^2$ might suffice, which a construction of Pomerance in the appendix of [Bl21] shows would be best possible. The thread and the proof-claim tab are empty. The community database record (teorth/erdosproblems) says proved (Lean), statement formalized, no formal-proof URL.

Status support. The status-defining source is Bloom's Theorem 3 (arXiv:2112.03726v2, p. 2; Theorem 1.3 on p. 2 of the published version): there is an absolute $C>0$ such that, for $N$ large, every $A\subseteq\{1,\ldots,N\}$ with

$$R(A)\ge C\,\frac{\log\log\log N}{\log\log N}\,\log N$$

contains $S\subseteq A$ with $R(S)=1$. For fixed $\delta>0$ the right side is below $\delta\log N$ once $N\ge N_0(\delta)$, so $R(A)>\delta\log N$ implies the hypothesis and the question has the answer yes. Acceptance: the paper appeared in J. Eur. Math. Soc. 27 (2025), 4563--4589 (submitted 1 February 2022, accepted 11 October 2023, online 11 July 2024, per the publisher's record), and its Appendix B reports a complete formal verification of the main results with Mehta. The library holds a complete rewritten proof of Theorem 3 through the explicit constant-$8$ variant of its Proposition 1 used by that formalization; the proof pages have not been independently reviewed.

Liu and Sawhney's Theorem 1.1 (arXiv:2404.07113v1, p. 1) lowers the threshold: for every $\varepsilon>0$ and $N$ large in terms of $\varepsilon$, $R(A)\ge(\log N)^{4/5+\varepsilon}$ forces a unit subsum. The paper appeared in Int. Math. Res. Not. 2026, no. 2, rnaf382 (received 28 October 2025, accepted 23 December 2025, online 14 January 2026, per the publisher's record); the published text has not been compared, so the locators are v1 locators. The rewritten proof in the library uses explicitly corrected forms of two of the paper's lemmas, recorded on its pages as compilation corrections, not as author errata.

What remains open. Writing $\lambda(N)$ for the largest reciprocal sum of a subset of $\{1,\ldots,N\}$ with no unit subsum, the two theorems give $\lambda(N)\ll(\log N)^{4/5+o(1)}$, while Pomerance's construction (Bloom's Theorem 4, Appendix A) gives $\lambda(N)\gg(\log\log N)^2$. The order of $\lambda(N)$ is not known; Bloom writes (p. 2) that he suspects $\lambda(N)\le(\log N)^{o(1)}$. This quantitative question lies beyond the yes-or-no question the page carries and does not affect the status.

Search scope. The problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit and the external Lean file it tags; the arXiv listings for 2112.03726 (v1 7 December 2021, v2 12 October 2023) and 2404.07113 (v1 only); the EMS and Oxford Academic article records; the Semantic Scholar citing-paper records for both papers (nine and five records; the 2025 and 2026 items are a stretched-exponential bound for approximating one from below by reciprocal subsums of multisets (arXiv:2607.04157), partitions with prescribed reciprocal sums (arXiv:2502.02200), best underapproximations (arXiv:2406.07218) and the count of unit-sum subsets (arXiv:2404.16016); none improves the exact-sum threshold); the arXiv API listing of the sixty most recent abstracts mentioning unit or Egyptian fractions (to 7 September 2026); and two general web searches. Not searched: MathSciNet, zbMATH, full-text search engines for scholarly literature, X. No later improvement of the $(\log N)^{4/5+o(1)}$ threshold was found; this is a bounded negative finding.

Remaining gaps. The published Liu–Sawhney text is uncompared. The rewritten proof of Bloom's Theorem 3 lacks independent review. The corrected Theorem 1.1 chain of Liu and Sawhney has the corpus's graded independent review of 2026-09-18, a fresh-context blind review with the verdict refutation-failed and a distinct grade recording PASS for the report contract and for independence, retained on the card's evidence index; this is the corpus's own review and so not reviewed evidence. The passage of [Er95] (item 8 of Part I, p. 6) carries the prize offer; the passage of [Er97c] (pp. 63--64) carries the $c\log n$ belief and is quoted on its result page; [Er96b] is not held and has no library home. The Lean files were not built, and the stronger Liu–Sawhney threshold has no formalization found.

Progress and known results

• Bloom's Theorem 3: the threshold $C\log N\log\log\log N/\log\log N$, with a complete rewritten proof in the library.
• Liu and Sawhney's Theorem 1.1: the threshold $(\log N)^{4/5+\varepsilon}$, with a complete rewritten proof in the library.
• Pomerance's construction, Bloom's Theorem 4: $\lambda(N)\gg(\log\log N)^2$, the barrier to Erdős's speculation.
• The density form of the question is Problem 298; the coloring form is Problem 46; the greedy consequence of Theorem 3 for disjoint unit-sum subsets is Problem 296.

Existing formalization

The formal-conjectures file ErdosProblems/47.lean, at the revision the Formalization link above pins, declares

erdos_47 : answer(True) ↔ ∀ δ : ℝ, 0 < δ → ∀ᶠ N : ℕ in atTop, ∀ A : Finset ℕ, A ⊆ Finset.Icc 1 N → δ * Real.log (N : ℝ) < A.reciprocalSum → ∃ S : Finset ℕ, S ⊆ A ∧ S.reciprocalSum = 1

under category research solved with proof sorry and the attribute formal_proof using lean4 at the file src/v4.29.1/ErdosProblems/Erdos47.lean of the collection plby/lean-proofs. That file (Erdos47.lean, at the pinned revision) names Bloom as informal author and Bhavik Mehta and Thomas Bloom as formal authors with the URL of the Bloom–Mehta repository, imports UnitFractions.Definitions and UnitFractions.FinalResults, states Theorem 3 as

unit_fractions_upper_log_density : ∃ C : ℝ, 0 < C ∧ ∀ᶠ N : ℕ in atTop, ∀ A : Finset ℕ, A ⊆ Icc 1 N → C * ((log (log (log N)) / log (log N)) * log N) ≤ rec_sum A → ∃ S ⊆ A, rec_sum S = 1

and derives erdos47_bloom and

erdos47 : ∀ δ > 0, ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ A : Finset ℕ, A ⊆ Finset.Icc 1 N → δ * log N < rec_sum A → ∃ S ⊆ A, rec_sum S = 1

without sorry; comments record #print axioms for both as propext, Classical.choice, Quot.sound. The original Bloom–Mehta Lean 3 development (b-mehta/unit-fractions, at the pinned revision) holds unit_fractions_upper_log_density at line 2042 of src/final_results.lean; Appendix B of the paper describes that verification. No formalization of the Liu–Sawhney threshold was found. The collection's README says its source subdirectories "build as a whole (last I checked)". This corpus has built, audited or kernel-checked none of it; the site's Lean suffix is a catalog label, and the community database records no formal-proof URL.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1992_my_forgotten_problems_number_theory
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• erdos_1980_survey_problems_combinatorial_number_theory
• erdos_1995_my_favourite_problems_number_theory_combinatorics
• erdos_1997_some_my_favorite_problems_results
• erdos_1997_some_my_favorite_problems_results / display_4_4
• bloom_2021_density_conjecture_about_unit_fractions
• bloom_2021_density_conjecture_about_unit_fractions / theorem_3
• bloom_2021_density_conjecture_about_unit_fractions / theorem_4
• liu_2024_further_questions_regarding_unit_fractions
• liu_2024_further_questions_regarding_unit_fractions / theorem_1_1