Problem 327

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claims/: The 2 claim pages of Problem 327, one per claimant's result; the problem's standing derives from them.

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Statement. Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd numbers?

What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\lvert A\rvert=o(N)$?

Formulation. The site's wording (the page shows no last-edited date), with the site's quotation marks. The page holds two questions, tracked separately here. Writing $f_k(N)$ for the largest $|A|$ with $A\subseteq\{1,\ldots,N\}$ and $a+b\nmid kab$ for all distinct $a,b\in A$, the first question asks whether $f_1(N)$ can exceed the odd numbers' $\lceil N/2\rceil$ by a positive proportion of $N$, and the second whether $f_2(N)=o(N)$. The link to unit fractions: $a+b\mid ab$ exactly when $\frac1a+\frac1b$ is a unit fraction (site commentary), and $a+b\mid2ab$ exactly when the average $\frac12(\frac1a+\frac1b)$ is a unit fraction ([Sa26], p. 1). The first condition is stronger than the freedom from two-term relations of Problem 302, since here the third denominator $ab/(a+b)$ need not lie in $A$. $f_1(N)$ is OEIS A384927 ($1,2,3,4,5,5,6,\ldots$, to $N=5000$).

Status. Open on the site: the label is OPEN (the page shows no last-edited date) and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is claimed, claim answered: one full claim, Della Pietra's AI-assisted manuscript of 29 July 2026 (its page), asserting a positive answer to the first question and an independent negative answer to the second, is pending and accepted by nobody. Second question: Sawin's Theorem 1 (arXiv:2607.15419v1, 16 July 2026) constructs sets of positive density with $a+b\nmid2ab$ for all distinct pairs, a negative answer, filed as a partial claim (its page); the source is an arXiv preprint with no journal acceptance, independent review or citing work found on 2026-09-17, so the claim is pending and derives nothing. No refereed source answers either question.

Source. erdosproblems.com/327, accessed 2026-09-17: the problem page (OPEN, marked as not resolvable by a finite computation; source key [ErGr80]; no last-edited date shown), its eleven-comment discussion thread (22 August 2025 to 2 July 2026) and its proof-claim tab with one partial and one full claim (18 and 29 July 2026). The site thanks Wouter van Doorn. Cite as: T. F. Bloom, Erdős Problem #327, https://www.erdosproblems.com/327, accessed 2026-09-17.

References.

• [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), p. 37 (the site gives no page; Sawin and Della Pietra cite p. 37). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Sa26] Sawin, W., Sets of unit fractions without two members whose average is a unit fraction. arXiv:2607.15419v1 (16 July 2026, the only version, 9 pages); preprint, no journal record. Theorem 1, p. 1; the AI-use declaration, p. 2. Library home: sawin_2026_sets_unit_fractions_without_two_members_average_unit_fraction.
• [vD25] van Doorn, W., Two-colouring and density lead to many solutions of $1/x+1/y=1/z$. Undated GitHub note (2025); Theorem 2, p. 3. Library home: doorn_2025_two_coloring_density_solutions_unit_fraction_equation.
• [Li26] Liu, Yu Leon, Upper bounds for Erdős Problem #327 and its $k$-variant. Public four-page manuscript (PDF dated 19 May 2026) linked from the site's thread (comment of 13 May 2026; the certificate script erdos327_cert.py was committed 20 May 2026 per the GitHub API); unrefereed; not filed in the library. Theorem 6, p. 2; Table 1 and the AI-use acknowledgment, p. 4.
• [DP26a] Della Pietra, D., A positive-density improvement over the odd numbers in Erdős Problem 327. Manuscript in the GitHub repository donalddellapietra/erdos-327-proof: version 1 of 29 July 2026 (14 pages, the release PDF) and the revised version of 30 July 2026 (15 pages, at the repository head), the version cited here; the companion [DP26b], Admissibility variants of Erdős Problem 327: the multipliers $k\ge2$ (16 pages), was added to the repository on 30 July 2026 (dates per the GitHub API, UTC); unrefereed; not filed in the library.
• [Ko26] Korsky, S., Large sets of integers with no harmonic triples. arXiv:2607.05823v1 (7 July 2026); [Sa26] says its construction answers Korsky's Question 1.2. Not held.
• [OEIS] Xie, Y., Sequence A384927, The On-Line Encyclopedia of Integer Sequences (2025; entry last modified 21 January 2026, server time): $f_1(n)$ for $n\le5000$ (b-file by S. Kesarwani), with a comment conjecturing that $a(n)/n\to1/2$; accessed.

Formalization. None recorded. No file ErdosProblems/327.lean exists in google-deepmind/formal-conjectures (2026-10-07); the community database records the statement as not formalized and no formal proof. The Lean development that [DP26a] describes is the author's own, linked from its claim page, and is not built or audited here, so it gives no formalized evidence.

Current assessment

The question (site formulation). The statement above; OPEN; no last-edited date. The commentary states the unit-fraction connection, credits Wouter van Doorn with an elementary argument showing that every $A\subseteq\{1,\ldots,N\}$ with $|A|\ge(25/28+o(1))N$ contains distinct $a,b$ with $a+b\mid ab$, refers for it to the discussion under Problem 301, and points to Problem 302. The thread (eleven comments, 22 August 2025 to 2 July 2026): an elementary link between the two questions (22 August 2025, below); Cambie's computations for $N=500$ (a largest set of size $350$ for the second question; the first question's initial values) and a construction remark (5 September 2025); suggestions by Tao and Bloom to submit the data to the OEIS (5 and 7 September 2025); a SAT and integer-programming computation by the account Sharvil Kesarwani (25 November 2025) extending A384927 to $N=5000$, with $f_1(N)/N$ staying above $0.7$ (least value $3521/4991$) and $f_2(N)/N$ above $0.69$ to $N=1000$ (least value $679/980$); the account leon2k2k2k's announcement of [Li26]'s bounds $0.7769$ and $0.7630$, found with an AI model's help and checked by hand according to the comment (13 May 2026), a reader's routine check finding no issue but noting the missing certificate (14 May 2026), and the certificate link (20 May 2026); and Sawin's greedy computation for the second question (density $0.6508$ at $N=5000$, done with an AI tool per the comment) with the observation that the greedy set contains the integers whose prime factors grow fast enough, of count $N/(\log N)^{0.3588\ldots+o(1)}$ (2 July 2026). The proof-claim tab lists two claims (below). The community database records open, not formalized, OEIS A384927.

Origin. Printed p. 37 of the 1980 monograph, right after the questions of Problems 301 and 302: "Of course, $\frac1t=\frac1x+\frac1y$ holds if and only if $x+y\mid xy$. Suppose $X\subseteq\{1,2,\ldots,n\}$ so that $x,y\in X$ implies $x+y\nmid xy$. Can $X$ be substantially more than the odd numbers? What if $x,y\in X$, $x\ne y$, implies $x+y\nmid2xy$? Must we have $|X|=o(n)$ in this case?" The site's statement is the passage with $a\ne b$ made explicit in the first condition, which the passage states only for the second. Read as the passage words it, with $x=y$ allowed, the first condition would exclude every even $x$, since $2x\mid x^2$, so the first question would be trivially no; the site, Sawin and Della Pietra read both conditions for distinct pairs.

Second question: Sawin's preprint (progress, preprint qualification). Theorem 1 of [Sa26] (p. 1): with $A_N$ the set of $a\le N$ such that $a+b\nmid2ab$ for every $b\le N$ with $b\ne a$ and $\Omega(b)\le\Omega(a)$, (1) $a+b\nmid2ab$ for all distinct $a,b\in A_N$ and (2) $|A_N|>cN$ for an absolute $c>0$ and all large $N$. Part (1) is immediate; part (2) is proved on pp. 2--9 by restricting to integers without very small prime factors and with controlled prime-factor counts and bounding the exceptions through a mean-value theorem of de la Bretèche and Tenenbaum, a strategy the paper attributes to Stef's 1992 thesis; the proof is not independently verified. The paper makes no effort to compute $c$ and says its method gives, for the first question, a lower bound worse than the odd numbers. Acceptance evidence: none beyond the site's proof-claim tab, where the author submitted the result on 18 July 2026, first as a full claim and, by his own comment of 29 July 2026, as a claim on the second part only, naming the AI systems ChatGPT 5.5 for data analysis and reference search and ChatGPT 5.6 for proofreading and keeping the proof strategy and the write-up as his own; the arXiv listing of 2026-09-17 shows one version and no journal reference, no citing work was found, and the site's status and commentary do not mention the preprint. Provenance recorded, not judged: the paper's own declaration (p. 2) says the author asked ChatGPT to look for patterns in Cambie's $N=500$ example and that it "was also used for reference search and proofreading". So the second question has a preprint-level negative answer and no refereed one; the result is a pending partial claim, Sawin's page, which derives no standing.

First question: what is established and what is claimed. Lower bound $f_1(N)\ge\lceil N/2\rceil$: for odd $a,b$ the sum $a+b$ is even and $ab$ odd. Upper bounds: a set $A$ with $a+b\nmid ab$ for all distinct $a,b\in A$ has no triple of distinct $x,y,z\in A$ with $1/x+1/y=1/z$ (such a triple has $x+y\mid xy$ with $x\ne y$), so van Doorn's Theorem 2 gives $f_1(N)<9N/10+(\log N)^3+1$. The site's sharper $f_1(N)\le(25/28+o(1))N$ holds: within $S_a=\{2a,3a,4a,6a,12a\}$ the pairs with $x+y\mid xy$ include $\{3a,6a\}$, $\{6a,12a\}$ and $\{4a,12a\}$ for every $a$ (a path through $3a,6a,12a,4a$; exactly these when $\gcd(a,210)=1$, and more pairs for other $a$, which only shrink pair-free sets), so a pair-free set meets each full $S_a$ in at most three elements and each truncated $\{2a,3a,4a,6a\}$ in at most three, the same counts as on the Problem 301 page, whose density computation then gives the omission of at least $(3/28-o(1))N$ elements. Claimed and unrefereed: [Li26]'s Theorem 6, $\alpha_1\le0.7769$ and $\alpha_2\le0.7630$ for $\alpha_k=\limsup f_k(N)/N$, by a smooth--rough decomposition that reduces to a finite computation over $P$-smooth integers up to $X$ ($P=\{2,3,5,7,11,13\}$, $X=2000$; Table 1, p. 4), with the acknowledgment (p. 4) that "Some computations and drafting were assisted by AI-based tools" and that the author "independently checked" the arguments. [Li26]'s bounds lie above $1/2$ for the first question and give no $o(N)$ bound for the second, so they settle neither question and have no claim page. [DP26a]'s Theorem 1.1 (p. 1): absolute constants $\varepsilon>0$ and $N_0$ with $f_1(N)\ge(1/2+\varepsilon)N$ for $N\ge N_0$, a positive answer to the first question, obtained by starting from nearly all odd integers and adding twice the members of a positive-density set that is admissible for the doubled condition and chosen by prime-factor count; its p. 14 lists two errata relative to a first version, says its Lean development "was not rebuilt for this revision", and discloses that "AI systems were used for data analysis, reference search, adversarial proof audits, independent reconstruction of intermediate estimates, formalization assistance, and proofreading"; the site's proof-claim tab carries it as a full claim submitted 29 July 2026, declaring the AI system GPT 5.6 Sol, whose notes say that the Lean theorem erdos327FullConclusion_unconditional builds without sorry and that the manuscript also proves the second question's negative answer independently. It is the pending full claim, Della Pietra's page, from which the standing claimed derives. [DP26b] claims $f_k(N)\ge(1/2+\varepsilon_k)N$ for odd $k$ and $f_k(N)\ge c_kN$ for all $k$, and says the method does not reach $f_2(N)\ge(1/2+\varepsilon)N$. None of these has acceptance evidence, independent review or a refereed version; none is accepted.

An elementary link between the two questions (forum). The thread's comment of 22 August 2025 (account Adenwalla) observes that $B\subseteq\{1,\ldots,N\}$ satisfies $a+b\nmid2ab$ for distinct $a,b$ exactly when $2B\subseteq\{1,\ldots,2N\}$ satisfies $x+y\nmid xy$, since $2a+2b\mid4ab$ is $a+b\mid2ab$; so a set for the second question is the same thing as an all-even set for the first. Sawin's construction therefore also gives even sets of positive density for the first condition, without by itself beating the odd numbers.

Data (not status). OEIS A384927 gives $f_1(n)$ to $n=5000$; the forum's least ratio is $3521/4991\approx0.7055$, above the odd numbers' $1/2$ and below the proved $25/28$; the second question's computed ratios to $N=1000$ stay above $0.69$ (forum, not in the OEIS). These finite values are consistent with every claim above and decide nothing.

Search scope (2026-09-17 UTC). The problem, discussion and proof-claim pages; the community database record; the formal-conjectures directory as listed on 2026-09-17 (no file); the arXiv listings for 2607.15419 (one version, no journal reference) and 2607.05823 (one version); the Semantic Scholar citation list for 2607.15419 (empty); arXiv API searches for abstracts on unit fractions and averages (Sawin's preprint and one unrelated record), on harmonic triples (Sawin's preprint and unrelated records) and for "Erdős problem" with unit fractions (none); the GitHub API for the head of donalddellapietra/erdos-327-proof and the commit of [Li26]'s certificate script; OEIS A384927; the primary sources [Sa26], [vD25], [ErGr80], [Li26], [DP26a] and [DP26b]. Not searched: MathSciNet, zbMATH, Google Scholar, X. Nothing found is refereed.

Remaining gaps. (1) The second question's negative answer is a preprint (Sawin) with a declared AI-assisted discovery step and no review; its proof is not independently verified, and its claim page is pending. (2) The first question is open between $1/2$ and $25/28$ in refereed-or-elementary terms, with an unrefereed AI-assisted full claim of a positive answer, pending on its claim page, and unrefereed computational upper bounds; none is independently verified. (3) The site's $25/28$ argument has no written source; it is recorded from the site. (4) [Ko26] is not held. (5) No formal statement was found on 2026-10-07.

Progress and known results

Second question: Sawin's Theorem 1 (preprint, 2026) gives $f_2(N)>cN$ for large $N$, so $|A|=o(N)$ is not forced; the constant is not computed. First question: $\lceil N/2\rceil\le f_1(N)\le(25/28+o(1))N$ (odd numbers; the site's argument), with van Doorn's Theorem 2 giving the weaker $9N/10+(\log N)^3+1$; claimed, unrefereed: $\alpha_1\le0.7769$, $\alpha_2\le0.7630$ ([Li26]) and $f_1(N)\ge(1/2+\varepsilon)N$ ([DP26a], the pending full claim). Sawin's result is the pending partial claim on its page. Related: the two-term relation inside a set is Problem 302, the all-length relation Problem 301, and sets of unit fractions without three-term arithmetic progressions are the subject of [Ko26].

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_1980_old_new_problems_results_combinatorial_number_theory
• doorn_2025_two_coloring_density_solutions_unit_fraction_equation
• doorn_2025_two_coloring_density_solutions_unit_fraction_equation / theorem_2
• sawin_2026_sets_unit_fractions_without_two_members_average_unit_fraction
• sawin_2026_sets_unit_fractions_without_two_members_average_unit_fraction / theorem_1