Problem 319

../

claims/: The 2 claim pages of Problem 319, one per claimant's result; the problem's standing derives from them.

---

Statement. What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that

$$\sum_{n\in A}\frac{\delta_n}{n}=0$$

and

$$\sum_{n\in A'}\frac{\delta_n}{n}\neq 0$$

for all non-empty $A'\subsetneq A$?

Formulation. The site's wording as accessed (the page shows no last-edited stamp). Write $c(N)$ for the largest size of a set $A\subseteq\{1,\ldots,N\}$ carrying signs $\delta$ whose signed reciprocals sum to zero while no nonempty proper subset of $A$, with the same signs, sums to zero: a minimal signed zero-sum relation among the reciprocals of $1,\ldots,N$. Trivially $c(N)\le N$, and $c(N)\ge4$ for $N\ge6$ because $A=\{1,2,3,6\}$ with $\delta(1)=1$ and $\delta=-1$ elsewhere works (checked here: the full sum is $1-\frac12-\frac13-\frac16=0$ and the fourteen proper nonempty subsums are nonzero). Adenwalla's bound $c(N)\ge(1-\frac1e+o(1))N$ (below, pending on its claim page) and the trivial bound would give $c(N)$ the order $N$; what the question then leaves open is the asymptotic: the limit of $c(N)/N$, if it exists, would lie in $[1-\frac1e,1]$, and whether $c(N)=(1-o(1))N$ is unknown. The formal-conjectures file also poses the $\Theta$-order variant, which the lower bound together with $c(N)\le N$ would answer with $N$.

Status. Open on the site: the label is OPEN (2026-10-07; the page shows no last-edited stamp) and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is open, claim none: both claim pages are partial and pending, and a partial claim derives nothing. The site's commentary credits Adenwalla with the lower bound $c(N)\ge(1-\frac1e+o(1))N$ from Croot's Main Theorem (Acta Arith. 99 (2001); refereed), written out below and recorded on Adenwalla's claim page; the credit is commentary on a problem the site labels OPEN, not an acceptance, so the bound is site-credited and pending. With the trivial upper bound $N$ it would give $c(N)$ the order $N$. The one claim on the site's proof-claim tab, an AI-assisted partial claim of 16 July 2026 that $c(N)=N-o(N)$, is pending. No determination of the asymptotic, and no proof that $c(N)=(1-o(1))N$, was found in the search whose scope the Current assessment records. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/319, accessed 2026-09-18: the problem page (OPEN, marked as not resolvable by a finite computation; source key [ErGr80]; no last-edited stamp; a formalized statement recorded), its empty discussion thread and its proof-claim tab with one partial claim; on 2026-10-07 the label was OPEN with the same commentary. The site cites [Cr01] in its commentary and thanks Sarosh Adenwalla and Hayato Egami. Cite as: T. F. Bloom, Erdős Problem #319, https://www.erdosproblems.com/319, accessed 2026-09-18.

References.

• [Cr01] Croot, III, Ernest S., On unit fractions with denominators in short intervals. Acta Arith. 99 (2001), no. 2, 99--114, DOI 10.4064/aa99-2-1; arXiv:math/9904181v1 (30 April 1999, the only arXiv version). The arXiv preprint is not held. The Main Theorem is on p. 100 of the published version and p. 1 of the preprint. Library home: croot_1999_unit_fractions_denominators_short_intervals; result page main_theorem.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. Statement only. The file ErdosProblems/319.lean of formal-conjectures at the linked commit (main) declares erdos_319 (N : ℕ) : IsGreatest { #A | (A) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } answer(sorry) under category research open with proof sorry; a variant erdos_319.variants.isTheta asking for the $\Theta$-order of the same maximum, also under category research open although variants.lb with the trivial bound answers it with $N$; and erdos_319.variants.lb, the lower bound $(1-1/e+o(1))N$ under category research solved, also with proof sorry, whose docstring credits Adenwalla and Croot [Cr01]. The formal set is the site's: subsets of $\{1,\ldots,N\}$ with a sign function and the two conditions. The file is a statement, not a proof, and no declaration carries a formal_proof attribute. The community database, records a formalized statement and no formal proof.

Current assessment

The question (site formulation accessed). The statement above; status OPEN; source key [ErGr80]. In its commentary the site attributes to Adenwalla the bound $|A|\ge(1-\frac1e+o(1))N$, deduced from Croot's main result [Cr01]: Croot's theorem yields integers near the top of $[1,N]$ whose reciprocals sum to $1$, a counting estimate shows that there are at least $(1-\frac1e+o(1))N$ of them, and signing them $-1$ and the integer $1$ as $+1$ gives a signed zero-sum set (written out under Known results). The commentary also notes that the formal-conjectures project has a Lean statement of the problem. The thread is empty; the proof-claim tab lists one claim (below). The community database (teorth/erdosproblems, data/problems.yaml,) records open since 31 August 2025, statement formalized, OEIS "possible", no formal proof.

Origin. Printed p. 43 of the 1980 monograph (PDF p. 39 of the public scan at https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf), in the chapter on unit fractions after the signed-sum questions of pp. 41--42: "How large can a set $A\subseteq\{1,2,\ldots,n\}$ be so that for some choice of $\delta_k$ [sic] $=\pm1$, $a\in A$, we have: (i) $\sum_{a\in A}\frac{\delta_a}{a}=0$; (ii) For every nonempty proper subset $A'\subset A$, $\sum_{a\in A'}\frac{\delta_a}{a}\ne0$?" The site's statement is this question with $N$ for $n$. The monograph gives no bound.

Known results. The lower bound, pending on Adenwalla's claim page, is the site's construction from Croot's Main Theorem (published p. 100; preprint p. 1; read depth on the result page: claims checked): for any rational $r>0$ and all $N'>1$ there are integers $N'<x_1<\cdots<x_k\le(e^r+O_r(\log\log N'/\log N'))N'$ with $r=\sum1/x_i$. Take $r=1$ and $N'$ with $(e+o(1))N'=N$, that is $N'=(\frac1e-o(1))N$: the denominators form a set $B\subseteq((\frac1e-o(1))N,N]$ with $\sum_{b\in B}1/b=1$. The reciprocal sum of all integers of that interval is $\log(e-o(1))=1+o(1)$ and each omitted integer costs at least $1/N$, so $B$ omits at most $o(N)$ of them and $|B|\ge(1-\frac1e-o(1))N$. With $A=B\cup\{1\}$, $\delta(1)=1$ and $\delta(b)=-1$, the full signed sum is $1-1=0$; a proper nonempty subset either omits $1$, so its sum is negative, or contains $1$ and misses some $b\in B$, so its sum is $1-\sum_{b\in B'}1/b>0$ with $B'\subsetneq B$. Hence $c(N)\ge(1-\frac1e+o(1))N$. The minimality check is written here; the site's commentary states the bound and the choice of signs. Croot's paper itself does not mention signed sums; the theorem is used only for the set $B$. The trivial upper bound $c(N)\le N$ is the only upper bound found. The gap is the factor between $1-1/e\approx0.632$ and $1$; whether $c(N)=(1-o(1))N$ is the question left.

The site's proof claim. The proof-claim tab lists one partial claim, submitted on 16 July 2026 by the account popular_12345 with the AI system OpenAI GPT-5.6 declared, asserting $c(N)=N-o(N)$: a fixed positive-signed block protected by a squarefree modulus, almost all modulus-coprime integers with small prime-power factors in an interval $(aN,N]$ on the negative side, a completion lemma that strips prime-power factors from the running denominator through a subset-sum theorem of Conlon and coauthors, and a reservoir of denominators from Croot's theorem. The claimant's notes describe the manuscript as an unrefereed candidate proof of the asymptotic, not an exact evaluation, and say that the accompanying Lean development proves one lemma relative to two assumed interfaces for published results and leaves the density input, the negative side's asymptotics, the minimality argument and the main theorem unformalized. The claim has its page, the pending density-one claim, with its links and the one comment under it; the site says appearing on the tab is no guarantee of correctness. The manuscript and the Lean files are unchecked by this corpus, and a partial claim derives no standing.

Search scope. The site's problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit (statement only); the arXiv listing for math/9904181 (one version, no journal reference) and the Crossref record of DOI 10.4064/aa99-2-1; the Semantic Scholar citation list of Croot's published paper (eleven records: Martin 2000, Croot's own Sums of $k$ unit fractions, a 2013 survey chapter, a 2024 counting result for Problem 297, van Doorn 2025, Korsky 2026, an errata list; none on signed zero sums); arXiv API searches for abstracts on reciprocals with signs, zero and subsets (one unrelated record on nowhere-zero flows) and on unit or Egyptian fractions with "signed" (one unrelated record), and the sixty most recent abstracts mentioning unit or Egyptian fractions (to 7 September 2026; none on this problem); the primary sources [Cr01] (both versions) and [ErGr80] p. 43. Not searched: MathSciNet, zbMATH, Google Scholar, X, the claim's repository. Nothing found gives an upper bound below $N$ or a proof of $c(N)=(1-o(1))N$.

Remaining gaps. (1) No upper bound better than $N$ and no source determining the asymptotic, so the limit of $c(N)/N$, if it exists, is known only to lie in $[1-\frac1e,1]$; the lower bound rests on Croot's theorem, compiled as a statement with a structural sketch, plus the elementary deduction above. (2) The one proof claim is partial, AI-assisted and unchecked; it is pending on its claim page. (3) The exact values $c(N)$ for small $N$ are not tabulated (the site's OEIS field says "possible"; the example above gives $c(6)\ge4$ only).

Progress and known results

• Lower bound $c(N)\ge(1-\frac1e+o(1))N$: the site's construction, credited to Adenwalla and pending on Adenwalla's claim page, from Croot's Main Theorem with $r=1$, written out above.
• Upper bound: the trivial $c(N)\le N$.
• Claimed, pending: $c(N)=N-o(N)$, the partial claim of 16 July 2026 (AI-assisted, unrefereed, unchecked), which would give $c(N)\sim N$.
• Related: the small signed sums of Problem 317 (a signed sum of the reciprocals of $1,\ldots,n$ of size below $c/2^n$, nonzero), the near-one subsums of Problem 311, the distinct subsums of Problem 320, and the other consequences of Croot's theorem on Problem 284, Problem 286 and Problem 295. The largest subset of $\{1,\ldots,N\}$ with no subset of reciprocal sum one has size $(1-\frac1e+o(1))N$ (Problem 300); the coincidence of main terms is noted here, not a relation.

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
• croot_1999_unit_fractions_denominators_short_intervals
• croot_1999_unit_fractions_denominators_short_intervals / main_theorem