Problem 318

Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that

\sum_{n\in S}\frac{f(n)}{n}=0?

What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$ ?

Source: erdosproblems.com/318

Number theory
Unit fractions

Wiki page

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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, with the three questions standing as follows. Arithmetic progressions: yes, by Sattler's paper on property $P_1$ for the arithmetical sequence (Indag. Math. (Proc.) 85 (1982), 347--352, refereed), which is not held here and is attested by the site and by the thread's reading of it. Positive density: no; any infinite set with exactly one even number fails, by a two-line argument written out below that the site records and that Sattler's companion paper credits to Erdős. Squares other than $1$ : yes, by Theorem 6 of Larsen's manuscript "Sufficiently abundant numbers are pseudoperfect" (GitHub, 1 February 2026; nine pages; unrefereed; its closing line acknowledges the AI systems Claude Opus 4.5 and ChatGPT 5.2 Pro for proofreading), which the site accepts and for which no journal record or independent review was found. Lean proofs of all three parts exist outside this corpus (recorded under Formalization below); none has been built or audited here, so they give no formalized evidence. The three answers are the claim pages Sattler's theorem on arithmetic progressions, the one-even-number observation and Larsen's Theorem 6, from which the standing derives part by part; the second-hand standing of the arithmetic-progression source and the preprint standing of the squares source are recorded below. The 1975 cases $A=\mathbb N$ and the odd numbers from $3$ have partial pages of their own, and a further page, the Lean proof of the progression question, records a pending independent machine proof of the first question.

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. 42. Library home: Erdős and Graham 1980.
ErSt75: Erdős, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (3) 23 (1975), 183. Not held; the case $A=\mathbb N$ , attested by the monograph and the site; the account does not rest on it. Claim page: the positive integers have property P1.
Sa75: Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (3) 23 (1975), 184--189. Not held; the odd numbers, attested likewise. Claim page: the odd numbers from 3 have property P1.
Sa82: Sattler, R., On Erdös property $P_1$ for the sequence of squarefree numbers. Indagationes Mathematicae (Proceedings) 85 (1982), no. 3, 341--346, DOI 10.1016/1385-7258(82)90025-7 (the site writes the series as Nederl. Akad. Wetensch. Indag. Math.). Not held; the Crossref record carries Elsevier's open-access user license dated 29 July 2013, which places the article in the publisher's free open archive. The site credits its opening page with the one-even-number observation.
Sa82b: Sattler, R., On Erdös property $P_1$ for the arithmetical sequence. Indagationes Mathematicae (Proceedings) 85 (1982), no. 3, 347--352, DOI 10.1016/1385-7258(82)90026-9. Not held; the same open-archive license (29 July 2013); the status-defining source for the first question, quoted second-hand.
La26: Larsen, D., Sufficiently abundant numbers are pseudoperfect. Manuscript, 9 pp., in the GitHub repository Larsen-Daniel/Erdos-318 (318.pdf at the commit of 1 February 2026 pinned on the claim page, the head on 2026-09-18; the text is undated and cites the site's pages as accessed 29 January 2026). Theorem 6, p. 8. Library home: Larsen 2026.
Bl21: Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726; J. Eur. Math. Soc. 27 (2025), 4563--4589. Its Theorem 2 is the input of a thread argument recorded below. Library home: Bloom 2021.
Ch26: Chojecki, P., Signed zero-sums of reciprocal squares on the squares: a computational attack plan. Four-page note dated 24 January 2026, the PDF linked from the thread's comment of that day (68,735 bytes). A lead, not filed.

Problem 318

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