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Claim. Let $A\subseteq\mathbb N$ be an infinite arithmetic progression and let $f:A\to\{-1,1\}$ be non-constant. Then some finite non-empty $S\subset A$ has $\sum_{n\in S}f(n)/n=0$; in the paper's terms, every arithmetic progression has Erdős's property $P_1$. This answers the first question of Problem 318 in the affirmative and contains the earlier cases $A=\mathbb N$ (Erdős and Straus, 1975) and the odd numbers (Sattler, 1975).

Covers. The first of the problem's three questions, infinite arithmetic progressions. It says nothing about sets of positive density or about the squares, which have their own claim pages.

Source and acceptance. R. Sattler, 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: a refereed journal article, which is the refereed evidence. The Crossref record dates the article to 1982 and gives no day except the start of its first license, 1 January 1982, which is the date this page carries. The site's curator, Thomas Bloom, credits the paper with the affirmative answer in the commentary of the problem page (label SOLVED, last edited 1 April 2026) and is independent of the author; that credit is the reviewed evidence. The article is not held in the library: its statement is attested by the site and by a thread comment of 17 August 2025 reporting the paper's content, and the Crossref record carries Elsevier's open-archive license of 29 July 2013, so the text is free to read in a browser. Since the paper is not held, its exact hypothesis is recorded at second hand, from the site's wording. Both of Sattler's 1982 papers announce a third paper on the squares other than $1$ that never appeared; that question is settled on Larsen's page. An independent Lean proof of the progression statement, with Codex and GPT-5.6 Sol as its formal authors and no reference to this paper, has its own pending page, the Lean proof of the progression question.