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Let , where is the th prime. Does the set of such that have positive density?
Let , where is the th prime. Does the set of such that have positive lower density?
Source: erdosproblems.com/968
An accepted solution exists. The statement is true.
OPEN, the site's label (page last edited 31 March 2026; proof-claims tab empty when accessed 2026-10-07); the site's commentary reads the question as asking for positive lower density. The OpenAI release's preprint of 25 September 2026 proves that for every fixed a positive proportion of the consecutive prime gaps exceed , uniformly over all large initial segments, and deduces that the with have positive lower asymptotic density, the question the precise Statement asks. The release's Lean declaration states both results; this corpus built it, checked its axioms and found it identical to the release's comparator challenge, so the result is accepted on its claim page (OpenAI, 2026) and the problem stands solved and proved here, with "density" read as the lower density Erdős and Prachar asked for. Whether the set has an asymptotic density, the stronger reading under Formulation, remains open. The discussion thread (as of 2026-10-07) holds one conditional result: the comment of 10 September 2025 (Tao) gives a positive answer assuming the Riemann hypothesis and a weak form of the pair correlation conjecture; a conditional thread post, it has no claim page.
The site's wording does not say which density it means: "positive density" can ask that the set have an asymptotic density and that it be positive, or only that its lower density be positive, and the two readings differ for a set whose density need not exist. The change replaces "positive density" by "positive lower density"; nothing else changes. The evidence is the posers' own text. Erdős and Prachar ([ErPr61], p. 256; library card: Erdős and Prachar 1961) ask whether the "untere Dichte" (lower density) of the with , and of the with , is positive, and close the paragraph with the remark that it seems hard to prove that the with have "positive untere Dichte". Erdős's 1965 survey ([Er65b], p. 204; library card: Erdős 1965), the source of the site's wording, puts and says: "We easily show that the density of the integers for which is positive. We cannot show that the same holds for the for which ." It says "density" without qualification and is consistent with the 1961 question, so the ambiguity is already in the poser's 1965 text. The site's commentary reads the question as asking for positive lower density, and the formal-conjectures statement asks for positive lower density; neither is the evidence for the change. No result about another reading of the site's wording is recorded.