Problem 968

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claims/: The 1 claim page of Problem 968, one per claimant's result; the problem's standing derives from them.

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Statement. Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density?

Statement (precise). Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive lower density?

Notes. 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: erdos_1961_satze_und_probleme_uber_german) ask whether the "untere Dichte" (lower density) of the $k$ with $p_k/k<p_{k+1}/(k+1)$, and of the $k$ with $p_k/k>p_{k+1}/(k+1)$, is positive, and close the paragraph with the remark that it seems hard to prove that the $k$ with $p_k/k<p_{k+1}/(k+1)$ have "positive untere Dichte". Erdős's 1965 survey ([Er65b], p. 204; library card: erdos_1965_recent_advances_current_problems_number_theory), the source of the site's wording, puts $u_k=p_k/k$ and says: "We easily show that the density of the integers $k$ for which $u_k>u_{k+1}$ is positive. We cannot show that the same holds for the $k$ for which $u_k<u_{k+1}$." 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.

Formulation. Positive lower density means that some $c>0$ has at least $cx$ of the $n\le x$ with $u_n<u_{n+1}$ for all large $x$. A stronger reading of the site's wording also asks that the set have an asymptotic density; whether it does is a separate question that no source here addresses, and it is open. On the same page Erdős and Prachar give the short argument for the companion case, the $k$ with $p_k/k>p_{k+1}/(k+1)$, which [Er65b] calls easy.

Status. 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 $C>0$ a positive proportion of the consecutive prime gaps $p_{n+1}-p_n$ exceed $C\log p_n$, uniformly over all large initial segments, and deduces that the $n$ with $u_n<u_{n+1}$ 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 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.

Source. erdosproblems.com/968, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #968, https://www.erdosproblems.com/968.

References.

• [Er65b] Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196--244; the $u_k$ passage on printed p. 204 (PDF p. 10 of the public scan https://users.renyi.hu/~p_erdos/1965-17.pdf). Library home: erdos_1965_recent_advances_current_problems_number_theory.
• [ErPr61] Erdős, P. and Prachar, K., Sätze und Probleme über $p\sb{k}/k$. Abh. Math. Sem. Univ. Hamburg 25 (1961/62), 251-256.

Formalization. Statement in formal-conjectures, asking for positive lower density, tagged research open with a sorry body and no formal_proof attribute at the linked revision. The release's Lean proof of the corollary, built and checked in this corpus, is described on the claim page above.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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_1961_satze_und_probleme_uber_german
• erdos_1961_satze_und_probleme_uber_german / question_p256
• erdos_1961_satze_und_probleme_uber_german / satz_1
• erdos_1961_satze_und_probleme_uber_german / satz_2
• erdos_1965_recent_advances_current_problems_number_theory
• openai_2026_positive_lower_density_large_prime_gaps
• openai_2026_positive_lower_density_large_prime_gaps / corollary_1_2
• openai_2026_positive_lower_density_large_prime_gaps / theorem_1_1