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Erdős and Prachar's question: do the increases of $p_n/n$ have positive lower density?

Theorem 1.1: for every fixed $C>0$ there are $c(C)>0$ and $N_0(C)$ with $\#\{n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N$ for all $N\ge N_0$ , with ordinary lower density and equal weight on prime indices. Corollary 1.2 (take $C=2$ ) answers Erdős-Prachar: the increases of $p_n/n$ have positive lower density. The main theorem is stronger than the posed question. The constant $c(C)$ is not made explicit or compared with the Poisson prediction $e^{-C}$ , and the further questions recorded with Problem #968 on $u_n<u_{n+1}<u_{n+2}$ and $u_n>u_{n+1}>u_{n+2}$ are not addressed.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Erdős and Prachar's question: do the increases of $p_n/n$ have positive lower density? · Erdos Problem #968: increases of p_n/n · Problem 968

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Erdős and Prachar's question: do the increases of $p_n/n$ have positive lower density?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1, Corollary 1.2 and its short proof were read against Erdős Problem #968. Lean: lean/docs/026.md links ComparatorChallenges/PrimeGaps (theorem OAI.Problem344.large_gaps_and_ratio_density, solution module OAI.NumberTheory.PrimeGaps.RatioCorollary, file exists at the pinned commit), not in formalization.yaml. The statement was read: for every $C>0$ there are $c>0$ and $N_0$ with at least $cN$ indices $n\le N$ having $p_{n+1}-p_n>C\log p_n$ for all $N\ge N_0$ , and the set of $n$ with $p_n/n<p_{n+1}/(n+1)$ has positive lower asymptotic density. That is the headline. Not rebuilt here.