Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 026-01. Positive lower density of large prime gaps
Manuscript map · lines 2955–2964
OpenAI@2026-10-07#rem:oai-p-026-01
Authors: OpenAI. Family 026.
Title: Positive lower density of large prime gaps.
Abstract: For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C{}log p$, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which $p_n/n$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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