Positive lower density of large prime gaps

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed C\log p_n, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where p_n/n increases have positive lower density, answering Erdős and Prachar. (Lean)

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Positive lower density of large prime gaps

September 25, 2026 · source · cite as \cite{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026}

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.

Result 026 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.