{"id":"openai-math/Positive-lower-density-of-large-prime-gaps","pdf":"https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026","tags":["number-theory","prime-numbers","analytic-number-theory"],"score":0,"title":"Positive lower density of large prime gaps","domain":"Number theory","family":"026","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Positive lower density of large prime gaps"}