Result 026, Number theory

Positive lower density of large prime gaps

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

Lean formalization Proof

The bigger picture

Why it matters

How often are neighboring primes unusually far apart? The manuscript reports that gaps larger than any fixed multiple of a logarithmic reference scale occur with a frequency bounded away from zero.

What changes?

A consecutive prime gap is the difference between a prime and the next prime. Fix any positive constant C. The manuscript claims that, in every sufficiently large initial segment of the prime sequence, a positive proportion of these gaps exceed C times the natural logarithm of the smaller prime. The lower bound on that proportion may depend on C. This guarantees more than infinitely many examples, but does not guarantee the same frequency in every short interval.

What does that help mathematicians do?

The stated consequence concerns the ratio of the nth prime to its position n in the prime sequence. A positive proportion of indices, bounded below throughout sufficiently large initial segments, are steps where this ratio increases. This answers the question attributed to Erdős and Prachar. It rules out a picture in which upward steps persist forever but become vanishingly rare, without asserting that the ratio eventually increases at every step.

Are there practical applications?

The immediate value is foundational: the result strengthens what number theorists can say about the frequency of large gaps, rather than merely their existence. It supplies a persistent frequency constraint for studying prime spacing and yields the stated consequence for the prime-to-index ratio. The supplied material does not describe a computational method or practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Positive lower density of large prime gaps

September 25, 2026 19 pages

For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed Clog⁡pC\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 pn/np_n/n increases have positive lower asymptotic density, answering a question of Erdős and Prachar.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026,
  author = {{OpenAI}},
  title = {{Positive lower density of large prime gaps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf}{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/026.md.

Positive lower density of large prime gaps

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized supplement proves that the indices nn for which pn/n<pn+1/(n+1)p_n/n<p_{n+1}/(n+1) have positive lower asymptotic density, where pnp_n is the nnth prime. Thus the normalized prime sequence has a positive-density set of increases. This is the prime-ratio corollary associated with the paper's theorem on a positive lower density of large prime gaps.

Comparator links

Result Comparator statement
Positive lower density of prime-ratio increases PrimeGaps.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.