Positive lower density of arbitrarily large normalized prime gaps
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
If $p_n$ is the nth prime, must the set of indices n with $p_{n+1}-p_n>C$ log $p_n$ have positive lower natural density for every fixed $C>0$? The positive density may depend on C.Short English statement
If $p_n$ is the nth prime, must the set of indices n with $p_{n+1}-p_n>C$ log $p_n$ have positive lower natural density for every fixed $C>0$? The positive density may depend on C.Sources
- S1 Original question source or mathematical context. https://doi.org/10.4171/owr/2008/14.
- S2 Original question source or mathematical context. https://mathworld.wolfram.com/CramerConjecture.html.
- S3 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2792
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 026; 2026-09-25).
Positive lower density of arbitrarily large normalized prime gaps
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 026; 2026-09-25), the asserted result is that each fixed threshold $C\log p$ is exceeded by a proportion of consecutive prime gaps bounded away from zero, which also gives a positive lower proportion of increases in $p_n/n$.Mathematical statement of the claimed result
Manuscript 1.
Writing $p_n$ for the $n$th prime, the claimed positive lower density is \[ \forall C>0\ \exists c(C)>0,N_0(C)\ \forall N\ge N_0(C): \quad|\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}|\ge c(C)N. \] The density constant and threshold may depend on the fixed gap multiplier, which does not grow with $N$.Outline of the argument
For manuscript 1 (family 026; 2026-09-25), a square of signed divisor sums gives appreciable weight to a prime in one short interval and little weight to a prime in the adjacent interval. Two second-moment bounds convert weighted mass to ordinary starts; bounded multiplicity then counts distinct consecutive gaps.Formal verification
For manuscript 1 (family 026; 2026-09-25), the documented Lean scope is positive lower density of increases in the sequence $p_n/n$; the full large-gap theorem is not part of that selected result.The selected formal statement has narrower scope than the full manuscript claim.
The linked family Comparator checks are PrimeGaps.lean.
Their named declarations include OAI.Problem344.large_gaps_and_ratio_density.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Positive lower density of large prime gaps, 2026-09-25; citation key
OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 026: Lean scope.
- Comparator PrimeGaps.json.
- Comparator PrimeGaps.lean.
- OpenAI catalogue entry.