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Prime Gaps at Most 186

For the sequence of primes pnp_n, the project deriveslim inf⁡n→∞(pn+1−pn)≤186.\liminf_{n\to\infty}(p_{n+1}-p_n)\le186.More precisely, assuming three explicit analytic/numerical inputs, it proves DHL[40,2]\mathrm{DHL}[40,2]: every admissible set of forty integer shifts has infinitely many translates containing at least two primes. Applying this to an explicit admissible 4040-tuple of diameter 186186 yields infinitely many consecutive prime gaps of size at most 186186. The Lean development verifies the deduction from the stated inputs; the two Kloosterman-type estimates and the finite physical-integral/cap bounds remain external assumptions.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Construction
Field
Analytic number theory
Posed by
Alphonse de Polignac (the twin prime conjecture); the bounded form since Goldston, Pintz and Yıldırım
Year posed
1849
Years open
177y
Solved
2026-09-03
Model
GPT 6 Astra
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Significance
62 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Assuming three explicit input statements, the project proves DHL[40,2]\mathrm{DHL}[40,2]: every admissible 4040-tuple contains at least two primes infinitely often after translation. An explicit admissible 4040-tuple of diameter 186186 then giveslim inf⁡n→∞(pn+1−pn)≤186.\liminf_{n\to\infty}(p_{n+1}-p_n)\le186.The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimates cited to Katz/Deligne and Fouvry--Kowalski--Michel; the third consists of finitely many numerical integral and cap inequalities backed by a Python/FLINT certificate.

What the AI did

The project metadata identifies GPT 6 Astra, operating through Codex, as the agent used for the formalization workflow. Its stated task was to formalize the source statements faithfully, simplify the proofs, and retain unresolved finite-field and numerical inputs as explicit axioms. The resulting development was refined through human-guided edits and automated proof checks. The repository metadata attributes both the project and its source article “Improved Gaps Between Primes” to OpenAI.

Verification

Lean-checked, not Lean-verified, and the distinction is the whole of it. The Lean development in openai/PrimeGaps186 reports zero sorry in its three main declarations, and Comparator and Lean's kernel accept them - but all three depend on three project-specific axioms: a rank-three hyper-Kloosterman bound, a rank-two Kloosterman correlation bound, and a package of 104 outer, 45 inner and 3 cap numerical inequalities. So the kernel has checked that those three statements imply DHL[40,2] and the bound; it has not checked them. The first two are tied to established literature and the third is recomputed by a Python and FLINT certificate that does not discharge its Lean axiom. No independent expert has read the paper on the record.

Sources

Submitted by VibeGene on

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