Breakthrough Tracker record
An Astra-attributed proof improves the classic lower bound for large prime gaps
An OpenAI preprint proves that the largest prime gap below X is bounded below, up to a constant, by log X times (log₂ X)² times log₄ X divided by (log₃ X)², improving Rankin's classical expression by a factor of log₂ X.
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- Stable ID
math-long-prime-gaps-iterated-log-bound-2026- Revision
math-long-prime-gaps-iterated-log-bound-2026.v1- Field
- Mathematics · Analytic number theory and formalized mathematics
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional formalized result
- Last checked
AI role
The paper attributes the proof to GPT-6 Astra. OpenAI published a complete Lean 4 development and independent kernel-checking instructions, but no external specialist correctness assessment was located.
Record details
- Problem or result
- Asymptotic lower bounds for unusually large gaps between consecutive primes
- Authors
- OpenAI; proof attributed to GPT-6 Astra
- Institutions
- OpenAI
- Result date
- Publicly announced September 3, 2026
Why it matters
The improved iterated-log factor changes a term in the classical large-gap scale that OpenAI says had resisted improvement for more than 80 years, while supplying a machine-checkable proof artifact.
Limits
This is an unreviewed institutional preprint, and the tracker did not independently rebuild the Lean repository. The theorem is asymptotic and hides an unspecified positive constant; it is not an explicit new largest numerical prime gap.
Sources
- Primary: OpenAI, GPT-6 Astra research announcement
- Primary: OpenAI, Improved Long Gaps Between Primes
- Artifacts: Lean 4 formalization of the large-gap bound
Correction and revision history
- 2026-09-07 — Added after primary-source, scope, status, AI-role and limitation review.
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