The Umans-Wang Arithmetic-Progression Divisor Conjecture
An -divisor set contains a multiple of every integer from 1 to . Umans and Wang proposed, as the arithmetic-progression form of their Strong -Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Combinatorial number theory
- Posed by
- Chris Umans, Sheng Wang
- Year posed
- 2025
- Years open
- 1y
- Solved
- 2026-08-07
- Model
- GPT-5.6 Sol (via Codex, reasoning effort ultra)
- Vendor
- OpenAI
- Collaborators
- Xinjie He, Amit Sahai
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 18 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
"The proof was discovered in an OpenAI Codex run using the gpt-5.6-sol model with reasoning effort set to ultra. Codex also produced the initial write-up. Subsequent human review verified the proof, reviewed the citations, and revised the exposition." The authors note the prompting strategy borrowed from the UCLA Moonshot Harness project.
Verification
A preprint days old with no independent review. The paper states the argument is self-contained, needs no computer-assisted calculation and no access to the model transcript, with the prime number theorem as its only analytic input, so it is checkable on its own terms.