What was already known
Erdős, Sárközy and Szemerédi conjectured a chain under the stronger condition of positive lower logarithmic density. Davenport–Erdős had proved a chain for positive lower natural density.
AI involvement
The authors credit a separate autonomous GPT-5.4 Pro run with the theorem. Eight human authors rewrote, explained and reviewed the final proof.
Section 11 explicitly describes a similar autonomous run establishing Theorem 1.6; the final exposition is human.
Importance
Resolves the divisibility-chain conjecture under a weaker density assumption and gives a quantitative chain bound.
Limitations
The autonomous run transcript is unavailable, so its exact steps cannot be compared with the human final proof. No Lean formalization of this theorem is claimed.
Technical verification
The authors report reviewing the final proof of Theorem 1.6; no Lean formalization of this theorem is claimed.
Formal checking
Generation and review of the final proof
- Checker
- Eight human authors of the primitive-sets paper
- Scope
- Theorem 1.6 quantitative infinite divisibility-chain conclusion
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Section 9 explains the invariant weight and Markov-chain construction; Remark 9.1 explains why finite chains alone cannot be glued into the asserted infinite chain.
Correction and publication history
Resolves the divisibility-chain conjecture under a weaker density assumption and gives a quantitative chain bound.
Read this published revision