Sárközy's Conjecture on Sums and Products Modulo a Prime
For let . Sárközy conjectured that for all large primes, every set of size at least has -like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Additive combinatorics over finite fields
- Posed by
- András Sárközy
- Year posed
- —
- Years open
- —
- Solved
- 2026-03-31
- Model
- Harmonic Aristotle
- Vendor
- Harmonic
- Collaborators
- Quanyu Tang
- Verification
- Lean-checked, statement unaudited
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
Aristotle produced formal Lean proofs of all four principal statements of the paper (the formalization is public), and "was also used to assist in the preparation of this paper." The counterexample construction itself builds on a classical projective-geometric orbit.
Verification
All four principal statements formalized and checked in Lean; Wouter van Doorn assisted with the formalization. No independent expert review yet. Tier: the formalization was produced within the project (Aristotle, with van Doorn assisting); no independent statement audit.