The Sum-Product Conjecture over the Reals
Erdos and Szemeredi conjectured that every finite set of reals satisfies . False: there are arbitrarily large , of algebraic integers in a number field of degree , with for an absolute . Variants give counterexamples in function fields of fixed positive characteristic.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Additive combinatorics
- Posed by
- Paul Erdos, Endre Szemeredi
- Year posed
- 1983
- Years open
- 43y
- Solved
- 2026-05-27
- Model
- GPT-5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Thomas F. Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zakharov
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
the model's contribution is one simplifying lemma; the authors state the main ideas are human
What the AI did
The limits here matter more than the headline, and the authors state them plainly: GPT-5.5 Pro was a sounding board in the early stages, but the final proof including all the main ideas was almost entirely human-generated, and everything in the paper was written by the authors. The single exception they name is Lemma 3.4, suggested by the model, which replaced a more complicated result of Schinzel with a short elementary argument. There is a second, indirect AI thread: the authors say they were inspired to revisit number fields of large degree by OpenAI's counterexample to the unit distance conjecture, and note their construction needed far less number-theoretic input than that one did.
Verification
arXiv preprint by four established additive combinatorialists; not yet peer-reviewed. Given the size of the claim this one deserves refereeing before it is treated as settled.
Source
arXiv:2605.28781 - The sum-product conjecture is false for real numbers