Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs
Inspired by the AI disproof of the unit-distance conjecture, Bloom, Sawin, Schildkraut and Zhelezov proved on 27 May 2026 that the Erdős–Szemerédi sum-product conjecture is false over the real numbers. They built sets A with |A+A| and |AA| ≤ |A|^(2−c). A July 2026 paper (arXiv 2607.20525) showed a GPT-5.5 Pro agent autonomously generated correct disproofs in 7 of 8 independent trials, some with new constructions.
Key facts
- Human paper: arXiv 2605.28781 (27 May 2026), 'inspired' by OpenAI's unit-distance disproof, which used related algebraic-number-theory ideas
- AI replication: GPT-5.5 Pro agent, three-stage prompting pipeline, correct disproofs in 7/8 runs; some avoid units by using L^p-type regions of algebraic integers
- The 1983 conjecture (max(|A+A|,|AA|) ≥ |A|^(2−ε)) remains open over the integers
Science result
- Field
- mathematics / additive combinatorics
- Problem
- Erdős–Szemerédi sum-product conjecture (over R) (open since 1983)
- Result
- Finite sets of reals with both sumset and product set of size at most |A|^(2−c), disproving the conjecture over R; later reproduced autonomously by an AI agent.
- AI system
- GPT-5.5 Pro (in the follow-up)
- Human role
- Original disproof human-led, inspired by an AI result; follow-up shows autonomous AI rediscovery
- Verification
- Human proof (preprint); AI proofs checked by authors
- Status
- confirmed
What happened
The number-theoretic idea behind the AI's unit-distance counterexample prompted human experts to attack a second famous Erdős conjecture, which fell within a week. A later study showed the AI could have done it alone.
Why it matters
It shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.
Changelog
- 2026-09-29: created
Related events
- OpenAI model disproves Erdős's 80-year-old unit distance conjecture ★★★★★
- GPT-5.5 Pro-assisted construction lowers the smallest known Borsuk counterexample dimension from 64 to 63 ★★★
Sources (2)
- paperThe sum-product conjecture is false for real numbers (arXiv 2605.28781)
- paperGPT-5.5 Pro agent disproofs of the sum-product conjecture over R (arXiv 2607.20525)
id: 2026-05-27-sum-product-conjecture-false-over-reals · updated 2026-09-29 · open in the interactive timeline