Erdős–Szemerédi sum-product conjecture shown false over the reals
A GPT-5.5 Pro agent re-disproves it in 7 of 8 runs
Event confirmedAwaiting review
Importance: major (4 of 5)The takeaway
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.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra27 days after its cutoff
- Claude Opus 5.5In its training data
- Gemini 4 Hydrogen57 days after its cutoff
- Grok 4.7In its training data
It is after the cutoff of GPT-6 Astra and Gemini 4 Hydrogen, and can be in the training data of Claude Opus 5.5 and Grok 4.7.
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
- AI disclosure (Huang, arXiv 2607.20525, abstract): ‘we present a simple agent built on GPT-5.5 Pro. Using a problem-agnostic, three-stage prompting pipeline -- proof-plan proposal, proof construction, and review -- the agent autonomously generated correct proofs that the sum-product conjecture is false over R in 7 of 8 independent trials; in the remaining trial, it identified an unresolved gap in its argument.’
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.
What is disputed or not yet verified
| Verification | Unrefereed preprint by specialists (Bloom, Sawin, Schildkraut, Zhelezov, arXiv 2605.28781); the autonomous AI rediscovery is a separate unrefereed preprint (Huang, arXiv 2607.20525); no formal proof |
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Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Status confirmed → pending (both papers are unrefereed preprints); added Huang’s AI disclosure verbatim from the arXiv abstract
- Filed