OpenAI model disproves Erdős's 80-year-old unit distance conjecture
On 2026-05-20 OpenAI announced that an internal model found a counterexample to Erdős's 1946 unit-distance conjecture using algebraic number theory — widely described as the first historically significant proof produced by an AI; Timothy Gowers said he would recommend it to the Annals of Mathematics 'without any hesitation'. A wave of AI-assisted Erdős-problem solutions followed through summer 2026.
Key facts
- Counterexample: a grid construction where g(N) exceeds a fixed multiple of N^(1+ε), ε ≈ 6.24×10^-38 (Physics World)
- Method: algebraic number theory (Golod–Shafarevich class field towers, building on Ellenberg–Venkatesh and Hajir–Maire–Ramakrishna)
- Same-day human exposition and verification (arXiv 2605.20695) by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, V. Wang and Matchett Wood
- Will Sawin made the exponent explicit (1.014, later 1.0318) and showed this method cannot exceed about 1.2143; Kevin Buzzard reports it was later formalised in Lean
- Gowers: 'quite an important moment in the history of mathematics'; Jozsef Solymosi: 'I was most surprised by the depth of the solution'
- Timothy Gowers: would recommend Annals of Mathematics publication 'without any hesitation'
- Erdős #728 (Jan 4 2026) solved by amateurs Barreto & Price with GPT-5.2 Pro, formally verified with Aristotle
- Erdős #1196 (May 2026): paper co-authored by Barreto, Price, Terence Tao, Jared Duker Lichtman and others
- Aug 1 2026: OpenAI said unreleased model 'Astra' made 10 further advances incl. three more Erdős problems
- erdosproblems.com status at Quanta's Aug 2026 article: 565 solved, 652 open
Science result
- Field
- mathematics / discrete geometry
- Problem
- Erdős unit distance conjecture (planar point sets: at most N^(1+o(1)) unit distances) (open since 1946)
- Result
- Construction of planar N-point sets with at least N^(1+δ) unit distances for a fixed tiny δ>0, disproving Erdős's conjectured upper bound, via algebraic number theory; humans improved the exponent within weeks.
- AI system
- OpenAI internal reasoning model
- Human role
- Autonomous discovery by the model; checked and refined by human mathematicians
- Verification
- Expert-checked (Timothy Gowers and others); human follow-up papers
- Status
- confirmed
- Why surprising
- Widely described as the first historically significant proof produced by an AI; Gowers said he would recommend it to the Annals 'without any hesitation'.
What happened
The unit distance problem asks how many pairs of points among N points in the plane can be exactly distance 1 apart; Erdős conjectured an upper bound of N^(1+o(1)). OpenAI's model constructed a counterexample. Nine leading mathematicians commented on the result. Meanwhile amateurs using GPT-5.x and teams with Terence Tao resolved other Erdős problems, and Google DeepMind reported solving 9 of 353 open problems at a few hundred dollars each.
Why it matters
This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics. (Confidence medium: primary OpenAI post not fetched; details from reputable press.)
Changelog
- 2026-09-29: created
- 2026-09-29: added science block, primary OpenAI and arXiv links, exponent follow-ups and quotes; (science & math tab)
Related events
- AI systems score a perfect 42/42 at IMO 2026, officially graded ★★★★★
- OpenAI researchers claim GPT-5 'solved' 10 Erdős problems; the solutions were already in the literature ★★★
- Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper ★★★★
- GPT-5.5 Pro finds counterexample disproving McKean's 1966 conjecture and the Gaussian completely monotone conjecture ★★★
- Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs ★★★★
Sources (8)
- officialOpenAI: model disproves discrete geometry conjecture
- paperHuman exposition of the counterexample (arXiv 2605.20695)
- discussionGil Kalai: Amazing — Erdős unit distance problem was disproved by AI
- pressQuanta: Why the legendary Erdős problems are falling to AI
- pressScientific American: AI just solved an 80-year-old Erdős problem
- pressPhysics World: AI-led solutions of Erdős problems spark debate
- pressMAA: AI solves an 80 year-old Erdős problem
- discussionSlate: Did A.I. really solve a math problem mathematicians couldn't?
id: 2026-05-20-ai-disproves-erdos-unit-distance-conjecture · updated 2026-09-29 · open in the interactive timeline