OpenAI model disproves Erdős’s 80-year-old unit distance conjecture
Result confirmed
Importance: historic (5 of 5)Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Expert-checked
- Importance
- Historic (5 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra20 days after its cutoff
- Claude Opus 5.5In its training data
- Gemini 4 Hydrogen50 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
- 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
Show 3 more
- 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
- Exponent 1.043171 (Eric Naslund, Hexagon 2610.00016, 2 Oct 2026): finite planar sets U_j with u(U_j)/|U_j|^1.043171 → ∞, using ‘the number-field construction of OpenAI and Sawin’ with fields from an infinite pro-2 tower over ℚ(√241); the largest exponent previously claimed was 1.0358, in Naslund’s own unpublished manuscript. Provenance: ‘agents developing the mathematics, writing the programs and writing the paper; I supervised, prompted, and coordinated them’; a Frobenius condition was found in an agent run of Meta’s Muse Spark; certificates and paper by Claude Opus 5.5 in Claude Code (also ChatGPT Astra 6, Sol 5.6); Palomar PALOMAR-
2026-10-01-000018; human understanding ‘some parts’
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.)
Sources
9 sources from 9 sites. Numbers match the chips in the text.
9 sources: 3 primary, 4 press, 2 reactions
Primary
- OpenAI: model disproves discrete geometry conjectureopenai.com, official
- Human exposition of the counterexample (arXiv 2605.20695)arxiv.org, paper
- Hexagon 2610.00016: An Exponent of 1.043 for the Unit Distance Problem (Naslund)hexagonmath.org, paper
Press
- Quanta: Why the legendary Erdős problems are falling to AIquantamagazine.org, press
- Scientific American: AI just solved an 80-year-old Erdős problemscientificamerican.com, press
- Physics World: AI-led solutions of Erdős problems spark debatephysicsworld.com, press
- MAA: AI solves an 80 year-old Erdős problemmaa.org, press
Reactions
- Gil Kalai: Amazing — Erdős unit distance problem was disproved by AIgilkalai.wordpress.com, discussion
- Slate: Did A.I. really solve a math problem mathematicians couldn’t?slate.com, discussion
Changes
- Added Naslund’s exponent 1.043171 (Hexagon 2610.00016, Palomar-registered; from the Hexagon backlog)
- Related: back-link to the sum-product disproof that grew out of this result
- Added video(s) fireship-summer-math-fell-to-the-machines
- Linked Google DeepMind’s AlphaProof Nexus paper published in Science (AlphaProof Nexus, published in Science 8 Oct 2026)
- Added science block, primary OpenAI and arXiv links, exponent follow-ups and quotes; (science & math tab)
- Filed
