---
id: "2026-05-20-ai-disproves-erdos-unit-distance-conjecture"
url: "https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-05-20"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Result confirmed]
verification: Expert-checked
sources: 9
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/

# 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
- 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.)

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 20 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): in its training data
- Gemini 4 Hydrogen (training cutoff March 2026): 50 days after its cutoff
- Grok 4.7 (training cutoff May 2026): in its training data

## Sources

1. [OpenAI: model disproves discrete geometry conjecture](https://openai.com/index/model-disproves-discrete-geometry-conjecture/) (openai.com, official)
2. [Human exposition of the counterexample (arXiv 2605.20695)](https://arxiv.org/abs/2605.20695) (arxiv.org, paper)
3. [Hexagon 2610.00016: An Exponent of 1.043 for the Unit Distance Problem (Naslund)](https://hexagonmath.org/abs/2610.00016v1) (hexagonmath.org, paper)
4. [Quanta: Why the legendary Erdős problems are falling to AI](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/) (quantamagazine.org, press)
5. [Scientific American: AI just solved an 80-year-old Erdős problem](https://www.scientificamerican.com/article/ai-just-solved-an-80-year-old-erdos-problem-and-mathematicians-are-amazed/) (scientificamerican.com, press)
6. [Physics World: AI-led solutions of Erdős problems spark debate](https://physicsworld.com/a/ai-led-solutions-of-erdos-problems-spark-debate-over-the-future-of-mathematics/) (physicsworld.com, press)
7. [MAA: AI solves an 80 year-old Erdős problem](https://maa.org/math-values/ai-solves-an-80-year-old-erdos-problem/) (maa.org, press)
8. [Gil Kalai: Amazing — Erdős unit distance problem was disproved by AI](https://gilkalai.wordpress.com/2026/05/21/amazing-erdos-unit-distance-problem-was-disproved-it-was-achieved-by-ai/) (gilkalai.wordpress.com, discussion)
9. [Slate: Did A.I. really solve a math problem mathematicians couldn't?](https://slate.com/technology/2026/06/math-chatgpt-erdos-problem-solved-open-ai.html) (slate.com, discussion)

## Changes

- 2026-10-29: Added Naslund's exponent 1.043171 (Hexagon 2610.00016, Palomar-registered; from the Hexagon backlog)
- 2026-10-19: Related: back-link to the sum-product disproof that grew out of this result
- 2026-10-19: Added video(s) fireship-summer-math-fell-to-the-machines
- 2026-10-10: Linked [Google DeepMind's AlphaProof Nexus paper published in Science](https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/) (AlphaProof Nexus, published in Science 8 Oct 2026)
- 2026-09-29: Added science block, primary OpenAI and arXiv links, exponent follow-ups and quotes; (science & math tab)
- 2026-09-29 (filed): Created

## Related

- 2026-10-08: [Google DeepMind's AlphaProof Nexus paper published in Science](https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/index.md)
- 2026-07-23: [AI systems score a perfect 42/42 at IMO 2026, officially graded](https://postcutoff.com/e/2026-07-23-imo-2026-ai-perfect-scores/index.md)
- 2026-05-27: [Erdős–Szemerédi sum-product conjecture shown false over the reals](https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/), [Jacob Tsimerman](https://postcutoff.com/person/jacob-tsimerman/), [Kevin Buzzard](https://postcutoff.com/person/kevin-buzzard/), [Timothy Gowers](https://postcutoff.com/person/timothy-gowers/)
- Archived: 2 videos, listed in https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.json