{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/","md":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","date":"2026-05-20","date_precision":"day","short_title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","deck":null,"takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"OpenAI: model disproves discrete geometry conjecture","url":"https://openai.com/index/model-disproves-discrete-geometry-conjecture/","type":"official","group":"primary","domain":"openai.com"},{"n":2,"title":"Human exposition of the counterexample (arXiv 2605.20695)","url":"https://arxiv.org/abs/2605.20695","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Hexagon 2610.00016: An Exponent of 1.043 for the Unit Distance Problem (Naslund)","url":"https://hexagonmath.org/abs/2610.00016v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":4,"title":"Quanta: Why the legendary Erdős problems are falling to AI","url":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/","type":"press","group":"press","domain":"quantamagazine.org"},{"n":5,"title":"Scientific American: AI just solved an 80-year-old Erdős problem","url":"https://www.scientificamerican.com/article/ai-just-solved-an-80-year-old-erdos-problem-and-mathematicians-are-amazed/","type":"press","group":"press","domain":"scientificamerican.com"},{"n":6,"title":"Physics World: AI-led solutions of Erdős problems spark debate","url":"https://physicsworld.com/a/ai-led-solutions-of-erdos-problems-spark-debate-over-the-future-of-mathematics/","type":"press","group":"press","domain":"physicsworld.com"},{"n":7,"title":"MAA: AI solves an 80 year-old Erdős problem","url":"https://maa.org/math-values/ai-solves-an-80-year-old-erdos-problem/","type":"press","group":"press","domain":"maa.org"},{"n":8,"title":"Gil Kalai: Amazing — Erdős unit distance problem was disproved by AI","url":"https://gilkalai.wordpress.com/2026/05/21/amazing-erdos-unit-distance-problem-was-disproved-it-was-achieved-by-ai/","type":"discussion","group":"reactions","domain":"gilkalai.wordpress.com"},{"n":9,"title":"Slate: Did A.I. really solve a math problem mathematicians couldn't?","url":"https://slate.com/technology/2026/06/math-chatgpt-erdos-problem-solved-open-ai.html","type":"discussion","group":"reactions","domain":"slate.com"}],"official":3,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","summary":"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'"],"key_numbers":[],"tags":["math","research","erdos","discovery"],"science":{"field":"mathematics","subfield":"discrete geometry","problem":"Erdős unit distance conjecture (planar point sets: at most N^(1+o(1)) unit distances)","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.","open_since":"1946","ai_system":["OpenAI internal reasoning model"],"human_role":"Autonomous discovery by the model; checked and refined by human mathematicians","verification":{"key":"expert","label":"Expert-checked","text":"Expert-checked (Timothy Gowers and others); human follow-up papers"},"status":"confirmed","shock":"Widely described as the first historically significant proof produced by an AI; Gowers said he would recommend it to the Annals 'without any hesitation'."},"body_md":"## What happened\n\nThe 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.\nNine 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.\n\n## Why it matters\n\nThis 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.)","disputed":[],"related":[{"id":"2026-10-08-alphaproof-nexus-science-paper","url":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/","date":"2026-10-08","date_precision":"day","short_title":"Google DeepMind's AlphaProof Nexus paper published in Science","deck":"LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","takeaway":"This is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":16,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Google DeepMind"]},{"id":"2026-07-23-imo-2026-ai-perfect-scores","url":"https://postcutoff.com/e/2026-07-23-imo-2026-ai-perfect-scores/","date":"2026-07-23","date_precision":"day","short_title":"AI systems score a perfect 42/42 at IMO 2026, officially graded","deck":null,"takeaway":"Olympiad math is now saturated as an AI benchmark just one year after the first gold-level results; attention shifts to research-level math (FrontierMath Tier 4, Erdős problems).","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":0,"filed":"2026-09-29","updated":"2026-10-07","orgs":["Huawei","Xiaohongshu (RedNote)"]},{"id":"2026-05-27-sum-product-conjecture-false-over-reals","url":"https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/","date":"2026-05-27","date_precision":"day","short_title":"Erdős–Szemerédi sum-product conjecture shown false over the reals","deck":"A GPT-5.5 Pro agent re-disproves it in 7 of 8 runs","takeaway":"It shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"},{"id":"jacob-tsimerman","name":"Jacob Tsimerman","url":"https://postcutoff.com/person/jacob-tsimerman/"},{"id":"kevin-buzzard","name":"Kevin Buzzard","url":"https://postcutoff.com/person/kevin-buzzard/"},{"id":"timothy-gowers","name":"Timothy Gowers","url":"https://postcutoff.com/person/timothy-gowers/"}],"posts":[],"videos":[{"id":"fireship-summer-math-fell-to-the-machines","title":"The summer Math fell to the machines...","url":"https://postcutoff.com/v/fireship-summer-math-fell-to-the-machines/"},{"id":"openai-the-erdos-breakthrough","title":"The Erdős Breakthrough","url":"https://postcutoff.com/v/openai-the-erdos-breakthrough/"}],"models":[],"changes":[{"date":"2026-10-29","type":"edit","text":"Added Naslund's exponent 1.043171 (Hexagon 2610.00016, Palomar-registered; from the Hexagon backlog)"},{"date":"2026-10-19","type":"edit","text":"Related: back-link to the sum-product disproof that grew out of this result"},{"date":"2026-10-19","type":"edit","text":"Added video(s) fireship-summer-math-fell-to-the-machines"},{"date":"2026-10-10","type":"edit","text":"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)"},{"date":"2026-09-29","type":"edit","text":"Added science block, primary OpenAI and arXiv links, exponent follow-ups and quotes; (science & math tab)"},{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":20,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":50,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":"https://postcutoff.com/s/openai-disproves-erdos"}