{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/","md":"https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/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":"2025-12-08-ai-erdos-problems-wave-late-2025","date":"2025-12-08","date_precision":"day","short_title":"Genuine AI-assisted solutions to Erdős problems begin","deck":"#124 (Aristotle), #1026 (48-hour human–AI collaboration)","takeaway":"In Nov–Dec 2025 AI tools produced the first genuinely new (if modest) solutions to Erdős problems.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Alexeev & Mixon on Erdős #707 (arXiv 2510.19804)","url":"https://arxiv.org/abs/2510.19804","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Terence Tao's wiki: AI contributions to Erdős problems","url":"https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems","type":"discussion","group":"reactions","domain":"github.com"},{"n":3,"title":"Terence Tao: The story of Erdős problem #1026","url":"https://terrytao.wordpress.com/2025/12/08/the-story-of-erdos-problem-126/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"},{"n":4,"title":"erdosproblems.com forum: problem #124","url":"https://www.erdosproblems.com/forum/thread/124","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":5,"title":"Xena Project: formalization of Erdős problems","url":"https://xenaproject.wordpress.com/2025/12/05/formalization-of-erdos-problems/","type":"discussion","group":"reactions","domain":"xenaproject.wordpress.com"}],"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Harmonic","Google DeepMind","OpenAI"],"title":"Genuine AI-assisted solutions to Erdős problems begin: #124 (Aristotle), #1026 (48-hour human–AI collaboration)","summary":"In Nov–Dec 2025 AI tools produced the first genuinely new (if modest) solutions to Erdős problems. Harmonic's Aristotle proved a version of #124 in Lean autonomously (29 Nov). Erdős #1026 (posed 1975) was fully solved within ~48 hours by humans combining Aristotle, AlphaEvolve, GPT and deep-research tools (7–9 Dec). Terence Tao warned these were 'long-tail' problems.","key_facts":["#124 (from a 1995 paper): Aristotle proved it autonomously in Lean from the formal statement; Bloom noted it was the easier of two variants, and Tao's wiki lists it as partial","#1026: Aristotle proved the key case c(k²)=1/k in Lean (7 Dec); full answer c(k²+2a+1) = k/(k²+a) assembled by 8–9 Dec","Tao on #1026: 'It was only through the combined efforts of all the contributors and their tools that all these key inputs were able to be assembled within 48 hours.'","#367: partial result by Alexeev, van Doorn and Tao with Aristotle and Gemini Deep Think (Nov 2025)","#707 ($1000 problem): Alexeev & Mixon disproved it with ChatGPT-assisted Lean checks, then found Marshall Hall Jr. had a counterexample in 1947","Tao: such results 'do not meet the hyped up goal of AI autonomously solving major mathematical open problems'"],"key_numbers":[],"tags":["math","erdos","lean","aristotle","alphaevolve"],"science":{"field":"mathematics","subfield":"combinatorics / number theory","problem":"Erdős problems #124, #367, #707, #1026 and others","result":"First AI-involved genuine new solutions of listed-open Erdős problems, including a full solution of #1026 and a Lean-verified autonomous proof for a variant of #124.","open_since":"1975","ai_system":["Aristotle","AlphaEvolve","Gemini Deep Think","GPT-5"],"human_role":"Mixed: #124 near-autonomous (formal statement given); #1026 human–AI collaboration","verification":{"key":"lean","label":"Lean-verified","text":"Formal proofs in Lean for key steps; expert-checked (Tao, Bloom)"},"status":"confirmed","shock":"Problems open for decades fell in days, but experts stressed they were obscure ones that few people had seriously attempted."},"body_md":"## What happened\n\nAfter the October 2025 fiasco, a distributed community of mathematicians and amateurs began systematically attacking the ~1,100 Erdős problems with AI tools, with results logged on Tao's wiki. The first genuinely new results arrived within weeks.\n\n## Why it matters\n\nErdős problems became the first large, public, verifiable scoreboard for AI in research mathematics, and set the stage for 2026's much larger results.","disputed":[],"related":[{"id":"2026-01-06-erdos-728-gpt-5-2-aristotle","url":"https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/","date":"2026-01-06","date_precision":"day","short_title":"Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof","deck":null,"takeaway":"It showed the combination of informal LLM reasoning with formal verification as a practical, trustworthy workflow for research maths that non-experts could run.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Harmonic"]},{"id":"2025-10-17-gpt-5-erdos-problems-controversy","url":"https://postcutoff.com/e/2025-10-17-gpt-5-erdos-problems-controversy/","date":"2025-10-17","date_precision":"day","short_title":"OpenAI researchers claim GPT-5 'solved' 10 Erdős problems","deck":"The solutions were already in the literature","takeaway":"The episode set the standard of scepticism for later AI maths claims, and led to Tao's public wiki tracking exactly what AI contributed to each Erdős problem.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"retracted","labels":["Retracted"]},"sources":3,"official":0,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"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":"2026-09-29","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":null,"in_training_data":true},{"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":null,"in_training_data":true},{"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/genuine-ai-assisted-solutions"}