{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/","md":"https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/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":"2026-01-06-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":"On 4–6 Jan 2026 amateur Kevin Barreto relayed an informal argument from GPT-5.2 Pro to Harmonic's Aristotle, which formalised it in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof (arXiv 2601.07421)","url":"https://arxiv.org/abs/2601.07421","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"The Decoder: Tao says GPT-5.2 Pro cracked an Erdős problem but warns the win says more about speed than difficulty","url":"https://the-decoder.com/terence-tao-says-gpt-5-2-pro-cracked-an-erdos-problem-but-warns-the-win-says-more-about-speed-than-difficulty/","type":"press","group":"press","domain":"the-decoder.com"},{"n":3,"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"}],"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Harmonic"],"title":"Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof","summary":"On 4–6 Jan 2026 amateur Kevin Barreto relayed an informal argument from GPT-5.2 Pro to Harmonic's Aristotle, which formalised it in Lean. It was widely accepted as the first Erdős problem solved essentially autonomously by AI with no prior solution in the literature. Terence Tao said the win 'says more about speed than difficulty'.","key_facts":["Jan 4: first run solved an ambiguous reading of the problem; Jan 5: GPT-5.2 Pro upgraded the argument to the intended statement; Jan 6: Aristotle formalised it","Tao: 'a near-autonomous solution that has not been reproduced in existing literature'","Human role: prompting and relaying only; Barreto clarified no mathematical hint was given","Write-up: arXiv 2601.07421"],"key_numbers":[],"tags":["math","erdos","lean","aristotle","gpt-5-2"],"science":{"field":"mathematics","subfield":"number theory / combinatorics","problem":"Erdős problem #728","result":"Full solution of the intended version of #728, generated by GPT-5.2 Pro and machine-checked in Lean by Aristotle.","open_since":"","ai_system":["GPT-5.2 Pro","Aristotle"],"human_role":"Near-autonomous: human relayed prompts between two AI systems","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean; endorsed by Terence Tao"},"status":"confirmed","shock":"An end-to-end AI pipeline, informal proof plus formal verification, closed an Erdős problem with essentially no human mathematics."},"body_md":"## What happened\n\nAn amateur used two commercial AI systems in tandem: one to find a proof and one to formally verify it. After fixing a misreading of the problem statement, the pipeline produced a Lean-checked solution.\n\n## Why it matters\n\nIt showed the combination of informal LLM reasoning with formal verification as a practical, trustworthy workflow for research maths that non-experts could run.","disputed":[],"related":[{"id":"2026-05-03-erdos-1196-primitive-sets","url":"https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/","date":"2026-05-03","date_precision":"day","short_title":"Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets","deck":"Tao co-authors the paper","takeaway":"It was the first AI solution to a well-known, decades-old Erdős conjecture that specialists had actively worked on, not just an obscure entry.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2025-12-08-ai-erdos-problems-wave-late-2025","url":"https://postcutoff.com/e/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":"Erdős problems became the first large, public, verifiable scoreboard for AI in research mathematics, and set the stage for 2026's much larger results.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Harmonic","Google DeepMind","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/erdos-problem-728-solved"}