{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/","md":"https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/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":"2026-05-03-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":"23-year-old amateur Liam Price gave GPT-5.4 Pro a single prompt.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"Primitive sets and von Mangoldt chains (arXiv 2605.00301)","url":"https://arxiv.org/abs/2605.00301","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Scientific American: Amateur armed with ChatGPT vibe-maths a 60-year-old problem","url":"https://www.scientificamerican.com/article/amateur-armed-with-chatgpt-vibe-maths-a-60-year-old-problem/","type":"press","group":"press","domain":"scientificamerican.com"},{"n":3,"title":"Terence Tao: Primitive sets and von Mangoldt chains — Erdős problem #1196 and beyond","url":"https://terrytao.wordpress.com/2026/05/03/primitive-sets-and-von-mangoldt-chains-erdos-problem-1196-and-beyond/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"}],"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper","summary":"23-year-old amateur Liam Price gave GPT-5.4 Pro a single prompt. In about 80 minutes it sketched a proof of Erdős problem #1196, the 1966 Erdős–Sárközy–Szemerédi conjectures on primitive sets and divisibility chains, using Markov chains with von Mangoldt weights. Professionals including Terence Tao and Jared Lichtman turned it into a paper (arXiv 2605.00301) that also gives a short new proof of the Erdős primitive set conjecture.","key_facts":["Problem open since 1966 (~60 years)","Proof sketch by GPT-5.4 Pro in ~80 minutes from one prompt by Liam Price; escalated by Kevin Barreto","Paper authors include Tao, Alexeev, Barreto, Lichtman, Price and others","Lichtman (who proved the Erdős primitive set conjecture in 2022) said the argument looked like it came 'from The Book'","erdosproblems.com lists #1196 as PROVED; formalisation reported underway","AI disclosure (abstract, arXiv 2605.00301): 'We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights.'"],"key_numbers":[],"tags":["math","erdos","number-theory","gpt-5-4","amateur"],"science":{"field":"mathematics","subfield":"analytic number theory","problem":"Erdős problem #1196 (Erdős–Sárközy–Szemerédi conjectures on primitive sets)","result":"Proof of the conjectured bounds for primitive sets and divisibility chains, plus a new short proof of the Erdős primitive set conjecture.","open_since":"1966","ai_system":["GPT-5.4 Pro"],"human_role":"AI-generated key idea from a non-expert's prompt; professional mathematicians verified and wrote the paper","verification":{"key":"expert","label":"Expert-checked","text":"Unrefereed arXiv preprint (2605.00301) by eight authors including the specialists Jared Duker Lichtman and Terence Tao (both co-authors, so not independent checks); no formal proof"},"status":"pending","shock":"A non-mathematician's one-shot prompt produced an elegant proof that a leading expert compared to one 'from The Book'."},"body_md":"## What happened\n\nAn amateur prompted a public model, which found a proof strategy via random divisibility chains. The problem's leading experts confirmed and extended it within days.\n\n## Why it matters\n\nIt was the first AI solution to a well-known, decades-old Erdős conjecture that specialists had actively worked on, not just an obscure entry. It came weeks before the unit-distance disproof.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint (2605.00301) by eight authors including the specialists Jared Duker Lichtman and Terence Tao (both co-authors, so not independent checks); no formal proof","sources":[]}],"related":[{"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","url":"https://postcutoff.com/e/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":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":3,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]},{"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"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending: Tao and Lichtman, previously named as checkers, are co-authors of the unrefereed preprint; added the abstract's AI disclosure verbatim"},{"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":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":3,"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":33,"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/amateur-gpt-5-4-pro-vibe-maths"}