Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper
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
Science result
- Field
- mathematics / analytic number theory
- Problem
- Erdős problem #1196 (Erdős–Sárközy–Szemerédi conjectures on primitive sets) (open since 1966)
- Result
- Proof of the conjectured bounds for primitive sets and divisibility chains, plus a new short proof of the Erdős primitive set conjecture.
- 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
- Expert-checked by Tao, Lichtman and others; arXiv preprint
- Status
- confirmed
- Why surprising
- A non-mathematician's one-shot prompt produced an elegant proof that a leading expert compared to one 'from The Book'.
What happened
An 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.
Why it matters
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. It came weeks before the unit-distance disproof.
Changelog
- 2026-09-29: created
Related events
- Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof ★★★★
- OpenAI model disproves Erdős's 80-year-old unit distance conjecture ★★★★★
- Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run ★★★★
Sources (3)
- paperPrimitive sets and von Mangoldt chains (arXiv 2605.00301)
- discussionTerence Tao: Primitive sets and von Mangoldt chains — Erdős problem #1196 and beyond
- pressScientific American: Amateur armed with ChatGPT vibe-maths a 60-year-old problem
id: 2026-05-03-erdos-1196-primitive-sets · updated 2026-09-29 · open in the interactive timeline