---
id: "2026-05-03-erdos-1196-primitive-sets"
url: "https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-05-03"
date_precision: day
category: science
importance: 4
confidence: high
status: [Event confirmed, Awaiting review]
verification: Expert-checked
sources: 3
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/

# Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets

Full title: 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
- 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.'

## 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.

## What is disputed or not yet verified

- Verification: 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 3 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): in its training data
- Gemini 4 Hydrogen (training cutoff March 2026): 33 days after its cutoff
- Grok 4.7 (training cutoff May 2026): in its training data

## Sources

1. [Primitive sets and von Mangoldt chains (arXiv 2605.00301)](https://arxiv.org/abs/2605.00301) (arxiv.org, paper)
2. [Scientific American: Amateur armed with ChatGPT vibe-maths a 60-year-old problem](https://www.scientificamerican.com/article/amateur-armed-with-chatgpt-vibe-maths-a-60-year-old-problem/) (scientificamerican.com, press)
3. [Terence Tao: Primitive sets and von Mangoldt chains — Erdős problem #1196 and beyond](https://terrytao.wordpress.com/2026/05/03/primitive-sets-and-von-mangoldt-chains-erdos-problem-1196-and-beyond/) (terrytao.wordpress.com, discussion)

## Changes

- 2026-10-19: Status confirmed → pending: Tao and Lichtman, previously named as checkers, are co-authors of the unrefereed preprint; added the abstract's AI disclosure verbatim
- 2026-09-29 (filed): Created

## Related

- 2026-05-20: [OpenAI model disproves Erdős's 80-year-old unit distance conjecture](https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.md)
- 2026-01-06: [Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof](https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/)