---
id: "2026-01-06-erdos-728-gpt-5-2-aristotle"
url: "https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-01-06"
date_precision: day
category: science
importance: 4
confidence: high
status: [Result confirmed]
verification: Lean-verified
sources: 3
editor: human editor
human_review: null
version: "2026-10-04"
---

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/2026-01-06-erdos-728-gpt-5-2-aristotle/

# Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof

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

## What happened

An 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.

## Why it matters

It showed the combination of informal LLM reasoning with formal verification as a practical, trustworthy workflow for research maths that non-experts could run.

## Your AI and this story

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

## Sources

1. [Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof (arXiv 2601.07421)](https://arxiv.org/abs/2601.07421) (arxiv.org, paper)
2. [The Decoder: Tao says GPT-5.2 Pro cracked an Erdős problem but warns the win says more about speed than difficulty](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/) (the-decoder.com, press)
3. [Terence Tao's wiki: AI contributions to Erdős problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems) (github.com, discussion)

## Changes

- 2026-09-29 (filed): Created

## Related

- 2026-05-03: [Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets](https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/index.md)
- 2025-12-08: [Genuine AI-assisted solutions to Erdős problems begin](https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/)