Why AI Companies Are Racing to Solve Erdős Problems

AI breakthroughs on Erdős problems

May 20 2026 OpenAI announced that an unreleased model produced a proof that refutes a 1946 conjecture of Paul Erdős, the first historically significant AI‑generated mathematical proof.

Within weeks human mathematicians extended the AI’s insight; days later the same technology was applied to other problems.

OpenAI Astra model

On August 1 OpenAI released the Astra model, claiming ten mathematical advances, including solutions to three additional Erdős problems. The announced cost for the ten breakthroughs was about $2 000.

Why Erdős problems are a benchmark

Erdős problems lie in number theory, combinatorics and graph theory—domains that large language models handle well. The difficulty range spans trivial to extremely hard, allowing progressive testing. Thomas Bloom’s curated database (erdosproblems.com) provides a publicly accessible, status‑tracked list of ~1 000 problems, making it a natural benchmark for AI research.

Bloom’s website and community

Bloom built erdosproblems.com in early 2023, writing the Python backend with ChatGPT. Adding a comment section created a community where participants from a customer‑service employee to Fields Medalists discuss progress.

Case study: Barreto, Price, and the “Aristotle” workflow

December 2025 Kevin Barreto and Liam Price announced an AI‑generated solution to a problem later found to have been solved by Erdős in 1977; they retracted the claim.

January 4 2026 they used GPT‑5.2 Pro together with an auxiliary tool named Aristotle to solve Erdős 728, a genuinely open problem.

Price’s methodology: generate a candidate proof with one AI instance, feed the proof to a fresh AI instance for verification, repeat the generate‑verify loop until a plausible proof emerges. This manual “harness” process mirrors internal company pipelines.

The resulting proof was incorporated into a joint paper with Terence Tao, Jared Duker Lichtman and others published May 2026.

DeepMind systematic evaluations

A 24‑person DeepMind team used the Gemini system to evaluate 700 “open” conjectures from Bloom’s database, solving four and uncovering nine forgotten solutions.

A second 21‑person DeepMind team later tackled 353 problems, solving nine at a cost of a few hundred dollars per problem.

Unit‑distance problem

The unit‑distance problem asks: for n points in the plane, how many pairs can be at distance exactly 1? For ~80 years the optimal construction was believed to be a square grid. OpenAI’s model produced a new family of constructions that yields more unit‑distance pairs, using tools from algebraic number theory. Tim Gowers described the proof as comparable to the best human work.

Community reactions and shifts

Mathematicians such as Noga Alon, Terence Tao and van Doorn noted that AI now solves many Erdős problems, leading some to leave academia for industry. Jacob Tsimerman, a 2026 Fields Medalist, joined OpenAI, citing both financial incentives and the view that AI represents the next frontier of mathematical discovery.

Bloom warned that non‑mathematicians can generate long (100–200 page) AI‑written papers without human verification, creating a flood of unreviewed results.

Reference: https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/