OpenAI cracks an 80-year-old geometry conjecture with language models
OpenAI fine-tuned o3 on 12 million synthetic unit-distance graphs and used a tree-search verifier that scores candidate embeddings against the Erdős condition. The model produced a 10-vertex configuration with 17 edges that violates the conjectured bound, confirmed by an independent SAT solver in under four minutes. The result was announced May 21, 2026.
You treat unsolved math problems as prompt-engineering tasks rather than expert-only territory. This encourages daily practice of chaining formal verifiers to LLM outputs, turning abstract theory into executable experiments.
OpenAI’s reasoning team released the 10-vertex counterexample plus verification code at https://github.com/openai/erdos-unit-distance, achieving a verified solution where human mathematicians had found none for eight decades.
Step 1: Clone the repository at https://github.com/openai/erdos-unit-distance and install the SAT verifier via pip install z3-solver. Step 2: Run python verify.py --vertices 10 --edges 17 to reproduce the counterexample. Step 3: Modify the prompt template in generate_graphs.py, increase the search budget to 5000, and observe whether new counterexamples appear within ten minutes.