OpenAI Model Autonomously Solves 80-Year Math Problem

OpenAI announced that an internal general-purpose reasoning model autonomously solved the planar unit distance problem, a question posed by Paul Erdős in 1946. External mathematicians reviewed and verified the proof.

The planar unit distance problem asks how many pairs of points at distance one can occur among points placed in the plane. The question has no definitive solution since Erdős posed it in 1946. OpenAI reported the model reached the result without step-by-step human guidance and used methods from algebraic number theory.

The company described the result as an example of an advanced system holding a long formal argument, combining methods from different areas of mathematics, and producing work that withstands expert review. Reviewers confirmed the proof used sophisticated techniques from algebraic number theory.

OpenAI presented the milestone as part of efforts to automate research. The lab said systems with similar capabilities could one day assist work in biology, physics, materials science and medicine. The announcement also noted that human judgment remains central, with researchers choosing which problems to pursue and interpreting results.

The news arrives as OpenAI prepares for a possible initial public offering and after a U.S. jury recently cleared the company in litigation filed by Elon Musk. Rival firms are advancing: Anthropic projects a profitable quarter with estimated revenue of $10.9 billion, and Andrej Karpathy has joined Anthropic to work on frontier model research. Finance executives report early workplace effects, with Ken Griffin, chief executive of Citadel, warning that agentic AI is replacing tasks that once took PhD-level teams months to complete within hours.

Mathematicians who examined the work said the proof applied advanced algebraic number theory and met standards for verification. OpenAI and the outside reviewers identified the case as a rare instance in which a model produced a resolution to a longstanding mathematical question that specialists accepted.

The development raises questions for universities, research labs and companies about how to manage work that can be partly or fully automated and how to combine model output with human oversight and domain expertise.