OpenAI general-purpose model solves 70-year-old Erdős unit distance conjecture

OpenAI's General-Purpose Model Disproves 70-Year-Old Erdős Conjecture

OpenAI has solved the planar unit distance problem, refuting a foundational conjecture in mathematics that Paul Erdős posed decades ago. The result marks a significant milestone for AI reasoning applied to pure mathematics—not as a tool assisting human mathematicians, but as an independent solver of novel problems.

The problem itself asks: given n distinct points on a 2D plane, what is the maximum number of pairs that can be exactly one unit apart? Erdős conjectured that this maximum scales as n to the power of (1 + o(1/log log n)), where o(1/log log n) approaches zero as n grows infinitely large. OpenAI's model discovered that this conjecture is false. For infinitely many values of n, the actual maximum exceeds Erdős's predicted bound, scaling instead as n to the power of (1 + some constant).

The solution is not a brute-force application of existing mathematical techniques. OpenAI's model generated an 18-page proof with novel ideas that mathematicians believe could be useful for tackling other open problems. The inference cost was modest—estimated at hundreds to thousands of dollars, not millions—and did not require exhaustive computational spend to find a breakthrough.

Why this matters beyond academia. The result demonstrates that current general-purpose models can generate mathematical reasoning that lies outside their training distribution. They are not merely retrieving or remixing existing knowledge; they are producing ideas that constitute genuine mathematical discovery. Terence Tao, one of the world's leading mathematicians, called the result "incredible." The Erdős conjecture represents the type of long-standing, decade-old problem that AI researchers have long flagged as a test case for machine reasoning. In that frame, this is comparable to AlphaFold's solution to protein folding—a real step change in what these systems can do.

There are roughly 1,200 Erdős problems in total, with around 500 to 600 solved by humans so far. OpenAI's solve of problem 90 came with a $500 prize, a detail the speakers noted with some humor as "non-dilutive financing"—though the inference bill almost certainly exceeded the payout.