AI cracks 80-year-old math problem on unit distances

For nearly 80 years, mathematicians have been stumped by a deceptively simple question posed by Paul Erdős in 1946: if you place points in a plane, how many pairs can be exactly one unit apart? Known as the planar unit distance problem, it’s one of combinatorial geometry’s most famous puzzles — easy to state, brutally hard to crack. The prevailing belief held that a ‘square grid’ arrangement was optimal for maximizing unit-distance pairs. But an OpenAI model has now shattered that assumption, producing an infinite family of examples that yield a polynomial improvement over the old record. The proof, verified by external mathematicians, uses unexpected tools from algebraic number theory to solve an elementary geometry question. What makes this truly historic is how it happened: a general-purpose AI reasoning model, not one trained specifically for math or the problem, autonomously produced the proof. Why it matters: This marks the first time an AI has independently solved a prominent open problem central to a subfield of mathematics, signaling a new era where AI can generate original ideas and carry them through to completion.