Last week, OpenAI revealed that one of its artificial intelligence models had done something mathematicians have been trying—and failing—to do for 80 years: it proved Paul Erdős wrong.
The Hungarian mathematician had posed what seemed like an elegant puzzle back in 1946. Imagine an infinite sheet of paper dotted with points. You can arrange them however you wish. The question: how many pairs of those points can you position at exactly one unit distance from each other? Erdős conjectured that the best arrangement would be something intuitive, something like a grid—the kind of regular, geometric pattern that feels obviously efficient. For decades, mathematicians believed his instinct was sound. They tried proof after proof to confirm it. But OpenAI's model found a counterexample, using tools from algebraic number theory to demonstrate that there exist patterns involving far more unit-distance pairs than any grid could produce. The improvements only emerge for incomprehensibly large numbers of points—the newest result, refined by US mathematician Will Sawin days after OpenAI's announcement, doesn't yield gains until you reach around 10 to the power of 2 million points—but they exist nonetheless.
This matters because it marks a turning point in how mathematics itself gets done. The breakthrough wasn't the work of a specialized AI designed for theorem-proving. It came from a general-purpose model, guided by a human prompt, that autonomously followed its reasoning to a genuine discovery. Fields Medallist Timothy Gowers, one of mathematics' highest honors, stated he would recommend publication to the prestigious journal Annals of Mathematics "without any hesitation." He also noted that no previous AI-generated proof had approached this level of sophistication.
What makes this moment remarkable isn't just that a machine solved a problem humans couldn't. It's that the machine did it with minimal human intervention beyond the initial prompt. The accompanying paper documents both the prompt itself and the "chain of thought" the model used—a transparent record of how it reasoned through the problem, step by step.
The planar unit distance problem, formally known as Erdős problem 90, had captivated mathematicians precisely because it seemed so simple on the surface yet resisted all attempts at resolution. Its difficulty revealed unexpected connections to incidence geometry, graph theory, and extremal combinatorics—areas of mathematics that seemed at first glance entirely separate from the original puzzle. That a machine could navigate these connections, synthesize insights from different mathematical traditions, and arrive at a genuinely novel conclusion suggests something fundamental about AI's role in the research process.
Within days of OpenAI's announcement, Google DeepMind released its own results, showing that AI had also resolved nine additional open problems left by Erdős. The momentum is accelerating. Yet researchers remain cautious about what these breakthroughs reveal. Current AI excels at two things mathematicians have always needed: holding vast reservoirs of existing knowledge and tirlessly exploring countless speculative paths, even dead ends, without human fatigue constraining the search. Where AI's limitations still lie—in generating the conceptual leaps that suddenly reorganize our understanding of a problem—remains an open question. The future of mathematics may well depend on learning which tasks AI augments best, and which still require the irreplaceable spark of human intuition.