Paul Erdős was one of the most prolific and beloved mathematicians of the twentieth century, famous for posing deceptively simple questions that resisted solution for decades.
Last week, one of his most enduring conjectures — open since 1946 — was overturned. Not by a human mathematician, but by an AI model from OpenAI.
The mathematical community’s reaction ranged from astonishment to genuine excitement. One Fields Medalist — the highest honour in mathematics — said he would have recommended the resulting paper for publication in one of the world’s most prestigious journals without hesitation.
No AI-generated proof, he added, had previously come anywhere close to this level of sophistication.
The Problem Itself
The question at the heart of this breakthrough sounds almost playful. If you scatter a number of points across an infinite plane and arrange them however you like, what is the maximum number of pairs of points you can place at exactly one unit of distance from each other?
An intuitive approach leads most people to a square grid — the regular spacing naturally creates many unit-distance pairs. For eighty years, mathematicians widely believed that grid-like arrangements were essentially optimal, and Erdős himself conjectured that no construction could meaningfully improve on them.
He was wrong. OpenAI’s model, using tools from an area of mathematics called algebraic number theory, found an arrangement that produces dramatically more unit-distance pairs than any grid — and does so for infinitely many values of the number of points. The improvement only kicks in for astronomically large numbers of points, but mathematically it is decisive. The conjecture is disproved.
Why This Matters Beyond the Result
The achievement is striking not just for what was solved but for how. This is the first significant open mathematical problem to be resolved by AI with minimal human input beyond the initial prompt. The model’s chain of reasoning has been published in full, showing exactly how it worked through the problem.
What appears to have made the difference is something AI is exceptionally well suited for: encyclopedic knowledge combined with tireless exploration. The key ideas needed for the proof were already scattered across the mathematical literature. A human expert, given the right hints, could likely have assembled them — but would first have needed to notice the connection between fields that don’t often speak to each other. The AI rifled through that literature without fatigue, following speculative lines of reasoning that a human researcher might have dismissed as unlikely dead ends, until the pieces clicked into place.
Within days of OpenAI’s announcement, a human mathematician following the same line of reasoning pushed the result even further. And separately, a Google DeepMind team used their own model to resolve nine lesser open problems left by Erdős. The pace of follow-on work suggests the breakthrough has opened new territory, not just closed an old question.
The Harder Question
All of this raises an uncomfortable and genuinely open question: what kind of mathematics can AI actually do?
The consensus emerging from mathematicians who have studied the result closely is nuanced. AI excels at two of the three things that drive mathematical research — deep knowledge of existing work, and the stamina to follow enormous numbers of speculative paths without human time constraints. These are formidable capabilities, and clearly sufficient to crack problems that have resisted human effort for generations.
What remains uncertain is the third ingredient: genuine conceptual leaps. The lightbulb moments that don’t just connect existing ideas but reframe a problem entirely — seeing it in a way nobody has seen it before. These are the breakthroughs that have historically defined the frontier of mathematics, and it’s not yet clear whether current AI models can produce them, or whether they are still exclusively human territory.
A Shift That’s Already Happening
Whether or not AI can make true conceptual leaps, the Erdős result marks something real and significant. For centuries, mathematical progress depended almost entirely on human creativity working alone. That is no longer quite true. Researchers now have access to systems that can autonomously explore vast intellectual spaces and contribute meaningfully to problems once thought to require purely human insight.
Mathematics is being done differently. The question of what that ultimately means for the discipline — and for our understanding of what intelligence actually is — is one that mathematicians, and the rest of us, are only beginning to grapple with.