An OpenAI Model Has Made A New Breakthrough In An 80-Year-Old Open Erdős Problem

OpenAI announced that one of its internal reasoning models solved the planar unit distance problem, an open question in combinatorial geometry first posed by Hungarian mathematician Paul Erdős in 1946. The breakthrough disproves Erdős’s long-standing conjecture that the maximum number of unit-distance pairs among *n* points in a plane could not exceed roughly *n*^(1+C/log log n). Instead, the model proved that for infinitely many values of *n*, arrangements can yield at least *n*^(1+δ) pairs, where δ is a fixed positive exponent—later refined to δ = 0.014 by Princeton’s Will Sawin. The proof’s significance lies in its method: it repurposed tools from algebraic number theory, such as infinite class field towers and Golod–Shafarevich theory, to solve a problem rooted in Euclidean geometry. Fields Medalist Tim Gowers called the result a milestone, stating he would have recommended acceptance for publication in *Annals of Mathematics* without hesitation. The solution was independently verified by external mathematicians, reinforcing its validity. This breakthrough is part of a broader trend of AI models solving long-standing mathematical problems. OpenAI’s systems have previously earned gold medals at the International Mathematical Olympiad, while other AI tools, like Harmonic (backed by Robinhood CEO Vlad Tenev), claimed solutions to other Erdős problems. Even Donald Knuth, a skeptic of AI, was surprised when Claude solved an open problem he had worked on for weeks. The field has also faced scrutiny, with Google DeepMind’s Demis Hassabis criticizing overstated claims in AI research. Despite skepticism, the latest achievement underscores AI’s growing capability in advancing mathematical discovery, blending unexpected theoretical connections with computational power.