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80-year-old geometry puzzle cracked by OpenAI using number theory | Aamir Khollam, Interesting Engineering A longstanding Paul Erdős conjecture has fallen after OpenAI connected the problem with deep algebraic number theory. For nearly 80 years, mathematicians believed they understood the limits of a famous geometry puzzle first posed by legendary Hungarian mathematician Paul Erdős. Now, an AI model developed by OpenAI has overturned that assumption and solved one of the field’s most stubborn open problems. The breakthrough centers on the “unit distance problem,” a deceptively simple question that asks how many pairs of points can sit exactly one unit apart on a flat plane. Despite its simplicity, the problem has challenged mathematicians since 1946 and became one of the best-known questions in combinatorial geometry. Decades-old puzzle Imagine placing dots on a sheet of paper. The challenge is to arrange those dots so that as many pairs as possible sit exactly one unit apart. For decades, mathematicians believed square-grid patterns offered the best possible solution. Erdős himself proposed that the number of unit-distance pairs could only grow slightly faster than linearly as more points were added. Researchers spent generations trying to prove or disprove that theory. The new AI-generated proof changes that picture entirely. According to OpenAI, the model discovered an infinite family of point arrangements that produce significantly more unit-distance pairs than the classic square-grid approach. Princeton mathematician Will Sawin later refined the result and showed the improvement could be expressed with a fixed exponent. What surprised researchers most was the method behind the proof. Instead of relying on traditional geometry tricks, the AI connected the problem to algebraic number theory, a deep branch of mathematics that studies number systems extending ordinary integers. The proof used advanced concepts such as infinite class field towers and Golod-Shafarevich theory, tools rarely associated with geometric puzzles. In simple terms, the AI found a way to use hidden symmetries inside exotic number systems to create many more one-unit distances between points. That connection stunned experts. Mathematicians take notice The proof underwent external review by mathematicians who also produced a companion paper explaining the argument and its broader importance. Fields Medal winner Tim Gowers called the achievement “a milestone in AI mathematics.” Number theorist Arul Shankar said the work shows AI systems can move beyond assisting mathematicians and begin generating genuinely original ideas. Researchers also noted that the result may influence other geometry problems long thought unrelated to number theory. Thomas Bloom, one of the mathematicians involved in the companion work, said the discovery suggests deep number theory may hold answers to several unsolved questions in discrete geometry. He added that many mathematicians will likely revisit older problems using these newly revealed connections. The result also highlights how rapidly AI reasoning systems are evolving. Unlike specialized theorem-proving software, OpenAI said this proof came from a general-purpose reasoning model. Engineers did not specifically train it on the unit distance problem or build dedicated search tools for this task. That detail matters because it hints at broader scientific applications. Researchers believe systems capable of managing long chains of reasoning could eventually assist in fields such as physics, biology, engineering, and medicine. For now, the unit distance breakthrough stands as a landmark moment. A problem that resisted human effort for nearly eight decades fell to an AI system that approached geometry from an entirely unexpected direction. interestingengineering.com/a…