News that large language models (LLM) have made major advances in solving Erdős problems – a set of problems formulated by the renowned 20th-century mathematician Paul Erdős – has created an amalgamation of uproar and interest amongst mathematicians. The past month alone has seen two significant LLM-generated solutions. The first relates to prime sets, a generalization of prime numbers, and was solved after Liam Price, an amateur mathematician from the US, fed the problem statement into GPT-5.4 Pro without other information. The second came last week when the company behind ChatGPT, OpenAI, announced that it had used artificial intelligence to disprove Erdős’ planar unit distance conjecture.
LLMs have solved Erdős problems before, but the one Price chose wasn’t just any Erdős problem. It was one that human mathematicians had worked on for 60 years without success. The nature of the solution was also unusual. While previous LLM solutions to Erdős problems used standard techniques, this one took an entirely different approach. Rather than starting from Erdős’ original probability-theory-based framing of the problem, as human mathematicians had, the LLM found an alternative route – one that led naturally, in less than a page, to a correct proof.
“Paul Erdős had a concept of ‘Proofs from The Book’, meaning that the argument is so compact and elegant that this is the proof God would’ve written down in ‘The Book,’” Jared Lichtman, a mathematician at Stanford University in the US, wrote on the social media site X after the proof was announced. “After reading the GPT5.4 proof of Erdős #1196, I would say this is a Book Proof of the result.”\
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