An AI disproves a 1946 conjecture by Paul Erdős, and the mathematicians who checked it say they'd accept it 'for any journal'
An OpenAI model found a way to place points on a plane with more pairs at distance one than was thought possible. Mathematicians including Noga Alon and Tim Gowers checked the proof. It is the most striking result in a wave of AI solving problems that had been open for decades.
In 1946 the Hungarian mathematician Paul Erdős asked a question that is easy to state: if you place many points on a plane, how many pairs of them can be exactly the same distance apart? Erdős conjectured that a square grid was essentially the best you could do. For 80 years hardly anyone doubted it.
An internal reasoning model at OpenAI has found that he was wrong: an infinite family of configurations that beats that bound, built with advanced tools from number theory.
Checked by leading mathematicians
A group of outside mathematicians reviewed the proof and wrote a companion paper:
- Noga Alon calls it “an outstanding achievement, settling a long-standing open problem.”
- Tim Gowers, a Fields medallist, calls it “a milestone in AI mathematics.”
- Jacob Tsimerman says he would “accept it for any journal without hesitation.”
- Arul Shankar says the model is “capable of having original ingenious ideas.”
Not a one-off
According to Quanta Magazine, the famous Erdős problems, a list of hundreds of open questions, are falling one after another with AI’s help: teams at Google DeepMind and OpenAI, and hobbyists using public models, have solved more than a dozen, checked on the community’s forum. Large language models are “changing dramatically the way mathematical research is being done,” Alon says.
The caveats
Not every announcement holds up: some problems declared solved already had published solutions, and many of those resolved are minor. The Erdős proof has not yet appeared in a journal, and human mathematicians improved it within weeks. The most ambitious claim, a blowup in the Navier-Stokes equations (one of the Millennium Prize Problems), is still under review, with disputes over its verification and credit.
