This decades-old math problem has been solved by Chinese doctor using ChatGPT
OpenAI's latest flagship model helped a Chinese doctor accidentally solve a math problem that had stumped the world's brightest minds for decades, as the proof for Crouzeix's conjecture has finally been found.
It was certainly a bold statement when OpenAI leadership revealed how the new flagship ChatGPT models were capable of solving some of the most notorious math problems, showing quite how much of a leap in capability had been achieved.
Few expected that performance to be mirrored in the real world so soon, however, yet Beijing-based neurosurgeon Jin Shanmu made history by cracking the code on a problem that has remained unsolved since 2004 despite the best efforts from leading mathematicians.
As reported by the South China Morning Post, Jin is a self-taught mathematician outside of his work as a scientist, and had consulted the flagship model in an attempt to solve an issue related to brain ultrasounds.
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After 16 straight hours of continuous autonomous work, however, GPT-5.6-Sol managed to find proof for Crouzeix's conjecture, ending decades of attempts without even trying to do so in the first place.
Proposed by Michel Crouzeix, the conjecture named after him proposes that applying a function to a matrix – a grid of numbers with rows and columns – never makes it grow larger than twice its expected regional limit.
Part of the process involved Jin's conversations with Anne Greenbaum and Alex Townsend, professor and associate professor at the University of Washington and Cornell University, respectively.
They discovered a paper published by Jin where he had used the model to prove the conjecture – something Townsend himself had been attempting for over a year now using various GPT models to no success – and the three experts, alongside Crouzeix himself, worked together to confirm the discovery.
"Crouzeix's conjecture had been open for more than two decades, so we initially read the preprint with some skepticism," Townsend explained after the fact, "but after a few hours we realized that the argument was the real deal."
Townsend added that "Jin explained to us by email that his route into matrix analysis began with his research on transcranial ultrasound. He encountered Crouzeix's conjecture while teaching himself the subject and was drawn to its unusually simple statement and to the visual geometry of the numerical range."
One surprising factor is that Jin has reportedly "been strikingly modest about the discovery," with the Chinese neurosurgeon emphasizing "that there was 'certainly an element of luck' in finding the key idea."
Whether luck plays a part is irrelevant to how significant this is as a discovery, as Jin has not only made history in a field outside of his primary research, but has also proved the potential of ChatGPT and other leading AI models when it comes to solving these types of conjectures.