GPT-5 and the future of mathematical discovery

The problem and why it mattered

Every significant math problem has a story—someone who posed a question, someone who tried to solve it, someone who could not, and eventually, maybe, someone who could. The story behind answering one frustratingly simple optimization theory question⁠(opens in a new window) is no different, except the researcher worked with a tool capable of quickly surfacing ideas and techniques from across a wide range of mathematical papers.

With 15 years in applied mathematics and optimization theory, Professor Ernest Ryu of the University of California, Los Angeles (UCLA), was curious about the large language model (LLM) everyone was talking about. In 2023, he decided to test ChatGPT‑3.5’s ability to solve simple math and logic problems, like scheduling meetings with multiple people across time zones. He noticed it would understand implicit constraints (like no one wanting to have a meeting between 12 a.m.–6 a.m.) but its ability to produce accurate results widely varied. It had many strengths, but in his opinion, it still had a long way to go.

When OpenAI unveiled GPT‑5 two years later, Ryu began hearing about its rapidly advancing capabilities in mathematics. He decided to try again now that the model had matured to see if it could handle a more complex problem. He did not expect that this would meaningfully contribute to solving a longstanding question in his field.

Ryu decided to tackle an “open” problem, meaning it was unsolved and recognized in the community as something of interest. His mathematical intuition told him it may admit a simple solution; a human just wasn’t able to find it yet.

The question: when an algorithm uses a phenomenon called the Nesterov Accelerated Gradient, or NAG, it becomes dramatically faster—but does the extra momentum added from NAG not affect the algorithm’s stability?

Exploring the problem with GPT‑5

“GPT‑5 was a very unusual collaborator,” he said, “in that it would propose something completely out of the blue.”