OpenAI Group PBC has published 722 math papers that it generated using an unreleased artificial intelligence model.
Some of the papers, which were posted to GitHub late Tuesday, prove long-running hypotheses. Others disapprove proposed explanations of mathematical phenomena or narrow down the possible answers to complex, yet-unsolved puzzles. Overall, OpenAI’s papers span about 20 mathematical subfields.
One of the most notable discoveries relates to the Riemann hypothesis. It’s a more than 150-year-old paper about prime numbers, numbers that can only be divided by themselves or one. The Riemann hypothesis argues that prime numbers follow a pattern described by a mathematical object called the Riemann zeta function.
Proving the conjecture is considered to be one of the most important challenges in mathematics. Its significance partly stems from the fact that many later math papers are based on the assumption the Riemann hypothesis is correct. In other words, proving the Riemann hypothesis would verify those papers.
OpenAI’s unreleased AI model didn’t manage to come up with a complete answer. However, it did prove an important piece of the puzzle called the quasi-Riemann hypothesis.
Theoretical computer science was another major focus of OpenAI’s research exercise. The company’s unreleased AI model produced more than 80 papers about the topic.
Three of the papers focus on matrix multiplications, the mathematical operations that AI models use to process data. For the past few decades, researchers have been working to develop faster, more hardware-efficient ways to perform such calculations. It’s believed that matrix multiplications can only be sped up so much before a limit is reached.
OpenAI’s model developed a clearer definition of that limit. In a separate paper, it developed a new algorithm for multiplying integers. An integer is a data structure that contains a whole number. Like matrix multiplications, integer multiplications are foundational building blocks of many programs.
The research breakthrough is also relevant for physicists. The company published more than a dozen proofs that relate to partial differential equations, or PDEs. Those are physics equations that are essential to chip design, architecture, quantum mechanics and a range of other fields.
The model proved a version of De Giorgi’s conjecture, a hypothesis that relates to an equation used to study metal alloys. It also clarified a number of questions related to the Navier–Stokes equations, which engineers use to study the flow of liquids.
In September, the AI model that produced the papers solved a different problem related to the Navier–Stokes equations. That puzzle ranked as one of the most difficult open questions in mathematics.
Many of OpenAI’s new papers contain Lean files. Those are code snippets that make it possible to quickly verify a newly published proof using a computer. Going forward, OpenAI plans to release Lean proofs for more of the papers. The company also intends to finance of research events and programs that will focus on reviewing AI-generated math discoveries.