OpenAI Publishes 722 AI-Generated Math Papers, Including Result on Riemann Hypothesis

The papers were posted to GitHub late Tuesday, according to SiliconANGLE. Some of them prove long-standing hypotheses, while others reject proposed explanations of mathematical phenomena or eliminate some of the possible answers to complex open puzzles.

The most prominent result concerns the Riemann hypothesis, a conjecture more than 150 years old about prime numbers, which can be divided only by themselves or by one. The hypothesis holds that primes follow a pattern described by a mathematical object called the Riemann zeta function. Proving it is considered one of the central challenges in mathematics, in part because many later papers rest on the assumption that it is correct; a proof would verify those results. OpenAI's model did not produce a complete proof, but it proved an important part of the problem known as the quasi-Riemann hypothesis.

Theoretical computer science was another focus of the exercise, accounting for more than 80 papers. Three of them address matrix multiplication, the operation AI models use to process data and one that researchers have spent decades trying to perform faster and more efficiently in hardware, on the belief that such calculations can be sped up only so far before a limit is reached. The model gave a clearer definition of that limit, and in a separate paper it set out a new algorithm for multiplying integers, a data structure containing a whole number that, like matrix multiplication, underpins many programs.

More than a dozen proofs relate to partial differential equations, or PDEs, physics equations used in chip design, architecture, quantum mechanics and other fields. The model proved a version of De Giorgi's conjecture, a hypothesis tied to an equation used to study metal alloys, and clarified questions connected to the Navier-Stokes equations, which engineers use to study the flow of liquids. In September, the AI model behind the papers solved a different problem related to the Navier-Stokes equations that ranked among the most difficult open questions in mathematics.

Many of the new papers include Lean files, code snippets that make it possible to verify a newly published proof quickly with a computer. OpenAI plans to release Lean proofs for more of the papers, and it intends to finance research events and programs focused on reviewing AI-generated mathematical discoveries.