The most exciting claims from OpenAI’s heap of new proofs
Scientists have called out several math and computer science results as the most significant in the company’s overwhelming new deluge of achievements
OpenAI's recent release of over 700 manuscripts containing proofs, solutions, and progress on 372 open problems across various fields of mathematics, theoretical computer science, and physics has generated significant excitement among scientists. While many of these proofs require further verification, experts highlight several key achievements.
One of the most notable results is progress on the Riemann hypothesis, a longstanding unsolved problem in mathematics. The Riemann hypothesis concerns the zeros of the Riemann zeta function, which are intimately connected to the distribution of prime numbers. OpenAI's model has produced two important findings: a proof of the "quasi-Riemann hypothesis," which offers substantial insights into the distribution of prime numbers, and evidence that none of the generalizations of the zeta function possess a "Siegel zero." The latter finding effectively settles the question.
Another groundbreaking result is the resolution of Hilbert's 10th problem. This problem asks whether there exists an algorithm capable of determining whether any given equation has whole-number solutions. Yuri Matiyasevich's work in 1970 proved that such an algorithm does not exist. However, mathematicians have since explored extending the result to equations with a broader range of values.
OpenAI's model has demonstrated that even when variables are allowed to be rational numbers, the set of equations with whole-number solutions remains unsortable.
OpenAI's work also sheds light on the Kakeya conjecture, which involves sliding a piece of chalk around on a surface to cover the smallest possible area while pointing in every direction. In three dimensions, the problem was resolved by Hong Wang and Joshua Zahl this summer. Now, OpenAI's model has extended the solution to four dimensions.
Other important results include proofs related to Artin's conjecture on primitive roots, which involves special number systems with a highest number, and progress on Paul Erdős' famous conjecture, as well as contributions to the Langlands program, a vast set of conjectures often referred to as the "Grand Unified Theory of Mathematics."
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