OpenAI Model Achieves Major Breakthroughs in Mathematics
OpenAI recently announced two landmark achievements by its AI model in pure mathematics and optimization theory. The model not only disproved the 80-year-old “Planar Unit Distance Conjecture” but also solved the 40-year-old problem of point convergence for Nesterov’s Accelerated Gradient (NAG) method, marking a new level of capability for AI in automated scientific discovery.
Disproving the “Planar Unit Distance Conjecture”
The “Planar Unit Distance Conjecture,” proposed by renowned mathematician Paul Erdős in 1946, is one of the most famous and easily understood open problems in discrete geometry. The conjecture seeks to determine the maximum number of pairs of points with a distance of exactly 1 among n points in a plane. For a long time, mathematicians widely believed that point arrangements based on a square grid were optimal or near-optimal for maximizing the number of unit distance pairs.
However, according to OpenAI’s paper, “Planar Point Sets with Many Unit Distances,” its AI model discovered a new family of point set constructions. With this new arrangement, the number of unit distance pairs can polynomially exceed that of traditional grid-based solutions, thus providing a clear counterexample that disproves the 80-year-old conjecture.
Reportedly, the model’s approach to constructing the proof is highly novel, centered on building an infinite tower of totally real fields and incorporating advanced algebraic tools like Galois groups and the Golod-Shafarevich theorem. This breakthrough has been acknowledged by several top mathematicians. Fields Medalist Timothy Gowers stated he would “have no hesitation in recommending that this paper be published.” Additionally, nine mathematicians, including Noga Alon and Thomas Bloom, co-authored a companion paper to verify and explain the AI’s proof.
Proving the Point Convergence of NAG
In optimization theory, the AI also solved a 40-year-old open problem. In 1983, Yuriy Nesterov introduced Nesterov’s Accelerated Gradient (NAG) method, which improves the convergence rate of gradient descent from O(1/k) to O(1/k²) by adding a “momentum” term. It is widely used in machine learning and large-scale optimization.
Although its convergence rate was proven, the question of whether the sequence of iterates converges to a specific point or oscillates indefinitely around the optimum—the “point convergence” problem—remained unsolved. Professor Ernest Ryu and his student Uijeong Jang from the University of California, Los Angeles (UCLA), successfully proved that the iterates of NAG do converge to a specific point using an advanced model from OpenAI (referred to as ChatGPT in their paper). The proof is detailed in their paper, “Point Convergence of NAG.”
A New Paradigm of AI-Driven Automated Theorem Proving
Further research demonstrates the growing autonomy of AI in mathematical proofs. In a subsequent paper titled “Nesterov Flow May Travel Infinitely Long,” Ryu explored the trajectory length of the continuous-time version of NAG. The paper’s abstract explicitly acknowledges: “All proofs of this work are due entirely to an internal model at OpenAI.”
This indicates that AI is no longer just a tool for verifying human ideas or providing computational support. In these cases, the AI model can independently explore complex mathematical spaces and construct entirely new proof paths that human researchers had not conceived. This heralds the arrival of a new paradigm of human-AI collaboration, or even AI-led mathematical research, which promises to accelerate the resolution of major problems in other fundamental scientific fields.
