A $30$ run of an evolutionary algorithm guided by an LLM discovered the exact values of three unsolved mathematical constants in combinatorics.
Zarankiewicz numbers are notoriously difficult to calculate and have stumped mathematicians for decades. This project used an LLM to guide an evolutionary search to find 3 new exact values and 41 new lower bounds. The entire process cost less than a nice dinner, proving that AI can be a low-cost engine for pure math discovery. It moves LLMs past simple coding assistants into the realm of hypothesis generators for hard science. Professional mathematicians can now use these models to explore massive search spaces that were previously too expensive to probe.
New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search
arXiv · 2605.01120
The Zarankiewicz number $\textbf{Z}(m, n, s, t)$ is the maximum number of edges in a bipartite graph $G_{m, n}$ such that there is no complete $K_{s, t}$ bipartite subgraph. We determine for the first time the exact values of three Zarankiewicz numbers: $\textbf{Z}(11, 21, 3, 3)=116$, $\textbf{Z}(11, 22, 3, 3)=121$, and $\textbf{Z}(12, 22, 3, 3)=132$. We further establish lower bounds for 41 more Zarankiewicz numbers, including several that are within one edge of the best known upper bound, and