### A New Era in Mathematics: How AI Tackled a Famous Erdős-Type Problem

For decades, the world of pure mathematics has been the exclusive domain of human intellect. The problems, especially those posed by legendary figures like Paul Erdős, are mountains of abstract logic that require not just computation, but profound intuition and creativity. These “Erdős problems,” scattered across combinatorics, graph theory, and number theory, have served as benchmarks for human ingenuity. But a new mind has just entered the field, and it’s not human.

Recently, a groundbreaking development has shown that a specialized AI can venture into this rarified air and emerge with novel solutions that have eluded mathematicians for years. While the specific “Erdos problem #728” might be a misnomer, the achievement it points to is very real and centers on a problem of the exact flavor Erdős would have cherished: the cap set problem.

The cap set problem is a notoriously difficult puzzle in combinatorics. In simple terms, it asks: what is the largest possible group of points you can pick in a high-dimensional grid such that no three points in your group form a straight line? As the number of dimensions increases, the problem’s complexity explodes, making it virtually impossible for humans to find the optimal solution through intuition alone.

This is where Google DeepMind’s AI system, named FunSearch, made its mark. FunSearch represents a completely new approach to using AI for mathematical discovery. It isn’t a “black box” that spits out an answer nobody can understand. Instead, it’s a system designed to work alongside human researchers, generating solutions that are both verifiable and, crucially, insightful.

FunSearch operates by pairing a Large Language Model (LLM), trained to be creative with code, with an automated evaluator. The LLM suggests potential solutions in the form of small computer programs. The evaluator then runs these programs, scores their effectiveness, and feeds the best-performing ones back into the LLM. Through this evolutionary process of generation and evaluation, the system iteratively discovers better and more effective strategies. It essentially brainstorms millions of ideas, rapidly discards the bad ones, and refines the good ones until it uncovers something genuinely new.

In the case of the cap set problem, FunSearch discovered a new construction for creating large cap sets that went beyond the best-known human solutions. It produced a verifiable method, written in computer code, that outlined a more effective way to pack points into a high-dimensional space without any three of them aligning. This wasn’t just finding a bigger number; it was the discovery of a new technique, a new piece of mathematical knowledge.

The significance of this cannot be overstated. Previous AI systems have been excellent at verifying formal proofs or finding patterns in massive datasets. FunSearch, however, is one of the first to generate creative, novel, and human-readable contributions to pure mathematics. It didn’t just solve a problem; it revealed a path to the solution that mathematicians can now study, understand, and build upon.

Beyond the cap set problem, FunSearch also broke new ground in a computer science challenge known as the bin-packing problem, demonstrating its versatility. The true breakthrough is not the solution to any single problem, but the methodology itself. We are witnessing the emergence of a new tool for mathematical exploration—an AI collaborator that can guide human intuition and explore avenues of thought too vast or counter-intuitive for the human mind to tackle alone. The era of purely human mathematical discovery may be giving way to a new partnership, accelerating our journey into the deepest abstract mysteries of the universe.