Human-AI Collaboration Proves Extremal Chowla Set Conjecture

A team of mathematicians and AI researchers used Google's Co-Scientist system to investigate extremal Chowla sets in finite groups. They defined a Chowla set as a nonempty subset S of a finite group G where every element has order greater than |S|, and denoted C(G) as the maximum cardinality of such a set. The team proved that C(G) is determined by the distribution of element orders. For cyclic groups, they derived an exact divisor formula and characterized integers n for which C(Z/nZ) equals Euler's totient φ(n). They showed that the liminf of C(Z/nZ)/φ(n) is 1, while the limsup is infinite, and computed corresponding limits under normalization by n. For finite abelian groups, they obtained an explicit formula using invariant-factor decomposition and a closed formula for abelian p-groups. The research demonstrates a novel human-AI collaborative approach in mathematics, with the AI system generating conjectures and proofs that were verified by human mathematicians.