AI System Theo-Conjecture Solves 35-Year-Old Math Problem, Discovers New Term

AI Solves Long-Standing Mathematical Conjecture

Theo-Conjecture, an automated discovery system leveraging a large language model, has successfully addressed a mathematical problem that originated in the 1980s. This problem, initially posed by the program Graffiti, captured the attention of mathematician Paul Erdős. Randy Davila presented the problem to Theo-Conjecture, which then processed and analyzed it.

Proof and Unexpected Discovery

The AI system returned a proof for the answer that Erdős and his collaborators had previously conjectured. In addition to confirming the existing hypothesis, Theo-Conjecture also identified an unexpected extra term that had not been predicted by human mathematicians. This outcome highlights the potential for AI agents to contribute to mathematical discovery.

The Graph Problem Explained

The problem involves constructing a graph from integers 2 through 'n', where dots represent numbers and lines connect numbers sharing a factor greater than one. Prime numbers in this graph form the largest possible independent set, meaning no two primes share a factor. This relationship is expressed by the formula α(Gₙ) = π(n), where α(Gₙ) is the size of the largest independent set and π(n) is the prime-counting function.

Implications for AI in Mathematics

The successful resolution of this problem by Theo-Conjecture demonstrates a collaborative approach between AI and human mathematicians. The system's ability to propose, test, and revise mathematical ideas in a loop, combined with its discovery of an unpredicted term, suggests a growing role for AI in complex mathematical research and problem-solving.