Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop

Yuri Chervonyi, Sergei Gukov, Thang Luong

While artificial intelligence has made rapid progress in mathematical reasoning, many open problems depend on structure that is revealed only through large-scale computational exploration and analytical synthesis. One such example is Lehmer's problem from 1933. It asks whether the Mahler measure of an integer polynomial, which measures how far its roots stray from the unit circle, can be made arbitrarily small. Previously, constructing irreducible polynomials with integer coefficients and small Mahler measure has been carried out systematically only for even degrees $d \le 180$. At higher degrees, Boyd's analytical one-parameter family frequently factors, skipping nearly 50% of target degrees and leaving no general construction for arbitrary $d$. Here, we address the high-degree regime through a human-AI discovery loop, in which we extend the conversational prompting with a fast numerical pipeline, an exact symbolic verifier, and large-scale distributed search. The AI agent prototypes numerical and symbolic solvers locally, launches parallel searches across distributed clusters and works with the human researcher to extract and verify mathematical hypotheses. Through this…