More progress in mathematics thanks to ML & AI: Lehmer's conjecture from 1933

Here is another example by Google researchers! How fast will ML & AI drive future progress regarding the queen of sciences? Very fast!

Notice this problem was apparently unresolved for almost 100 years. Either the problem was not very relevant or low priority or it was hard to solve.

From the abstract:

"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≤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 interactive loop, we derive a five-parameter construction, which we call the Generalized Boyd algorithm.

It eliminates degree-skipping and constructs irreducible polynomials with Mahler measure strictly below Boyd's limit point α1≈1.255434 for every target degree.

Evaluated across all even degrees up to 2000, our approach improves upon six previously published values and holds 99.6% of all best known records.

With this work we release the verified dataset of polynomials with minimal Mahler measure across all studied even degrees."

[2610.30277] Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop