Human–AI search reports new high-degree constructions in Lehmer’s problem
A new preprint uses AI-assisted numerical search and exact symbolic checks to construct low-Mahler-measure polynomials across a broad range of even degrees.
Using a search loop to probe a long-running number-theory problem
A preprint submitted on October 21 describes a human–AI workflow for finding integer polynomials with unusually small Mahler measure, a quantity determined by how far a polynomial’s roots lie from the unit circle. The target is part of the landscape around Lehmer’s problem, which asks whether non-cyclotomic integer polynomials can have Mahler measures arbitrarily close to 1. The paper does not claim to settle that problem.
From numerical candidates to an exact construction
The authors combine conversational AI assistance with numerical and symbolic programs, distributed searches and exact verification of candidate polynomials. Their search led them to a five-parameter family they call the Generalized Boyd algorithm. Unlike an earlier one-parameter construction that can factor at many higher even degrees, the proposed family is intended to produce irreducible examples at every even target degree.
The authors report tests through degree 2,000. They say the family produces a polynomial with Mahler measure below Boyd’s stated limit point at every tested even degree, improves six previously published values and matches or holds nearly all of the best-known records in their dataset. Those are the authors’ reports for their construction, calculations and comparison set; MathsAI has not independently rerun the searches or checked the proofs.
Discovery support is not a proof of an open conjecture
The paper is a useful example of where AI can contribute in mathematical research: proposing computational experiments, helping connect patterns to a symbolic family and accelerating the search for examples that humans can analyse. But an extensive table of constructions is different from a proof or disproof of Lehmer’s problem itself. This remains an unreviewed preprint, and the significance of its record claims depends on independent inspection of the exact verification, prior literature and subsequent mathematical review.