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

Yuri Chervonyi, Sergei Gukov, Thang Luong

The authors report irreducible integer polynomials of small Mahler measure at every even degree from 10 to 2000. Their five-parameter Generalized Boyd family attains or ties 991 of the 995 benchmark records at degrees 12–2000 and gives a value below Boyd’s limit point, about 1.255434, at each tested degree. The search improves six previously published records.

Sources for this record

What was already known

Systematic low-Mahler-measure searches covered even degrees through 180, with some published constructions through 250. Boyd’s one-parameter family can lose target degrees when cyclotomic factors are removed. The new search and five-parameter family fill the tested high-degree range.

AI involvement AI-centered

Internal Gemini models, including Gemini 4 Argon, co-designed search methods, generated and searched polynomial candidates, built numerical and symbolic verification pipelines, and helped extract the five-parameter family from computed optima. Chervonyi, Gukov and Luong supplied mathematical objectives and priors, steered the iterative search and developed the analytical interpretation. The retained contribution is the jointly discovered polynomial constructions, not autonomous resolution of Lehmer’s problem.

Importance Notable

Extends a concrete construction benchmark to degree 2000 and improves six published values. The authors explain how cyclotomic factorization compresses the search results into a short family.

Limitations

Lehmer’s conjecture remains open. Best known benchmark records are not proven global minima. The retained numerical coverage is the tested finite range; exact cyclotomic degree consistency for every even degree is separate from universal irreducibility or Mahler-measure guarantees. Numerical root calculations have no documented rigorous error enclosure in the reviewed protocol.

Technical verification Partial

The paper reports Mathematica irreducibility tests over the rationals, numerical root-based Mahler measures and exact rational Sturm root counts. These support the finite constructions and tested root structure, but do not certify the numerical Mahler inequalities or global optimality over all integer polynomials. No complete independent audit of the dataset or all-degree irreducibility claim was established.

Human understanding Full

The named human authors explain the search-to-formula synthesis, cyclotomic degree bookkeeping and root-distribution mechanism in Sections 2–6 and the appendices. This supports understanding of the retained finite constructions without proving every universal extension.

Correction and publication history