Small Mahler measures of integer polynomials by degree
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Lehmer’s problem asks whether Mahler measures of integer polynomials can approach 1 from above. A concrete search target is the smallest known measure of an irreducible integer polynomial at each degree.
Why it matters and what counts as progress
Why it matters: Degree-specific constructions supply benchmarks and structured families even when they do not settle the universal gap near 1.
What counts as progress: A rigorously verified smaller measure at a specified degree, a new parametric family, or a theorem about the gap above 1; finite searches do not resolve Lehmer’s problem.
How the problem developed
- 1933 · Open problem [1]
D. H. Lehmer Asked whether integer polynomials can have Mahler measures arbitrarily close to 1 from above, now known as Lehmer's problem.
- July 2026 · Data resource [2]
Michael Mossinghoff The public Mahler Measure Database recorded best-known small measures by even degree through degree 200.
- 21 Oct 2026 · Construction [3]
Yuri Chervonyi, Sergei Gukov and Thang Luong Reported a five-parameter generalized Boyd algorithm producing irreducible integer polynomials below Boyd's limit for every target degree, updating six records and 99.6% of best-known even-degree values through 2000.
AI contribution
Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop
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
Website timeline
Publication dates and editorial changes.
Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop