AI

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

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Based on a feed excerpt; full article content has not been acquired. Source evidence retrieved 23/10/2026, 04:03:24 UTC.

Briefings

A human-AI discovery loop successfully constructed irreducible polynomials with Mahler measure below 1.255434 for all even degrees up to 2000, surpassing previous limits. The Generalized Boyd algorithm eliminates degree-skipping issues, improving upon six prior values and holding 99.6% of best known records. The work extends Lehmer's problem research by enabling systematic construction for high degrees previously inaccessible to analytical methods.

Key points

• The Generalized Boyd algorithm constructs irreducible polynomials with Mahler measure strictly below 1.255434 for every target even degree up to 2000.
• The approach eliminates degree-skipping and improves upon six previously published values, holding 99.6% of all best known records.

Context

Lehmer's problem asks if Mahler measure can be arbitrarily small for integer polynomials. Prior work limited constructions to even degrees d ≤ 180, where Boyd's analytical family frequently factors. This breakthrough enables systematic construction for high degrees using a human-AI discovery loop.

Limitations

Evidence is preprint arXiv:2610.30277v1, not peer-reviewed. Results are evaluated only across even degrees up to 2000. No data on odd degrees or degrees beyond 2000 is provided.

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