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

A worked example of an agent plus exact verifier plus distributed search producing new mathematical constructions, a reusable pattern for AI-driven discovery.

• Lehmer's 1933 problem asks whether an integer polynomial's Mahler measure, how far its roots stray from the unit circle, can be made arbitrarily small. Systematic constructions of irreducible small-measure polynomials had covered only even degrees up to 180.
• At higher degrees, Boyd's one-parameter family often factors, skipping nearly 50% of target degrees. The authors' five-parameter Generalized Boyd algorithm removes the skipping and stays strictly below Boyd's limit point of about 1.255434 for every target degree.
• The loop extended conversational prompting with a fast numerical pipeline, an exact symbolic verifier and distributed search. The agent prototyped solvers and launched parallel cluster searches, and worked with the human researcher to extract and verify hypotheses.
• Across all even degrees up to 2000, the approach improves six previously published values and holds 99.6% of best known records. The authors release the verified dataset of minimal-measure polynomials.

• AI agent — An AI system that doesn't just answer once but works toward a goal in a loop — taking actions, reading the results, and deciding what to do next.

• articleThe Problem Is the Problem: Towards Scalable Mathematical DiscoveryZeyu Zheng, Shengtong Zhang, Jeremy Avigad, Prasad Tetali, Sean Welleck
• articleAgentic Data Engineering for LLMs: System Design and a Controlled Empirical StudyRuomeng Ding, Qianli Shen, Zhaoyang Han, Meixin Chen, Daoyuan Chen, Yaliang Li
• articleInduction and Inquiry via Probabilistic Reasoning over Language and CodeWasu Top Piriyakulkij, Sam Acquaviva, Cassidy Langenfeld, Joshua Tenenbaum, Kevin Ellis

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