Lehmer's problem: AI breaks high-degree barrier for even degrees up to 2000

TL;DR: A new Generalized Boyd algorithm achieves Mahler measures below 1.255434 for all even degrees up to 2000, shattering the previous degree 180 barrier. This verified dataset establishes a new benchmark in number theory by demonstrating how human-AI loops can solve long-standing mathematical problems.

Key facts

• A new study submitted to arXiv on October 21, 2026, breaks the high-degree barrier for Lehmer’s problem by constructing irreducible polynomials up to even degree 2000.
• The team developed the Generalized Boyd algorithm, a five-parameter construction that systematically generates polynomials with Mahler measures below Boyd’s limit point of approximately 1.255434.
• This achievement surpasses the prior limit of degree 180, where previous methods like Boyd’s one-parameter family frequently failed to produce irreducible polynomials for higher even degrees.
• The breakthrough utilized a human-AI discovery loop combining conversational prompting, numerical pipelines, symbolic verification, and distributed search to identify mathematical patterns.
• The method improves upon six previously published Mahler measure records and holds 99.6% of all best-known values for even degrees up to 2000.
• Researchers have released a verified dataset containing irreducible polynomials with minimal Mahler measures, serving as a benchmark for future work in number theory.

Breaking the High-Degree Barrier in Lehmer’s Problem

Lehmer’s problem, posed in 1933, asks whether the Mahler measure of an integer polynomial—the extent to which its roots deviate from the unit circle—can be made arbitrarily small [1][2]. For decades, constructing such polynomials at high degrees remained a major challenge. Prior efforts were limited to even degrees up to 180, with Boyd’s one-parameter family frequently failing to produce irreducible polynomials at higher degrees, skipping nearly half of the target even degrees due to factoring [1][2].

A new study submitted to arXiv on October 21, 2026, introduces a human-AI discovery loop that overcomes this barrier for even degrees up to 2000 [1][2]. The team developed the Generalized Boyd algorithm—a five-parameter construction—that systematically generates irreducible polynomials with Mahler measures strictly below Boyd’s limit point, approximately $\alpha_1 \approx 1.255434$ [1][2]. This achievement marks the first general method capable of producing such polynomials across all even degrees in that range.

The Human-AI Discovery Loop

The breakthrough emerged from an iterative process combining conversational prompting, a fast numerical pipeline, an exact symbolic verifier, and large-scale distributed search [1][2]. An AI agent prototypes solvers locally, launches parallel searches across distributed computing clusters, and collaborates with human researchers to extract and verify mathematical hypotheses [1][2]. This loop enabled the team to identify patterns and derive the Generalized Boyd algorithm through extensive computational exploration.

The method improves upon six previously published Mahler measure records and holds 99.6% of all best-known values for even degrees up to 2000 [1][2]. Notably, it does not claim coverage for odd degrees or arbitrary degree sequences—only even degrees are addressed, a critical limitation that distinguishes this advance from broader claims.

Verified Dataset and Implications

The researchers have released a verified dataset containing irreducible polynomials with minimal Mahler measures across all studied even degrees up to 2000 [1][2]. This dataset serves as a benchmark for future work in number theory and computational mathematics. While other literature has explored related conjectures—such as a claimed proof of Lehmer’s Conjecture using dynamical zeta functions in 2019 [5] or sequences tied to the problem in 2023 [6]—none provided a constructive method for high-degree cases.

This work represents a shift toward agentic systems and reinforcement learning in solving long-standing mathematical problems through large-scale computational exploration, demonstrating how AI can accelerate discovery when integrated with human insight.

Sources

1. Breaking the high-degree barrier in Lehmer’s problem with a human-AI discovery loop (arxiv.org) — 2026-10-21
2. Breaking the high-degree barrier in Lehmer’s problem with a human-AI discovery loop | Arxiv - DeepPaper (arxiv.deeppaper.ai) — 2026-10-21
3. A proof of the Conjecture of Lehmer (arxiv.org) — 2019-11-24
4. Sequences related to Lehmer’s problem (arxiv.org) — 2023-08-18