# Researchers use human-AI loop to break high-degree barrier in Lehmer's problem

Digest AI · Research · published 2026-10-23T04:00:00Z

Canonical: https://digestai.news/story/researchers-use-human-ai-loop-to-break-high-degree-barrier-in-lehmer-s

## Summary

A team of researchers has used a human-AI discovery loop to make progress on Lehmer's problem, a long-standing question in number theory from 1933. The problem asks whether the Mahler measure of an integer polynomial can be made arbitrarily small. Previous systematic constructions of irreducible polynomials with small Mahler measure only reached even degrees up to 180, and existing analytical methods frequently failed at higher degrees.

The researchers combined conversational prompting with a fast numerical pipeline, an exact symbolic verifier, and large-scale distributed search. The AI agent prototyped solvers locally, launched parallel searches across clusters, and worked with the human researcher to extract and verify hypotheses. This loop produced a five-parameter construction called the Generalized Boyd algorithm, which eliminates degree-skipping and constructs irreducible polynomials with Mahler measure strictly below Boyd's limit point (≈1.255434) for every target degree. Evaluated across all even degrees up to 2000, the approach improves upon six previously published values and holds 99.6% of all best-known records. The verified dataset of polynomials with minimal Mahler measure across all studied even degrees has been released.

## Key points

- Human-AI loop combines prompting, numerical pipeline, symbolic verifier, and distributed search
- Generalized Boyd algorithm constructs irreducible polynomials for every even degree up to 2000
- Method improves six prior records and holds 99.6% of best-known values, dataset released

## Why it matters

Demonstrates AI-assisted mathematical discovery can overcome barriers where purely analytical or computational methods stalled, producing verified results across a vast search space.

## Sources

1. [Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop](https://arxiv.org/abs/2610.30277) (arXiv cs.AI, 2026-10-23, primary source)

## Cite

Digest AI, "Researchers use human-AI loop to break high-degree barrier in Lehmer's problem", 23 October 2026, https://digestai.news/story/researchers-use-human-ai-loop-to-break-high-degree-barrier-in-lehmer-s

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