# P34: Lehmer's Mahler measure problem

- ID: `P34`
- Reference: `lehmers-mahler-measure-problem`
- Page: https://theoremdb.org/statements/P34
- Record maturity: Reviewed problem with recorded work

## Problem

There exists \(\varepsilon>0\) such that every monic polynomial \(P\in\mathbb{Z}[x]\) whose Mahler measure satisfies \(M(P)>1\) also satisfies \(M(P)\ge1+\varepsilon\).

### Context

Lehmer found a polynomial with Mahler measure about 1.17628 and asked whether measures above 1 can approach 1.

### Problem setup

- **Definition (For a polynomial with leading coefficient a and complex roots alpha_i, its Mahler measure).** For a polynomial with leading coefficient a and complex roots alpha_i, its Mahler measure is |a| times the product over i of max(1, |alpha_i|).
- **Definition (Kronecker's theorem characterizes monic integer polynomials of Mahler measure 1 using roots of unity and zero roots).** Kronecker's theorem characterizes monic integer polynomials of Mahler measure 1 using roots of unity and zero roots.
- **Remark.** Lehmer found a polynomial with Mahler measure about 1.17628 and asked whether measures above 1 can approach 1.

### What counts as a solution

- Prove a uniform positive gap above 1 for all qualifying monic integer polynomials, or construct a sequence of such polynomials whose Mahler measures decrease to 1.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Dobrowolski gives a degree-dependent lower bound above 1 for noncyclotomic algebraic integers, while the gap shrinks with degree. Lehmer's degree-10 example remains the smallest known Mahler measure above 1. Exact unresolved remainder: Prove a constant c>1 such that every noncyclotomic monic integer polynomial P satisfies M(P)>=c, or disprove such a uniform gap. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Dobrowolski gives a degree-dependent lower bound above 1 for noncyclotomic algebraic integers, while the gap shrinks with degree. Lehmer's degree-10 example remains the smallest known Mahler measure above 1. Exact unresolved remainder: Prove a constant c>1 such that every noncyclotomic monic integer polynomial P satisfies M(P)>=c, or disprove such a uniform gap.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Dobrowolski gives a degree-dependent lower bound above 1 for noncyclotomic algebraic integers, while the gap shrinks with degree. Lehmer's degree-10 example remains the smallest known Mahler measure above 1.

Exact unresolved remainder: Prove a constant c>1 such that every noncyclotomic monic integer polynomial P satisfies M(P)>=c, or disprove such a uniform gap.

### Background and intake notes

- Original intake status: The cited survey lists Lehmer's problem among open Diophantine questions. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The gap-above-one formulation and status were checked against the cited survey on 2026-07-22.
- A stronger common form asserts that Lehmer's degree-ten polynomial gives the smallest possible measure above 1.

### Open directions

- **Route 1** (reported): Prove a uniform positive gap above 1 for all qualifying monic integer polynomials, or construct a sequence of such polynomials whose Mahler measures decrease to 1. [1](#reference-1)

### Computational notes

- Searches through bounded degree and coefficient ranges can locate small measures without proving a universal gap.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `lehmers-mahler-measure-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Michel Waldschmidt, Open Diophantine problems, Moscow Mathematical Journal 4(1) (2004), 245-305. DOI 10.17323/1609-4514-2004-4-1-245-305. Michel Waldschmidt, Moscow Mathematical Journal 4 (2004), discussion of Lehmer's problem https://www.mathnet.ru/eng/mmj150
   - Also cited at Discussion of Lehmer's problem in the Mahler-measure section
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The cited survey lists Lehmer's problem among open Diophantine questions. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - States Lehmer's uniform-gap problem and surveys the surrounding degree-dependent results.
   - Source named by the research packet.