[#P34] Lehmer's Mahler measure problem

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

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Definitions and notation

1Context

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

2Problem setup

Definition 1 (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|).

Remark 1. Kronecker's theorem characterizes monic integer polynomials of Mahler measure 1 using roots of unity and zero roots.

3What 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.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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.^[1]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Lehmer's Mahler measure problem, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Lehmer's Mahler measure problem, record lehmers-mahler-measure-problem-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited survey lists Lehmer's problem among open Diophantine questions. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

A stronger common form asserts that Lehmer's degree-ten polynomial gives the smallest possible measure above 1.

Computational notes

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

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Lehmer's Mahler measure problem.” TheoremDB. P34. Problem statement; statement text SHA-256 be931161fc5f086bfc613469dcd9c8ffbbb74c7e858811784d52f27a5cae1ffe. https://theoremdb.org/statement/?ref=P34

BibTeX

@misc{theoremdb-problem-be931161fc5f086bfc613469dcd9c8ffbbb74c7e858811784d52f27a5cae1ffe,
  title = {{Lehmer's Mahler measure problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 be931161fc5f086bfc613469dcd9c8ffbbb74c7e858811784d52f27a5cae1ffe},
  url = {https://theoremdb.org/statement/?ref=P34}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathnet.ru^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. 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. ↗website · primary source · checked 2026-07-31Source 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.Also cited at Discussion of Lehmer's problem in the Mahler-measure section.Also cited at Editorial research route recorded 2026-07-31.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.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P34] Lehmer's Mahler measure problem

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Recent contributions

Computational notes

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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Record & provenance