[#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 \(M(P)\ge1+\varepsilon\).
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|).
Definition 2 (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 1. Lehmer found a polynomial with Mahler measure about 1.17628 and asked whether measures above 1 can approach 1.
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
Current 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]
1Records
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
How to cite
TheoremDB contributors, “Lehmer's Mahler measure problem,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/lehmers-mahler-measure-problemThis page as plain text: lehmers-mahler-measure-problem.md
This problem includes 2 records joined by 2 typed links, sourced from mathnet.ru[1], current as of July 31, 2026.
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.