Lehmer's Mahler measure problem

Beat Lehmer's 1.17628 with an integer polynomial of smaller Mahler measure above 1, or prove the measures stay away from 1.

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1	# Lehmer's Mahler measure problem
2
3	Can an integer polynomial have Mahler measure larger than 1 but arbitrarily close to it? Lehmer asked in 1933, and his own degree-10 example, with measure 1.17628..., is still the smallest known; an integer polynomial below it would be checked in seconds.
4
5	## Problem
6
7	For a polynomial $f(x) = a_n \prod_{i=1}^{n} (x - \alpha_i)$ with complex coefficients, the Mahler measure is
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9	$$M(f) = \|a_n\| \prod_{i=1}^{n} \max\{1, \|\alpha_i\|\}.$$
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11	For $f \in \mathbb{Z}[x]$ one has $M(f) \ge 1$, and by Kronecker's theorem $M(f) = 1$ exactly when $f$ is, up to sign, a product of cyclotomic polynomials and powers of $x$. Let
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13	$$L = \inf \{ M(f) : f \in \mathbb{Z}[x],\ M(f) > 1 \}.$$
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15	Lehmer's question asks whether $L = 1$. Lehmer's polynomial
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17	$$\ell(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$
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19	has $M(\ell) = 1.176280\ldots$, so $1 \le L \le 1.176280\ldots$. Determine $L$: prove $L > 1$, or find integer polynomials with measure closer to 1.
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21	## Background
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23	Lehmer met the question in 1933 while studying the factorization of certain cyclotomic functions. Smyth (1971) answered it for non-reciprocal polynomials: if $f \in \mathbb{Z}[x]$ is not reciprocal and $f(0) \ne 0$, then $M(f) \ge M(x^3 - x - 1) = 1.324717\ldots$, so polynomials with measure close to 1 must be reciprocal. For an algebraic number of degree $d$ that is not a root of unity, Dobrowolski proved that its Mahler measure is at least $1 + (1-\epsilon)(\log\log d / \log d)^3$ for large $d$, a bound which tends to 1 and so leaves the question open.
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25	## What is known
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27	- Upper bound: $L \le M(\ell) = 1.176280\ldots$ (Lehmer, 1933). Borwein, Dobrowolski and Mossinghoff (2007) note that this is still the smallest known measure larger than 1 of an integer polynomial.
28	- Lower bound: only the trivial $L \ge 1$ is known in general.
29	- Partial results: Smyth's bound for non-reciprocal polynomials (1971), Dobrowolski's degree-dependent bound, and results for restricted classes such as polynomials with odd coefficients (Borwein, Dobrowolski and Mossinghoff, 2007).
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31	## What would count as progress
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33	- An integer polynomial $f$ with $1 < M(f) < 1.176280$. Checking it takes seconds: compute its roots with certified error bounds (for example in ball arithmetic) and confirm it has no cyclotomic factor.
34	- A proof that $L > 1$ (Lehmer's conjecture), or an explicit lower bound for a natural infinite class of reciprocal polynomials not covered before.
35	- Whether Lehmer's polynomial attains the minimum is also open. Both statements are in Lean in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/LehmerMahlerMeasureProblem.lean).
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37	## References
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39	1. D. H. Lehmer, "Factorization of certain cyclotomic functions", Annals of Mathematics 34 (1933), 461–479. [doi:10.2307/1968172](https://doi.org/10.2307/1968172)
40	2. C. J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer", Bulletin of the London Mathematical Society 3 (1971), 169–175. [doi:10.1112/blms/3.2.169](https://doi.org/10.1112/blms/3.2.169)
41	3. P. Borwein, E. Dobrowolski and M. J. Mossinghoff, "Lehmer's problem for polynomials with odd coefficients", Annals of Mathematics 166 (2007), 347–366. [doi:10.4007/annals.2007.166.347](https://doi.org/10.4007/annals.2007.166.347)
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43	## Source
44
45	Imported from [Optimization Constants in Mathematics, constant 40a](https://teorth.github.io/optimizationproblems/constants/40a.html), curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.