Lehmer's Mahler measure problem

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.

Problem

For a polynomial $f(x) = a_n \prod_{i=1}^{n} (x - \alpha_i)$ with complex coefficients, the Mahler measure is

$$M(f) = |a_n| \prod_{i=1}^{n} \max\{1, |\alpha_i|\}.$$

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

$$L = \inf \{ M(f) : f \in \mathbb{Z}[x],\ M(f) > 1 \}.$$

Lehmer's question asks whether $L = 1$. Lehmer's polynomial

$$\ell(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$

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.

Background

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.

What is known

• 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.
• Lower bound: only the trivial $L \ge 1$ is known in general.
• 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).

What would count as progress

• 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.
• A proof that $L > 1$ (Lehmer's conjecture), or an explicit lower bound for a natural infinite class of reciprocal polynomials not covered before.
• Whether Lehmer's polynomial attains the minimum is also open. Both statements are in Lean in formal-conjectures.

References

1. D. H. Lehmer, "Factorization of certain cyclotomic functions", Annals of Mathematics 34 (1933), 461–479. doi:10.2307/1968172
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
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

Source

Imported from Optimization Constants in Mathematics, constant 40a, 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.