Open problem, Arithmetic and number theory, Algebra, posed 1933

Lehmer's Mahler measure problem

Open

Does there exist a constant $\mu > 1$ such that every integer polynomial $P(x) \in \mathbb{Z}[x]$ with Mahler measure $M(P) > 1$ satisfies $M(P) \ge \mu$ ? More specifically, is Lehmer's number $\lambda \approx 1.17628$ , the Mahler measure of $L(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$ , the smallest Mahler measure strictly greater than $1$ ?

Research frontier as of 2026

As of 2026, Lehmer's Mahler measure problem remains open. Smyth's 1971 theorem completely resolves the non-reciprocal case by establishing the sharp lower bound $\theta_0 \approx 1.32471$ , so any counterexample must be a reciprocal polynomial of even degree. Dobrowolski's 1979 inequality and its subsequent refinements rule out Mahler measures approaching $1$ too rapidly as the degree $d$ grows, and Dimitrov (2019) proved the Schinzel–Zassenhaus conjecture on the maximum modulus of conjugates, a long-standing sibling problem. Nevertheless, no method has eliminated the possibility of high-degree reciprocal polynomials (or Salem numbers) with Mahler measure strictly between $1$ and $\lambda \approx 1.17628$ .

Best known results

Smyth (1971): every non-reciprocal integer polynomial $P(x)$ with $P(0) \ne 0$ satisfies $M(P) \ge \theta_0 \approx 1.32471795$ .
Dobrowolski (1979): for any non-cyclotomic irreducible integer polynomial of degree $d$ , $M(P) > 1 + c (\log \log d / \log d)^3$ .
Exhaustive computation (Mossinghoff, Rhin, and Wu 2008) verifies that $\lambda \approx 1.1762808$ is the minimum Mahler measure $> 1$ for all degrees $d \le 44$ .

Tools and where they stop

Tool	Achieved	Where it stops
Auxiliary polynomials and resultant bounds (Dobrowolski's method)	Gives the best general asymptotic lower bound $M(P) > 1 + c (\log \log d / \log d)^3$ in terms of the degree $d$	The factor $(\log \log d / \log d)^3$ decays to $0$ as $d \to \infty$ , failing to yield a degree-independent constant $\mu > 1$
Holomorphic / Hankel determinant techniques (Dimitrov's method)	Resolves the Schinzel–Zassenhaus conjecture on the largest conjugate modulus (house) of algebraic integers	Requires non-reciprocal structure or power series non-rationality that breaks down when conjugates cluster near the unit circle in reciprocal polynomials

Open questions

Is there a universal constant $\mu > 1$ bounding $M(P)$ away from $1$ across all degrees $d$ ?
Is Lehmer's number $\lambda \approx 1.17628$ the smallest Salem number?

References

D. H. Lehmer (1933). Factorization of certain cyclotomic functions · DOI:10.2307/1968172
C. J. Smyth (1971). On the product of the conjugates outside the unit circle of an algebraic integer · DOI:10.1112/blms/3.2.169
E. Dobrowolski (1979). On a question of Lehmer and the number of irreducible factors of a polynomial · DOI:10.4064/aa-34-4-391-401
Enrico Bombieri, Walter Gubler (2006). Heights in Diophantine Geometry · DOI:10.1017/CBO9780511542879