Lehmer’s problem on approaching Mahler measure 1

Determine whether 1 can be approached arbitrarily closely by Mahler measures of algebraic integers that are not roots of unity.

Background

For an algebraic number, the Mahler measure is at least 1, and it equals 1 precisely for roots of unity. The paper explains that Lehmer asked whether Mahler measures of algebraic integers outside the roots of unity can come arbitrarily close to 1. This question is presented as an unresolved problem and is central to the study of small Mahler measures, with Lehmer’s polynomial providing the smallest currently known Mahler measure greater than 1.

References

In (Lehmer, 1993), Lehmer then wonders whether one can approach 1 as closely as desired by Mahler measures of algebraic integers which are not roots of unity. This problem is, to date, an open problem.

— Finding New Limit Points of Mahler Measure by Methods of Missing Data Restoration  (2609.20468 - Sac-Épée et al., 17 Sep 2026) in Section 1, Introduction, page 1

In particular, \textbf{Lehmer's conjecture} is that the {\it smallest} real Salem number is \textbf{Lehmer's number} $\lambda_{\rm Lehmer}\approx 1.17628$, which is the largest real eigenvalue of the Coxeter element obtained from the first non-trivial even self-dual Lorentzian lattice $\mathrm{II}_{1,9}$ .

— Spacetime Quasicrystals and Computational Complexity Theory in the Road Towards Quantum Gravity  (2610.09855 - Mygdalas, 7 Oct 2026) in Chapter 5, Section 5.2, 'Hyperbolic Coxeter Eigensystems: Salem Numbers and Irrational Newman--Penrose Frames'