Math Conjectures

Lehmer’s Mahler-Measure Conjecture

OPENMajorExact constant problemProposed 1933 · Standard version

Canonical statement

There exists a real constant c>1c>1 such that every algebraic integer α≠0\alpha\ne0 that is not a root of unity satisfies
M(α)≥c, M(\alpha)\ge c,
where, if a0∏j=1d(x−αj)∈Z[x]a_0\prod_{j=1}^d(x-\alpha_j)\in\mathbb Z[x] is the minimal polynomial of α\alpha, then M(α)=∣a0∣∏j=1dmax⁡(1,∣αj∣)M(\alpha)=|a_0|\prod_{j=1}^d\max(1,|\alpha_j|).
View source LaTeX
There exists a real constant \(c>1\) such that every algebraic integer \(\alpha\ne0\) that is not a root of unity satisfies
\[
  M(\alpha)\ge c,
\] where, if \(a_0\prod_{j=1}^d(x-\alpha_j)\in\mathbb Z[x]\) is the minimal polynomial of \(\alpha\), then \(M(\alpha)=|a_0|\prod_{j=1}^d\max(1,|\alpha_j|)\).
NT-TRANS-018:[Lehmer1933Measure][Dobrowolski1979][Hironaka2009Lehmer][LaishramPrasad2025]
Number theoryDynamical systems

Notes

The Mahler measure of an algebraic integer α\alpha with minimal polynomial a0∏j(x−αj)∈Z[x]a_0\prod_j(x-\alpha_j)\in\mathbb Z[x] is M(α)=∣a0∣∏jmax⁡(1,∣αj∣)M(\alpha)=|a_0|\prod_j\max(1,|\alpha_j|); by Kronecker's classical theorem it equals 11 exactly when α\alpha is zero or a root of unity. In a 1933 paper on factorizations of cyclotomic-type polynomials, D. H. Lehmer asked whether M(α)M(\alpha) can otherwise come arbitrarily close to 11 [Lehmer1933Measure]. The canonical form of the conjecture posits a uniform gap: a constant c>1c>1 with M(α)≥cM(\alpha)\ge c for every nonzero algebraic integer that is not a root of unity. Lehmer exhibited a degree-1010 polynomial whose measure, about 1.176281.17628, is still the smallest known value above 11, and the sharper claim that this value is optimal also circulates under the same name.

The best general lower bound is Dobrowolski's, of the shape 1+c(log⁡log⁡d/log⁡d)31+c(\log\log d/\log d)^{3} in the degree dd; it tends to 11 as the degree grows and so falls short of a uniform gap [Dobrowolski1979]. Bounds under additional arithmetic hypotheses continue to be developed [LaishramPrasad2025], and the problem has well-known connections to dynamics, surveyed in expository accounts of Lehmer's number [Hironaka2009Lehmer].

A degree-independent gap above 11 remains unproved, and the problem is open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.