Definition
Lehmer's conjecture, also known as Lehmer's problem, is an unsolved hypothesis in number theory that asserts the existence of a universal constant $c>1$ such that the Mahler measure $M(f)$ of any non‑cyclotomic monic polynomial $f(x)$ with integer coefficients satisfies $M(f) \ge c$. Equivalently, the conjecture states that there is a positive lower bound, strictly greater than 1, for the absolute Mahler measure of all non‑cyclotomic algebraic integers.
Overview
The conjecture originates from a 1933 observation by the American mathematician D. H. Lehmer, who discovered a monic, reciprocal polynomial
$$ f_{L}(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1, $$
whose Mahler measure equals approximately 1.17628, now called Lehmer's number. No polynomial with integer coefficients and Mahler measure between 1 and this value has been found, leading Lehmer to ask whether a positive gap exists above 1 for all non‑cyclotomic polynomials. The conjecture has profound connections to several areas, including algebraic dynamics, height theory, and the distribution of algebraic numbers.
Despite extensive computational searches and numerous partial results, the conjecture remains open. The strongest general lower bound known is due to Dobrowolski (1979), who proved that for a non‑cyclotomic algebraic integer $\alpha$ of degree $d$,
$$ M(\alpha) > 1 + c \left(\frac{\log \log d}{\log d}\right)^{3}, $$
for a positive absolute constant $c$. This bound tends to 1 as the degree grows, and thus does not settle Lehmer's conjecture.
Etymology / Origin
The conjecture is named after Derrick Henry Lehmer (1905–1991), an influential number theorist noted for his work on computational methods and integer factorization. Lehmer introduced the problem in a short note titled “Factorisation of certain cyclotomic functions” (Bulletin of the American Mathematical Society, 1933), where he recorded the exceptional polynomial $f_{L}$ and posed the question of whether smaller Mahler measures exist.
Characteristics
| Aspect | Description |
|---|---|
| Statement | ∃ $c>1$ such that ∀ monic $f \in \mathbb{Z}[x]$ non‑cyclotomic, $M(f) \ge c$. |
| Mahler measure | For a polynomial $f(x)=\prod_{i=1}^{n}(x-\alpha_i)$, $M(f)= |
| Known lower bounds | Dobrowolski’s bound (1979); subsequent refinements give slightly better constants for restricted families (e.g., totally real fields). |
| Equivalent formulations | - Minimal absolute Weil height of non‑roots of unity. - Minimal entropy of an algebraic self‑map of the torus. - Minimal Salem number conjecture (Lehmer’s number is the smallest known Salem number). |
| Partial results | Proven for special families: (i) reciprocal polynomials of prime degree, (ii) polynomials with prescribed Galois group, (iii) polynomials arising from certain dynamical systems. |
| Computational evidence | Exhaustive searches up to degree 40 and coefficients of modest size have found no counterexample; the smallest Mahler measure encountered remains Lehmer’s number. |
| Open status | The conjecture has neither been proved nor disproved; it is listed as an open problem in major compilations (e.g., “Unsolved Problems in Number Theory” by R. K. Guy). |
Related Topics
- Mahler measure – A measure of the complexity of a polynomial, central to the conjecture.
- Cyclotomic polynomial – Polynomials whose roots are roots of unity; they have Mahler measure 1.
- Lehmer’s number – The Mahler measure of Lehmer’s polynomial, the smallest known value > 1.
- Salem numbers – Real algebraic integers > 1 all of whose conjugates lie on or within the unit circle; Lehmer’s number is the smallest known Salem number.
- Height of algebraic numbers – A notion of arithmetic size closely related to Mahler measure.
- Algebraic dynamics – The conjecture appears in the study of entropy of toral automorphisms.
- Dobrowolski’s theorem – Provides the best general lower bound currently known for Mahler measures.
Lehmer's conjecture remains a central, widely cited problem in analytic and algebraic number theory, influencing research on polynomial dynamics, transcendence theory, and computational number theory.