Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients satisfies one of the following properties:
- The Mahler measure of is greater than or equal to .
- is an integral multiple of a product of cyclotomic polynomials or the monomial , in which case . (Equivalently, every complex root of is a root of unity or zero.)
There are a number of definitions of the Mahler measure, one of which is to factor over as
and then set
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
Let be an irreducible monic polynomial of degree .
Dobrowolski  improved this to
Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier obtained C ≥ 1/4 for D ≥ 2.
Let be an elliptic curve defined over a number field , and let be the canonical height function. The canonical height is the analogue for elliptic curves of the function . It has the property that if and only if is a torsion point in . The elliptic Lehmer conjecture asserts that there is a constant such that
- for all non-torsion points ,
where . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:
Stronger results are known for restricted classes of polynomials or algebraic numbers.
If P(x) is not reciprocal then
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