Lehmer's conjecture

Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.^[1] The conjecture asserts that there is an absolute constant ${\displaystyle \mu >1}$ such that every polynomial with integer coefficients ${\displaystyle P(x)\in \mathbb {Z} [x]}$ satisfies one of the following properties:

• The Mahler measure^[2] ${\displaystyle {\mathcal {M}}(P(x))}$ of ${\displaystyle P(x)}$ is greater than or equal to ${\displaystyle \mu }$.
• ${\displaystyle P(x)}$ is an integral multiple of a product of cyclotomic polynomials or the monomial ${\displaystyle x}$, in which case ${\displaystyle {\mathcal {M}}(P(x))=1}$. (Equivalently, every complex root of ${\displaystyle P(x)}$ is a root of unity or zero.)

There are a number of definitions of the Mahler measure, one of which is to factor ${\displaystyle P(x)}$ over ${\displaystyle \mathbb {C} }$ as

${\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D}),}$

and then set

${\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}$

The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"

${\displaystyle P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1\,,}$

for which the Mahler measure is the Salem number^[3]

${\displaystyle {\mathcal {M}}(P(x))=1.176280818\dots \ .}$

It is widely believed that this example represents the true minimal value: that is, ${\displaystyle \mu =1.176280818\dots }$ in Lehmer's conjecture.^[4][5]