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 such that every polynomial with integer coefficients satisfies one of the following properties:
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"
^Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris (eds.). Number Theory and Polynomials. Cambridge University Press. pp. 322–349. ISBN978-0-521-71467-9.