Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. Derrick Henry Lehmer

Derrick Henry Lehmer

Derrick Henry "Dick" Lehmer was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes...

 conjectured that the Mahler measure of any integral polynomial

P(x),

that is not a multiple

Multiple (mathematics)

In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...

 of cyclotomic polynomial

Cyclotomic polynomial

In algebra, the nth cyclotomic polynomial, for any positive integer n, is the monic polynomial:\Phi_n = \prod_\omega \,where the product is over all nth primitive roots of unity ω in a field, i.e...

s, is bounded below.

More specifically


Essentially, to disprove this conjecture, one would try to find a polynomial


with (each coefficient is an integer), such that


is minimized, and where is not divisible by


This can also be stated in terms of the Mahler measure of an algebraic number

Algebraic number

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

, where the Mahler measure of an algebraic number is simply the Mahler measure of its minimal polynomial

Minimal polynomial (field theory)

In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...

.

Some active research consists of computational techniques for searching through polynomials of some degree trying to find those with smallest Mahler measure.