Lehmer's Mahler measure problem is an unsolved problem that asks whether the Mahler measures greater than
1 of integer polynomials are bounded away from
1 (Lehmer 1933). Its strong form asserts that every univariate
polynomial 
with integer coefficients and 
satisfies
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
,
denoted 
by Lehmer (1933) and 
by Hironaka (2009), is the largest positive root
of this polynomial. The roots
of this polynomial, plotted in the left figure above,
are very special, since 8 of the 10 lie on the unit circle
in the complex plane. The roots
of the polynomials (represented by half their coefficients)
giving the two next smallest known Mahler measures
are also illustrated above (Mossinghoff 1998, p. S11).
Smyth (1971) proved that a nonreciprocal irreducible polynomial 
with integer coefficients and 
satisfies 
, where 
is the real root of 
. Equality is attained by 
. Generalizations of Smyth's result have been constructed
by Lloyd-Smith (1985) and Dubickas (1997).
In general, the smallest Mahler measures occur for integer polynomials that are small in absolute value. The histogram above shows the distribution
of measures for random 
-polynomials of random
orders 1 to 10. Mossinghoff (1998) gives a table of the smallest known Mahler
measures for polynomial degrees up to 
.
Mossinghoff et al. (2008) verified that 
is the smallest Mahler measure
greater than 1 among all integer polynomials
of polynomial degree at most 54.
Chervonyi et al. (2026) developed a generalized Boyd construction using AI-assisted search and exact symbolic verification. The construction extends the family obtained
by substituting signed powers of a single variable into Boyd's two-variable polynomial,
removes cyclotomic factors, and checks for
irreducible quotients of the requested
polynomial degree. They report noncyclotomic
irreducible polynomials with integer coefficients
and Mahler measure below the limit
point 
considered by Boyd (1981, p. 459) for every even polynomial
degree 
with 
. Their searches improve previously published
records at polynomial degrees 178, 188, 190,
202, 242, and 250. These constructions give upper bounds
on the smallest Mahler measure greater than 1 at
each fixed polynomial degree among irreducible
polynomials with integer coefficients, rather than proofs of optimality.
is a Salem constant.
