Lehmer's Conjecture

Claim: for every ε > 0, only finitely many irreducible monic integer polynomials P have Mahler measure M(P) < 1 + ε. Equivalently: inf{M(P) : M(P) > 1} > 1 attained by Lehmer's polynomial.

Lehmer found his degree-10 Salem polynomial while studying factorisation tests. Dobrowolski (1979) proved the landmark general lower bound m(α) ≥ (log log d / log d)^3 · c/d — decaying to zero, so the conjecture says the decay never bites infinitely often. The problem now drives arithmetic dynamics: equidistribution of preperiodic points (Baker–Hsia), Bogomolov-type statements, and explicit class field constructions all orbit around it.