  We present analytic forms of three-loop four-gluon planar amplitudes in pure
Yang-Mills theory in this letter. Gauge invariant bases and a set of proper
master integrals are chosen such that the amplitudes are explicitly invariant
under cyclic permutations of external particles. The $D$-dimensional unitarity
method and integration-by-parts reductions with cut conditions are performed to
determine the coefficients of the master integrals. Helicity amplitudes in the
conventional dimension regularization scheme are obtained by setting gauge
invariant bases to helicity configurations. After renormalizing the ultraviolet
divergences, remaining divergences of the amplitudes agree with known infrared
divergence structures exactly. Our results provide an essential piece of
three-loop amplitudes required for the ${\rm N^3LO}$ order corrections to the
production of two jets at hadron colliders.
