  We study the reduced descendent Gromov-Witten theory of K3 surfaces in
primitive curve classes. We present a conjectural closed formula for the
stationary theory, which generalizes the Bryan-Leung formula. We also prove a
new recursion that allows to remove descendent insertions of $1$ in many
instances. Together this yields an efficient way to compute a large class of
invariants (modulo the conjecture on the stationary part). As a corollary we
conjecture a surprising polynomial structure which underlies the Gromov-Witten
invariants of the K3 surface.
