  We produce infinitely many distinct irreducible smooth 4-manifolds
homeomorphic to #(2m+1)(CP^2 # -CP^2) and #(2n+1)(S^2 x S^2), respectively, for
each m>3 and n>4. These provide the smallest exotic closed simply connected
4-manifolds with signature zero known to date, and in each one of these
homeomorphism classes, we get minimal symplectic 4-manifolds. Our novel exotic
4-manifolds are derived from fairly special small Lefschetz fibrations we build
via positive factorizations in the mapping class group, with spin and non-spin
monodromies.
