Negative virial coefficients and the dominance of loose packed diagrams for D-dimensional hard spheres

We study the virial coefficients B(k) of hard spheres in D dimensions by means of Monte-Carlo integration. We find that B(5) is positive in all dimensions but that B(6) is negative for all D greater than or equal to 6. For 7 less than or equal to k less than or equal to 17 we compute sets of Ree-Hoover diagrams and find that either for large D or large k the dominant diagrams are loose packed.'' We use these results to study the radius of convergence and the validity of the many approximations used for the equations of state for hard spheres.