  Using the method of maximal cuts, we obtain a form of the three-loop
four-point scattering amplitude of N=8 supergravity in which all ultraviolet
cancellations are made manifest. The Feynman loop integrals that appear have a
graphical representation with only cubic vertices, and numerator factors that
are quadratic in the loop momenta, rather than quartic as in the previous form.
This quadratic behavior reflects cancellations beyond those required for
finiteness, and matches the quadratic behavior of the three-loop four-point
scattering amplitude in N=4 super-Yang-Mills theory. By direct integration we
confirm that no additional cancellations remain in the N=8 supergravity
amplitude, thus demonstrating that the critical dimension in which the first
ultraviolet divergence occurs at three loops is D_c=6. We also give the values
of the three-loop divergences in D=7,9,11. In addition, we present the
explicitly color-dressed three-loop four-point amplitude of N=4
super-Yang-Mills theory.
