  A first quantised approach to loop amplitudes based on the pure spinor
particle is applied to the systematics of four-particle amplitudes in maximally
supersymmetric field theories. Counting of fermionic zero modes allows the
identification of momentum factors multiplying R**4 in the case of supergravity
(and F**4 in the Yang--Mills case) thereby making manifest their ultraviolet
properties as a function of dimension, D. For L=2,3,4 loops the leading
supergravity divergence is in D=4+6/L dimensions and proportional to d*2L R**4,
in line with earlier field theory calculations. However, at five loops there is
a radical change in the systematics, suggesting the presence of a contribution
with an explicit L=5 logarithmic ultraviolet divergence when D=24/5 that is
proportional to d*8 R**4. We further argue that d*8 R**4 should receive
contributions from all loops, which would imply that N=8 supergravity (with
D=4) is not protected by supersymmetry from a seven-loop divergence.
