  Consider a general diagonal system of Diophantine equations. Contingent on
existence of an appropriate mean value bound (and local solvability), an
asymptotic formula for the number of prime solutions is found. This is used in
conjunction with the Vinogradov Mean Value Theorem to give an asymptotic
formula for solutions of Vinogradov systems, and in conjunction with Hooley's
work on seven cubes to give a conditional result for the Waring-Goldbach
problem on seven cubes of primes, contingent on Hooley's form of Riemann
hypothesis.
