Recent progress on Vinogradov's mean value theorem has resulted in improved
estimates for exponential sums of Weyl type. We apply these new estimates to
obtain sharper bounds for the function H(k) in the Waring--Goldbach problem.
We obtain new results for all exponents k≥8, and in particular establish
that H(k)≤(4k−2)logk+k−7 when k is large, giving the first improvement
on the classical result of Hua from the 1940s
