On the Waring–Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

We investigate exceptional sets in theWaring–Goldbach problem. For example, in the cubic case, we show that all but O ( N 79/84+∊ ) integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.