On the Waring-Goldbach problem for fourth and sixth powers

Abstract

We consider the Waring–Goldbach problem for fourth and sixth powers. In particular, we establish that every sufficiently large positive integer under a natural congruence condition can be represented as a sum of 13 fourth powers of prime numbers. This improves upon the earlier result of Kawada and Wooley [‘On the Waring–Goldbach problem for fourth and fifth powers’, Proc. London Math. Soc. (3) 83 (2001) 1–50].