Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes

Abstract

Let $p_{1},p_{2},…,p_{6}$ be prime numbers. First we show that, with at most $O(N^{1/12+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$, which improves the previous result $O(N^{1/4+\varepsilon})$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O(N^{5/12+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$. Bibliography: 21 titles.