On the Waring-Goldbach problem for two squares and four cubes

Abstract

Let NN be a sufficiently large integer. In this article, it is proved that, with at most O(N112+ε)O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to NN can be represented in the form p12+p22+p33+p43+p53+p63{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}+{p}_{6}^{3}, where p1,p2,p3,p4,p5{p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5}, and p6{p}_{6} are prime variables. This result constitutes a large improvement upon the previous result of Liu [On a Waring-Goldbach problem involving squares and cubes, Math. Slovaca. 69 (2019), no. 6, 1249–1262].

Keywords

• waring-goldbach problem
• hardy-littlewood method
• exceptional set
• 11p05
• 11p32
• 11p55