On Waring--Goldbach problem with mixed powers

Abstract

It is proved that for every sufficiently large odd integer $n$, the equation $$n=p_1^2+p_2^2+p_3^3+p_4^4+p_5^5+p_6^6+p_7^8$$ is solvable in primes. This refines the work of Cai and Mu in 2015.