On Waring-Goldbach Problem for Squares, Cubes and Higher Powers

Abstract

Let ${\mathcal{P}}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan for ternary admissible exponent. Moreover, we use the refined admissible exponent to prove that, for $3⩽k⩽14$ and for every sufficiently large even integer $n$, the following equation \begin{equation*} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{equation*} is solvable with $x$ being an almost-prime ${\mathcal{P}}_{r(k)}$ and the other variables primes, where $r(k)$ is defined in Theorem. This result constitutes a deepening upon that of previous results.

24 pages. arXiv admin note: text overlap with arXiv:1708.04484