On the Waring-Goldbach Problem for One Square and Five Cubes

Abstract

Let ${\mathcal{P}}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^3 \end{equation*} is solvable with $x$ being an almost-prime ${\mathcal{P}}_6$ and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with ${\mathcal{P}}_{36}$ in place of ${\mathcal{P}}_6$.

16 pages. arXiv admin note: substantial text overlap with arXiv:1708.04484