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

Abstract

<abstract><p>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 $, the following equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} N = p_1^2+p_2^2+x^3+p_3^3+p_4^3+p_5^3 \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>is solvable with $ x $ being an almost–prime $ \mathcal{P}_7 $ and the other variables primes. This result constitutes a deepening upon that of previous results.</p></abstract>