Bulletin of the
Korean Mathematical Society BKMS

Let $P_r$ 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{gather*} N=x^2+p_1^2+p_2^3+p_3^3+p_4^4+p_5^4 \end{gather*} is solvable with $x$ being an almost-prime $P_{4}$ and the other variables primes. This result constitutes an improvement upon that of L\"{u} \cite{7}.

Keywords: Waring-Goldbach problem, Hardy-Littlewood method, almost-prime, sieve theory

MSC numbers: 11P32, 11N36