On the 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 (Bull Lond Math Soc 17(1):17–20, 1985) for ternary ‘admissible exponent’. Moreover, we use the refined ‘admissible exponent’ to prove that, for \(3\leqslant k\leqslant 14\) and for every sufficiently large even integer n, the following equation $$\begin{aligned} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{aligned}$$is solvable with x being an almost-prime \(\mathcal {P}_{r(k)}\) and the other variables primes, where r(k) is defined in Theorem 1.1. This result constitutes a deepening of previous results.