On the Waring–Goldbach problem for one square and five cubes

Abstract

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with [Formula: see text] in place of [Formula: see text].