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

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 N = x(2) + P-1(3) + P-2(3) + P-3(3) + P-4(3) + P-5(3) is solvable with x being an almost-prime P-6 and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with P-36 in place of P-6.