The Waring–Goldbach problem: one square and five cubes

Abstract

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 $$N=x^2+p^3_1+p^3_2+p_3^3+p_4^3+p_5^3 $$ is solvable with x being an almost-prime P 36 and the p j ’s primes. This result constitutes a refinement upon that of G.L. Watson.