On the Waring-Goldbach Problem for One Square and Five Cubes

TL;DR

This paper proves that large even integers can be expressed as a sum of one almost-prime with at most six prime factors, a square, and five prime cubes, improving previous results with fewer prime factors in the almost-prime.

Contribution

The paper improves the bound on the almost-prime's prime factors from 36 to 6 in the Waring-Goldbach problem involving one square and five cubes.

Findings

Successfully represents large even integers with the new bound.

Reduces the almost-prime prime factor count significantly.

Advances the understanding of additive representations involving primes and almost-primes.

Abstract

Let $\mathcal{P}_r$ 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{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^3 \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_6$ and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with $\mathcal{P}_{36}$ in place of $\mathcal{P}_6$.