Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers

TL;DR

This paper proves that large odd integers can be expressed as a sum involving one square, five cubes, one higher power, and an almost-prime, improving previous results in Waring-Goldbach type problems.

Contribution

It establishes new solvability results for a mixed Waring-Goldbach equation with almost-primes and primes, extending prior work by Lü and Mu.

Findings

Equation solvable for sufficiently large odd N

Almost-prime with bounded prime factors used for x

Improves previous bounds on r(b)

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 $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^4+p_6^b \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(b)}$ and the other variables primes, where $r(b)$ is defined in the Theorem. This result constitutes an improvement upon that of L\"u and Mu.