A Note on Waring–Goldbach Problem: One Square, Four Cubes and One kth Power

School of Mathematics and Data Science, Jiangnan University, Wuxi 214122, Jiangsu, China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 19 March 2026
Accepted: 23 April 2026

Abstract

This paper establishes that for k ≥ 4, all sufficiently large even integers n ≤ N, with at most O(N^{1/2 − ϑ(k) + ε}) exceptions, admit representations as the sum of one square of a prime, four cubes of primes and one kth power of a prime, where the exponent ϑ(k) relies on k. This result sharpens the bound previously obtained by [J. J. Li, F. Xue and M. Zhang, Bull. Aust. Math. Soc. 107 (2023) 416–431].