# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2022; 59(3): 439-448

Published online May 1, 2022 https://doi.org/10.4134/JKMS.j190679

## The exceptional set of one prime square and five prime cubes

Yuhui Liu

Jiangnan University

### Abstract

For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.

Keywords: Waring-Goldbach problem, exceptional set, Hardy-Littlewood method

MSC numbers: 11P05, 11P32

Supported by: Project supported by the National Natural Science Foundation of China (grant no. 11771333) and by the Fundamental Research Funds for the Central Universities (grant no. JUSRP122031).