## Abstract

Let $\mathcal{\mathcal{R}}\left(n\right)$ denote the number of representations of the positive integer $n$ as the sum of two squares and three biquadrates of primes and we write $\mathcal{\mathcal{E}}\left(N\right)$ for the number of positive integers $n$ satisfying $n\le N$, $n\equiv 5,53,101\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}120\right)$ and

$$\left|\mathcal{\mathcal{R}}\left(n\right)-\frac{{\mathrm{\Gamma}}^{2}\left(\frac{1}{2}\right){\mathrm{\Gamma}}^{3}\left(\frac{1}{4}\right)}{\mathrm{\Gamma}\left(\frac{7}{4}\right)}\frac{\mathfrak{\U0001d516}\left(n\right){n}^{\frac{3}{4}}}{{log}^{5}n}\right|\ge \frac{{n}^{\frac{3}{4}}}{{log}^{\frac{11}{2}}n},$$

where $0<\mathfrak{\U0001d516}\left(n\right)\ll 1$ is the singular series. In this paper, we prove

$$\mathcal{\mathcal{E}}\left(N\right)\ll {N}^{\frac{15}{32}+\mathit{\epsilon}}$$

for any $\mathit{\epsilon}>0$. This result constitutes a refinement upon that of Friedlander and Wooley (2014).

## Citation

Li Zhu. "Waring–Goldbach problem: two squares and three biquadrates." Rocky Mountain J. Math. 50 (1) 355 - 367, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.355

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