Waring-Goldbach Problem: Two Squares and Three Biquadrates

Assume that $\psi$ is a function of positive variable $t$, monotonically increasing to infinity and $0 \lt \psi(t) \ll \log t/(\log \log t)$. Let $\mathcal{R}_{3}(n)$ denote the number of representations of the integer $n$ as sums of two squares and three biquadrates of primes and we write $\mathcal{E}_{3}(N)$ for the number of integers $n$ satisfying $n \leq N$, $n \equiv 5, 53, 101 \pmod{120}$ and \[ \left| \mathcal{R}_{3}(n) - \frac{\Gamma^{2}(1/2) \Gamma^{3}(1/4)}{\Gamma(7/4)} \frac{\mathfrak{S}_{3}(n) n^{3/4}}{\log^{5}n} \right| \geq \frac{n^{3/4}}{\psi(n) \log^{5}n}, \] where $0 \lt \mathfrak{S}_{3}(n) \ll 1$ is the singular series. In this paper, we prove \[ \mathcal{E}_{3}(N) \ll N^{23/48+\varepsilon} \psi^{2}(N) \] for any $\varepsilon \gt 0$. This result constitutes a refinement upon that of Friedlander and Wooley [2].