On the quadratic Waring-Goldbach problem with primes in Piatetski-Shapiro sets

Abstract

In this paper, it is proved that, for any ${\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4,{\gamma }_5\in (\frac{28}{29},1)$, every sufficiently large integer $n$ subject to $n\equiv 5(\mathrm{mod}24)$ can be represented as the sum of five squares of primes, i.e., \begin{equation*} n=p_1^2+p_2^2+p_3^2+p_4^2+p_5^2, \end{equation*} such that $p_i=\lfloor m_i^{1/{\gamma }_i}\rfloor$ for some $m_i\in {\mathbb{N}}^{+}$ for each $1⩽i⩽5$. This result constitutes an improvement upon the previous result of Zhang and Zhai [29].

9 pages