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

TL;DR

This paper proves that sufficiently large integers congruent to 5 mod 24 can be expressed as the sum of five squares of primes within Piatetski-Shapiro sets, extending previous results in additive prime number theory.

Contribution

It establishes the representation of large integers as sums of five Piatetski-Shapiro primes squared, improving upon prior work by Zhang and Zhai.

Findings

Every large integer n ≡ 5 mod 24 can be written as sum of five squares of primes in Piatetski-Shapiro sets.

Primes p_i are of the form ⌊m_i^{1/γ_i}⌋ with γ_i in (28/29, 1).

The result advances understanding of Waring-Goldbach problems with primes in special 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 $n\equiv5\pmod{24}$ 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 $1\leqslant i\leqslant 5$. This result constitutes an improvement upon the previous result of Zhang and Zhai [29].