A Remark on the Piatetski-Shapiro-Hua Theorem

TL;DR

This paper improves the understanding of prime representations by showing that large numbers congruent to 5 mod 24 can be expressed as the sum of five prime squares, with one prime in a specific subset, extending previous results.

Contribution

It establishes a new result on prime representations involving a subset of primes, refining earlier theorems by Zhang and Zhai for certain large numbers.

Findings

Large numbers congruent to 5 mod 24 can be represented as five prime squares.

One prime in the representation belongs to a specific subset of primes.

The result applies for all sufficiently large N within the given conditions.

Abstract

In this paper, we prove that for any fixed $205/243<\gamma\leqslant1$, every sufficiently large $N$ satisfying $N\equiv 5 \pmod {24}$ can be represented as five squares of primes with one prime in $\mathcal{P}_\gamma$, which improves the previous result of Zhang and Zhai.