Quadratic residues and non-residues for infinitely many Piatetski-Shapiro primes

TL;DR

This paper proves that for infinitely many primes of the form [n^c], every nonempty finite subset of positive integers can be realized as quadratic residues or non-residues, generalizing previous results.

Contribution

It provides a quantitative proof that finite sets are quadratic residues or non-residues for infinitely many Piatetski-Shapiro primes, extending earlier work by S. Wright.

Findings

Finite subsets are quadratic residues for infinitely many primes of the form [n^c].

Similar results hold for quadratic non-residues under certain assumptions.

Generalizes previous results on quadratic residues for special prime sets.

Abstract

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by S. Wright in certain aspects.