Quadratic non-residues and non-primitive roots satisfying a coprimality condition

TL;DR

This paper proves the existence of certain quadratic non-residues that are not primitive roots and satisfy a coprimality condition for a broad class of primes, extending understanding of residue properties under specific modular constraints.

Contribution

It establishes new conditions under which quadratic non-residues that are not primitive roots exist, linking prime congruences, coprimality, and residue properties.

Findings

Existence of such quadratic non-residues under specified conditions

Conditions involving prime congruences and Euler totient ratios

Results applicable to a wide class of primes with certain bounds

Abstract

Let $q\geq 1$ be any integer and let $ \epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.