Quadratic residues that are not primitive roots

TL;DR

This paper proves that for primes with a certain property related to their totient function, there exist two consecutive quadratic non-residues that are not primitive roots.

Contribution

It establishes the existence of specific consecutive quadratic non-residues that are not primitive roots for primes satisfying a particular totient condition.

Findings

Existence of two consecutive quadratic non-residues that are not primitive roots for certain primes.

Primes with (p-1)  (p-1)/4 satisfy the property.

Provides a new characterization of quadratic residues and primitive roots.

Abstract

We prove that any prime $p$ satisfying $\phi(p-1)\leq (p-1)/4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$.