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

Abstract

Let $q\ge 1$ 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 $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-ε} \mbox{ and } \frac{ϕ(p-1)}{p-1} \leq \frac{1}{2} - ε, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g,\frac{p-1}{q}\right)=1$.

to appear in Bulletin of the Australian Mathematical Society