Primitive values of quadratic polynomials in a finite field

Abstract

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field ${\mathbb{F}}_q$ such that $Q(g)$ is also a primitive root, where $Q(x)=a{x}^2+bx+c$ is a quadratic polynomial with $a,b,c\in {\mathbb{F}}_q$ such that $b^2-4ac\ne 0$.

12 pages; code available at https://arxiv.org/src/1803.01435v2/anc/