Consecutive non-square non-primitive pairs in a finite field

Abstract

Let $q$ be an odd prime power and write \[ \theta_q := \frac{\phi(q-1)}{q-1}. \] If $\theta_q < \tfrac{1}{3}$, or if $\theta_q = \tfrac{1}{3}$ and $q \notin \{7,13,19,25,37\}$, then the finite field $\F$ contains a pair of consecutive elements that are both non-square and non-primitive. This extends a result of Jarso and Trudgian for prime fields $\Fp$, where the same conclusion was obtained under the stronger condition $\theta_p \le \tfrac{1}{4}$. More generally, let $\ell$ be the least odd prime divisor of $q-1$. If $\theta_q \le \tfrac{1}{3}$, then $\F$ contains a pair of consecutive elements that are non-squares and $\ell$th powers, with the sole exceptions $q \in \{7,13,19,25,37,43\}$.