On Consecutive Non-primitive Elements over Finite Fields

Abstract

In this article, we establish a bound on $\theta_q$ that guarantees the existence of a pair of consecutive non-primitive elements in $\mathbb{F}_q$, with the exceptions $q=4$ and $q=8$. We first derive a sufficient condition for the existence of such a pair using character sums and then obtain the stated bound by considering several cases according to the least prime divisor of $q-1$.