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

Stephen D. Cohen

Let $q$ be an odd prime power and write \[ θ_q := \frac{ϕ(q-1)}{q-1}. \] If $\hat{I}{¸}_q<\frac{1}{3}$, or if $\hat{I}{¸}_q=\frac{1}{3}$ and $q\notin \{7,13,19,25,37\}$, then the finite f…