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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.21429 |
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Table of Contents:
- Let $q$ be an odd prime power and write \[ θ_q := \frac{ϕ(q-1)}{q-1}. \] If $θ_q < \tfrac{1}{3}$, or if $θ_q = \tfrac{1}{3}$ 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 $θ_p \le \tfrac{1}{4}$. More generally, let $\ell$ be the least odd prime divisor of $q-1$. If $θ_q \le \tfrac{1}{3}$, then $\F$ contains a pair of consecutive elements that are non-squares and $\ell$th powers, with the sole exceptions $q \in \{7,13,19,25,37,43\}$.