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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.21429 |
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| _version_ | 1866917431071473664 |
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| author | Cohen, Stephen D. |
| author_facet | Cohen, Stephen D. |
| 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 $q \in \{7,13,19,25,37,43\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21429 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Consecutive non-square non-primitive pairs in a finite field Cohen, Stephen D. Number Theory 11T30 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 $q \in \{7,13,19,25,37,43\}$. |
| title | Consecutive non-square non-primitive pairs in a finite field |
| topic | Number Theory 11T30 |
| url | https://arxiv.org/abs/2604.21429 |