发表机构
College of Science, North China University of Technology（华北理工大学科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有限域中连续非平方非本原元组问题，通过得到非平方\(\ell\)次幂连续元组存在性定理并结合有限计算，证明\(\theta_q<4/15\)保证有三个连续NSNP元素，且\(4/15\)最优。
AI 中文摘要
设\(q\)为奇素数幂，令\(\theta_q=\frac{\varphi(q - 1)}{q - 1}\)。\(\mathbb{F}_q\)表示含\(q\)个元素的有限域，若\(\mathbb{F}_q\)中的元素既是非平方元又是非本原元，则称为非平方非本原元（NSNP）。首先得到非平方\(\ell\)次幂连续元组的一般存在性定理，其中\(\ell\)是\(q - 1\)的奇素因子。通过与有限计算相结合，证明当\(\theta_q<4/15\)时保证存在三个连续的NSNP元素，在边界\(\theta_q = 4/15\)时，仅\(\mathbb{F}_{31}\)、\(\mathbb{F}_{61}\)、\(\mathbb{F}_{121}\)为例外，常数\(4/15\)是最优的。
英文摘要
Let $q$ be an odd prime power and put \[ θ_q=\frac{φ(q-1)}{q-1}. \] An element of $\Fq$ is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let $\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\ell$th powers. Combining this estimate with a finite computation, we prove that $θ_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $θ_q=4/15$, the only exceptions are $q\in\{31,61,121\}$. As a further application, we show that for $\operatorname{char}\Fq>3$, the inequality $θ_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $θ_q=8/35$. Thus both constants are best possible.
