AI 中文总结
研究模奇素数\(p\)时长度为二和三的连续二次剩余与非剩余模式的精确频率，利用有限域特征和基本性质推导计数公式，对的频率由\(p \pmod 4\)分类，三元组频率依赖\(p \pmod 8\)及雅可比和\(T_p\)。
AI 中文摘要
本文确定了由奇素数\(p\)的二次特征形成的长度为二和三的所有连续符号模式的精确频率。利用有限域上特征和的基本性质，我们推导了在\(\{\pm 1\}\)中呈现特定序列模式的对\((n, n + 1)\)和三元组\((n - 1, n, n + 1)\)的精确计数公式。对的频率完全由\(p\pmod 4\)分类，而三元组对\(p\pmod 8\)和雅可比和\(T_p=\sum_{x\in\mathbb F_p}\chi(x^3 - x)\)表现出更复杂的依赖性。
英文摘要
This paper determines the exact frequency of all consecutive sign patterns of lengths two and three formed by the quadratic character modulo an odd prime $p$. Using basic properties of character sums over finite fields, we derive exact counting formulas for pairs $(n, n+1)$ and triples $(n-1, n, n+1)$ exhibiting specific sequence patterns in \(\{\pm 1\}\). The frequencies of pairs are classified entirely by \(p \pmod 4\), whereas triples exhibit a more complex dependency on \(p \pmod 8\) and the Jacobsthal sum $T_p=\sum_{x\in\mathbb F_p}χ(x^3-x)$.
