Jacobsthal和与四次剩余的排列

模素数下不同双二次剩余之和

Sum of Distinct Biquadratic Residues Modulo Primes

本文证明了关于模素数下双二次多项式不同输出值之和的两个猜想，给出依赖 p 模 8 余数和 c 二次剩余性的八情形公式，并据此分类了所有可能和值的集合。

模素数的连续二次剩余和非剩余模式的精确频率

Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime

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\(T_p\)。

Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $ε> 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(ε)$, the number of prime cubic residues $q < p^{1/2+ε}$ exceeds $p^{ε/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ pr…