Graphical patterns in quadratic residues

Sums of Quadratic residues and nonresidues

It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.

The largest cycles consist by the quadratic residues and Fermat primes

This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.

On the density of primes with a set of quadratic residues or non-residues in given arithmetic progression

Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$ is at least 2 and $U$ is the union formed by taking certain arithmetic progressions of length $s$ from each element of $\mathcal{A}$, we calculate the asymptotic density of the set of all prime numbers $p$ such that $U$ is a set of quadratic residues (respectively, quadratic non-residues) of $p$.

On consecutive quadratic non-residues: a conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than the square root of p. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds the square root of p.

Products of quadratic residues and related identities

In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.

Two_Generalizations_for_Quadratic_Residue_Codes_over_Finite_Fields

It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper generalizes the constructions for the quadratic residue code with length $p$ to be the length $n=p_{1}\cdots p_{g}$, and to be the case $m$-th residue codes with length $p$ over finite fields, where $m\geq 2$ is a positive integer. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, and then the corresponding …

Quadratic Residues and Non-residues in Arithmetic Progression

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this are also established, and some problems connected with these results that are worthy of further study are discussed.

Quadratic Non-residues in Short Intervals

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

Quadratic Residues and Non-Residues: Selected Topics

Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the Disquisitiones deals with quadratic residues and nonresidues. Beginning with these fundamental contributions of Gauss, the study of quadratic residues and nonresidues has subsequently led directly to many of the key ideas …

The distribution of spacings between quadratic residues, II

We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper extends the result to arbitrary q.

Conditional bounds for the least quadratic non-residue and related problems

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $L$-functions at $s=1$. In particular, we derive explicit upper and lower bounds for $L(1,χ)$ and $ζ(1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. Finally, we improve the best known theoretical bo…

Quadratic residues and quartic residues modulo primes

In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{cases}-(A^2+4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=1, \\(-A^2-4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2+5ij+2j^2…

Applications of Lerch's theorem and permutations concerning quadratic residues

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary $k$-th power residue modulo $p$ and primitive roots of a power of $p$. Finally, we discuss some permutation problems con…

Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $ ε\in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-ε} \mbox{ and } \frac{ϕ(p-1)}{p-1} \leq \frac{1}{2} - ε, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.

Quadratic residues and difference sets

It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, …

二次 Weyl 群多重 Dirichlet 级数的留数

Residues of quadratic Weyl group multiple Dirichlet series

本文研究有限不可约根系的 Chinta-Gunnells 平均在单短根极除子处的留数，给出由正交根子系平均表示的显式公式，并推广到有理函数域与 Gauss 域上的二次 Weyl 群多重 Dirichlet 级数。

Characterizations of the $d$th-power residue matrices over finite fields

In a recent paper of the author with D. Dummit and H. Kisilevsky, we constructed a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and proved a simple criterion characterizing such matrices. We then analyzed the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively. In this paper, the goal is to construct and study the fi…

最小二次剩余与二次型表示的整数

The least quadratic residue and integers represented by quadratic forms

该研究针对模n的最小非平凡既约二次剩余ℓ(n)，从n的大小与素因子数维度建立近乎最优界，构造出ℓ(n)异常大的模，并将结果应用于推导有界判别式二元二次型表示不超过N的正整数的速率界。

Fekete polynomials, quadratic residues, and arithmetic

Fekete polynomials associate with each prime number $p$ a polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo $p$. These polynomials were first considered in the 19th century in relation to the studies of Dirichlet $L$-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomial…

$(1-2u^3)$-constacyclic codes and quadratic residue codes over $\mathbb{F}_{p}[u]/\langle u^4-u\rangle$

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^4$ is def…

Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues

In 2000, A. Tripathi used generating functions to obtain a formula for the number of non-negative solutions (x,y) of the equation ax + by = n where a, b and n are given positive integers. We generalize this procedure for the number of solutions of the equation ax + by + cz = n. The formula leads us to a surprising connection between the number of solutions of the equation ax + by + cz = n and quadratic residues. As a consequence of our work, we are able to prove the equivalence between two fundamental results by Gauss and Sylvester in the nineteenth century which are generally viewed to be ind…

实二次域的算术 Dijkgraaf-Witten 不变量、二次剩余图与密度公式

Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

本文针对实二次域整数环，利用素数间的二次剩余图简化了 Hirano 的模 2 算术 Dijkgraaf-Witten 不变量公式，回答了 Ken Ono 的问题，并进一步给出了相应不变量的密度公式。

Prescribing the binary digits of squarefree numbers and quadratic residues

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cu…

Quadratic residues and a new infinity of orders for which a conjecture of Ryser about Circulant Hadamard matrices holds

For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $ω(h) =k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result implies that for all of the odd numbers $h$, with $1< h < 10^{13}$ there is no Circulant Hadamard matrix of order $n=4h^2.$

幂剩余、数字展开与相对类数

Power residues, digit expansions and relative class numbers

本文综述了模素数幂剩余分布与相对类数的联系，重点在二次和六次幂剩余，狄利克雷类数公式及几何在推广中起关键作用。

广义Sato-Tate与二次剩余

Generalized Sato-Tate and quadratic residues

本文证明广义Sato-Tate猜想可用于精确刻画大素数模下连续二次剩余的分布规律。

A new theorem on quadratic residues modulo primes

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac ap\right)=\varepsilon\right\}\right|=\frac{3-(\frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with $\{x^2+b\}_p>\{ax^2+b\}_p$, and $\{m\}_p$ with $m\in\mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

We consider the problem of bounding the dimension of Hilbert cubes in a finite field $\mathbb{F_p}$ that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is $O_ε(p^{1/8+ε})$ for any $ε> 0$, matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

When the number of divisors is a quadratic residue

Let $q > 2$ be a prime number and define $λ_q := \left( \fracτ{q} \right)$ where $τ(n)$ is the number of divisors of $n$ and $\left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When $τ(n)$ is a quadratic residue modulo $q$, then $\left( λ_q \star \mathbf{1} \right) (n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function $λ_q \star \mathbf{1}$ to the well known average order of $τ$. The proof reveals that the results depend heavily on the value of $\left( \frac{2}{q} \right)$. A bound for short sums in the case $q=5$ is a…

Distribution of residues modulo $p$ using the Dirichlet's class number formula

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number formula for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$.

算术级数中二次型的零剩余类偏向

Biases Towards the Zero Residue Class for Quadratic Forms in Arithmetic Progressions

本文研究了二元二次型表示整数向零剩余类的偏向，证明该偏向多源于渐近展开的次级项，对未证情形提出猜想，并给出相关分布结果与数值数据。

超越黎曼假设界限：算术级数中最小素数与最小二次非剩余的对关联方法

Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue

研究算术级数中最小素数与最小二次非剩余问题，结合广义黎曼假设中零点水平位置与对关联视角，在相关假设下建立更精确估计，为狄利克雷L函数对关联现象与经典问题提供新联系。

Sumsets in quadratic residues

We describe all sets $A \subseteq \F_p$ which represent the quadratic residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\hat{+} A$. Also, we consider the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$ and prove that $A$ has a structure in the situation.

Classicality of overconvergent Hilbert eigenforms: Case of quadratic residue degree

Let $F$ be a quadratic real field, $p$ be a rational prime inert in $F$. In this paper, we prove that an overconvergent $p$-adic Hilbert eigenform for $F$ of small slope is actually a classical Hilbert modular form.

A computable formula for the class number of the imaginary quadratic field $Q(\sqrt{-p}), \ p=4n-1$

A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are also found.

A conjecture of Sárközy on quadratic residues, II

Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. Sárközy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with $|\mathcal{A}|,|\mathcal{B}|\geqslant2$ satisfy $\mathcal{A}+\mathcal{B}=\mathcal{R}_p$. In this paper, we show that if such subsets $\mathcal{A},\mathcal{B}$ do exist, then there are at least $(\log 2)^{-1}\sqrt p-1.6$ elements in $\mathcal{A}+\mathcal{B}$ that have unique representations and one should have \begin{align*} \frac{1}{4}\sqrt{p}< |\mathcal{A}|,|\mat…

A note on Linnik's Theorem on quadratic non-residues

We present a short, self-contained, and purely combinatorial proof of Linnik's theorem: for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that for any $N$, there are at most $C_\varepsilon$ primes $p \leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^\varepsilon$.

若干涉及模素数二次剩余的行列式

Some determinants involving quadratic residues modulo primes

该研究计算了若干模素数二次剩余相关的行列式，证明了一类含正切函数的矩阵行列式在特定素数和整数条件下的精确取值公式，并提出了后续研究的相关猜想。

Quadratic residue patterns and point counting on K3 surfaces

Quadratic residue patterns modulo a prime are studied since 19th century. We state the last unpublished result of Lydia Goncharova, reformulate it and prior results in terms of algebraic geometry, and prove it. The core of this theorem is an unexpected relation between the number of points on a K3 surface and that on a CM elliptic curve.

Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…

The stochasticity parameter of quadratic residues

Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of $\mathbb{Z}/M\mathbb{Z}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper we study the stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$. We present a method which allows to find the asymptotics of $S(R_M)$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,\mathbb{Z}/M\mathbb{Z})$ the average value of $S(U)$ over all subsets $U\subseteq \mathbb{Z}/M\mathbb{Z}$ of size $k$, whic…

某些扩展二次剩余码中的特殊设计

Exceptional designs in some extended quadratic residue codes

本文证明了长度为14的扩展三元二次剩余码和长度为18的扩展四元二次剩余码中存在3-设计。

On Cubic Residue Matrices

The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices with interesting and predictable determinants.

Counting Additive Decompositions of Quadratic Residues in Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. Sárközy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

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…

Fourier优化、最小二次非剩余与算术级数中的最小素数

Fourier optimization, the least quadratic non-residue, and the least prime in an arithmetic progression

本文提出一种Fourier优化框架，在GRH假设下改进了估计模素数的最小二次非剩余和算术级数中最小素数这两个数论经典问题的渐近界。

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\(T_p\)。

Proof of some conjectures involving quadratic residues

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le j<k\le(p-1)/2}(e^{2πiaj^2/p}+e^{2πiak^2/p}) \\=&\begin{cases}1&\text{if}\ p\equiv1\pmod 8,\\\left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}&\text{if}\ p\equiv5\pmod8,\end{cases} \end{align*} and that \begin{align*}&\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \{aj^2\}_p>\{ak^2\}_p\right\}\right| \\&+\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \{…

On a properness of the Hilbert eigenvariety at integral weights: the case of quadratic residue fields

Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is 1 or 2. In this paper, we show that the eigenvariety for Res_{F/Q}(GL_2), constructed by Andreatta-Iovita-Pilloni, is proper at integral weights for p>=3. We also prove a weaker result for p=2.

论二次剩余的分布

On the distribution of quadratic residues

该研究运用加性组合方法探究素数域平方数集合的分布，得出当前该集合间隔数的最佳上界，且将结果拓展至小倍增集合，推进了二次剩余分布相关研究。

二次剩余与多米诺骨牌平铺

Quadratic residues and domino tilings

该研究发现多米诺骨牌平铺数公式与Legendre符号公式的相似性，探究二者关联并证明了用多米诺骨牌平铺表示Jacobi符号的公式，搭建起数论与组合数学间的新联系。

长度为4和5的二次剩余模式

Quadratic residue patterns of length 4 and 5

该研究将二次剩余模式结果推广到特定字母表上长度为4和5的任意单词，通过低亏格超椭圆曲线的弗罗贝尼乌斯迹得到模式出现次数公式，并利用广义佐藤 - 泰特方法描述l = 4时归一化误差项的极限分布。

实二次域中的三次二次剩余符号

On Triple Quadratic Residue Symbols in Real Quadratic Fields

针对窄类群平凡的实二次域，引入三次二次剩余符号，通过模2算术三重Milnor不变量定义，并刻画其在8次二面体扩张中的分解律，推广了Rédei的三次符号。

Density of non-residues in Burgess-type intervals and applications

We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.