## 数学代写|数论作业代写number theory代考|MATH 1001

2022年7月26日

couryes-lab™ 为您的留学生涯保驾护航 在代写数论number theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数论number theory代写方面经验极为丰富，各种代写数论number theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
couryes™为您提供可以保分的包课服务

## 数学代写|数论作业代写number theory代考|Wilson’s Theorem

John Wilson (1741-1793) was an outstanding British mathematician at the University of Cambridge. However, the following theorem that bears his name was not discovered by him.

Theorem 3.6 (Wilson) A number $p>1$ is prime if and only if
$$(p-1) ! \equiv-1 \quad(\bmod p) .$$
In 1770 Edward Waring first published in Meditationes algebraicae on p. 288 the implication $\Rightarrow$ without any proof and attributed it to John Wilson. He literally wrote that if $p$ is a prime number, then the sum of $(p-1) !+1$ is divisible by $p$. At that time, the concept of congruences had not been introduced yet. According to Hardy and Wright $[138$, p. 81$]$ the implication $\Rightarrow$ was already known to Gottfried Wilhelm Leibniz (1646-1716) in somewhat modified form. The converse implication $\Leftarrow$ was later proved by Joseph-Louis Lagrange in 1773. Therefore, sometimes Theorem $3.6$ is called the Wilson-Lagrange Theorem.

Let us further note that the assumption of $p>1$ is omitted in many textbooks. For $p=1$ the congruence (3.5) is satisfied, but 1 is by definition not a prime number.
Proof of Theorem 3.6. $\Rightarrow$ : If $p=2$, then congruence (3.5) obviously holds. So let $p>2$ be a prime and let $a$ be an arbitrary positive integer less than $p$. By Theorem $3.4$ there exists exactly one positive integer $b<p$ such that $a b \equiv 1(\bmod p)$. From Theorem $3.5$ we get that if $a b \equiv 1$ (mod $p$ ) and $h \equiv a$ (mod $p$ ), then either $a \equiv 1$ $(\bmod p)$, or $a=p-1(\bmod p)$. From this it follows that the integers $2,3, \ldots, p-$ 2 can be reordered as the progression $a_{2}, a_{3}, \ldots, a_{p-2}$ so that in pairs we have
$$a_{i} a_{i+1} \equiv 1 \quad(\bmod p)$$
for $i=2,4,6, \ldots, p-3$. Between 2 and $p-2$ there are exactly $p-3$ numbers, which is an even number. Therefore,
$$(p-1) ! \equiv 1 \cdot(p-1) a_{2} \cdots a_{p-2} \equiv(p-1) 1^{(p-3) / 2} \equiv-1 \quad(\bmod p)$$

## 数学代写|数论作业代写number theory代考|Dirichlet’s Theorem

In 1837 Peter Gustav Lejeune Dirichlet (1805-1859) published an interesting theorem that uses very sophisticated analytical methods in number theory.

Theorem 3.10 (Dirichlet) Let $a, d \in \mathbb{N}$ be coprime integers. Then there exist infinitely many primes in the arithmetic progression
$$a, a+d, a+2 d, a+3 d, \ldots$$
A proof of this statement is in the seminal paper by Peter Gustav Lejeune Dirichlet [94]. Theorem $3.10$ can be equivalently formulated so that the set
$$S={p \in \mathbb{P} ; p \equiv a(\bmod d)}$$
has infinitely many elements. Moreover, the density of $S$ in the set of primes $\mathbb{P}$ is equal to $1 / \phi(d)$, where $\phi$ is the Euler totient function, i.e.,
$$\lim _{x \rightarrow \infty} \frac{\mid{p \in \mathbb{P} ; p \equiv a \quad(\bmod d) \text { and } p \leq x} \mid}{|{p \in \mathbb{P} ; p \leq x}|}=\frac{1}{\phi(d)} .$$
A proof of this statement can be found e.g. in Ireland and Rosen [151, pp. 251261]. However, one of the most beautiful and at the same time most surprising mathematical results from the beginning of the 21 st century is the Green-Tao theorem published in Annals of Mathematics, see Ben Green and Terence Tao [126].

Theorem 3.11 (Green-Tao) For any positive integer $k$ there exists an arithmetic progression of length $k$ consisting solely of primes.

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Wilson’s Theorem

$$(p-1) ! \equiv-1 \quad(\bmod p) .$$
1770 年，Edward Waring 首次在 Meditationes algebraicae 上发表。288的寓意 $\Rightarrow$ 没有 任何证据并将其归因于约翰威尔逊。他从字面上写道，如果 $p$ 是一个㸹数，那么和
$(p-1) !+1$ 可以被 $p$. 当时，还没有引入同余的概念。根据哈代和赖特 $[138$ ，页。81]含 义 $\Rightarrow$ Gottfried Wilhelm Leibniz (1646-1716) 已经知道它的形式有所修改。反义词后来 由约瑟夫-路易斯拉格朗日在 1773 年证明。因此，有时定理3.6称为威尔逊-拉格朗日定 理。

$$a_{i} a_{i+1} \equiv 1 \quad(\bmod p)$$

$$(p-1) ! \equiv 1 \cdot(p-1) a_{2} \cdots a_{p-2} \equiv(p-1) 1^{(p-3) / 2} \equiv-1 \quad(\bmod p)$$

## 数学代写|数论作业代写number theory代考|Dirichlet’s Theorem

1837 年，Peter Gustav Lejeune Dirichlet (1805-1859) 发表了一个有趣的定理，该定理在 数论中使用了非常巹杂的分析方法。

$$a, a+d, a+2 d, a+3 d, \ldots$$

$$S=p \in \mathbb{P} ; p \equiv a(\bmod d)$$

$$\lim _{x \rightarrow \infty} \frac{\mid p \in \mathbb{P} ; p \equiv a \quad(\bmod d) \text { and } p \leq x \mid}{|p \in \mathbb{P} ; p \leq x|}=\frac{1}{\phi(d)} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。