# 数学代写|数论作业代写number theory代考|MAST90136

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

By the Euler-Fermat Theorem $2.21$ and Theorems $2.18$ and $2.22$ we have: if $a$ and $n$ are coprime positive integers, then $\operatorname{ord}{n} a \mid \phi(n)$ and there exists a positive integer $b$ such that $\operatorname{ord}{n} b=\phi(n)$ for $n=2$ or 4 or $p^{k}$ or $2 p^{k}$, where $p$ is an odd prime and $k \in \mathbb{N}$. As we shall see later, the maximum possible order modulo $n$ can be significantly lower than $\phi(n)$, if $n$ is not a power of a prime number (cf. Table 13.3). To this end we introduce the Carmichael lambda function $\lambda(n)$, which first appeared in [62]. Its definition is actually a mere modification of the Euler totient function $\phi(n)$.

For any positive integer $n$ the Carmichael lambda function $\lambda(n)$ is defined as follows:
\begin{aligned} \lambda(1) &=1=\phi(1) \ \lambda(2) &=1=\phi(2) \ \lambda(4) &=2=\phi(4) \ \lambda\left(2^{k}\right) &=2^{k-2}=\frac{1}{2} \phi\left(2^{k}\right) \text { for } k \geq 3 \ \lambda\left(p^{k}\right) &=(p-1) p^{k-1}=\phi\left(p^{k}\right) \text { for any odd prime } p \text { and } k \geq 1 \ \lambda\left(p_{1}^{k_{1}} p_{2}^{k_{2}} \cdots p_{r}^{k_{r}}\right) &=\left[\lambda\left(p_{1}^{k_{1}}\right), \lambda\left(p_{2}^{k_{2}}\right), \ldots, \lambda\left(p_{r}^{k_{r}}\right)\right] \end{aligned}
where $p_{1}, p_{2}, \ldots, p_{r}$ are different primes and $k_{i} \in \mathbb{N}$ for $1 \leq i \leq r$.
The values of $\lambda(n)$ for $n \leq 32$ are listed in Table 13.3. By Theorems $2.22$ and $2.23$, if $p$ is a prime and $k \in \mathbb{N}$, then $\lambda\left(p^{k}\right)$ is equal to the maximum possible order modulo $p^{k}$. From the definition of $\lambda(n)$ we also see that
$$\lambda(n) \mid \phi(n)$$ for all $n \in \mathbb{N}$ and that $\lambda(n)=\phi(n)$ if and only if $n \in\left{1,2,4, p^{k}, 2 p^{k}\right}$, where $p$ is an odd prime and $k \in \mathbb{N}$ (cf. Theorem 2.22).

Nótice that $\lambda(n)$ cân bé much smallér than $\phi(n)$ if $n$ has many primé faccồs. Fôr instance, let $n=2^{6} \cdot 11 \cdot 17 \cdot 41=490688$. Then
$$\lambda(n)=\left[\lambda\left(2^{6}\right), \lambda(11), \lambda(17), \lambda(41)\right]=[16,10,16,40]=80,$$
while
$$\phi(n)=\phi\left(2^{6}\right) \phi(11) \phi(17) \phi(41)=32 \cdot 10 \cdot 16 \cdot 40=204800$$

## 数学代写|数论作业代写number theory代考|Legendre and Jacobi Symbol

The French mathematician Adrien-Marie Legendre (1752-1813) introduced a very useful symbol $\left(\frac{a}{p}\right) \in{-1,0,1}$ into number theory. Before we define it and discuss its properties, we define quadratic residues and nonresidues modulo a prime.
Let $n \geq 2$ and $a$ be coprime integers. If the quadratic congruence
$$x^{2} \equiv a \quad(\bmod n)$$
has a solution $x$, then $a$ is called a quadratic residue modulo n. Otherwise, $a$ is called a quadratic nonresidue modulo $n$.
Let $p$ be an odd prime. Then the Legendre symbol $\left(\frac{a}{p}\right)$ is defined as follows
1 , if $a$ is a quadratic residue modulo $p$,
$-1$, if $a$ is a quadratic nonresidue modulo $p$,
0 , if $p \mid a .$
The quadratic character $a$ with respect to a prime number $p$ is thus expressed via the Legendre symbol. It is obvious that the sequence $\left(\left(\frac{a}{p}\right)\right)_{a=0}^{\infty}$ is periodic. For example,\begin{aligned} &\left(\frac{0}{7}\right)=0,\left(\frac{1}{7}\right)=1,\left(\frac{2}{7}\right)=1,\left(\frac{3}{7}\right)=-1,\left(\frac{4}{7}\right)=1,\left(\frac{5}{7}\right)=-1,\left(\frac{6}{7}\right)=-1 \ &\left(\frac{7}{7}\right)=0,\left(\frac{8}{7}\right)=1,\left(\frac{9}{7}\right)=1, \ldots . \end{aligned}
If $p$ is an odd prime, then the number of quadratic residues is equal to the number of quadratic nonresidues, which is equal to $\frac{1}{2}(p-1)$ (for a proof see e.g. [327, p. 279]). Values of the Legendre symbol for $p=17$ are marked in Fig. 2.6.

The Legendre symbol plays an important role in testing prime numbers and in calculating primitive roots. In Sects. 5.2, 7.13, 11.5, etc., we shall see some of its practical applications. Leonhard Euler proposed a simple method how to calculate the Legendre symbol $\left(\frac{a}{p}\right)$.

# 数论作业代写

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

$$\lambda(1)=1=\phi(1) \lambda(2) \quad=1=\phi(2) \lambda(4)=2=\phi(4) \lambda\left(2^{k}\right) \quad=2^{k-2}$$

$$\lambda(n) \mid \phi(n)$$

$$\lambda(n)=\left[\lambda\left(2^{6}\right), \lambda(11), \lambda(17), \lambda(41)\right]=[16,10,16,40]=80,$$

$$\phi(n)=\phi\left(2^{6}\right) \phi(11) \phi(17) \phi(41)=32 \cdot 10 \cdot 16 \cdot 40=204800$$

## 数学代写|数论作业代写number theory代考|Legendre and Jacobi Symbol

$\left(\frac{a}{p}\right) \in-1,0,1$ 进入数论。在我们定义它并讨论它的性质之前，我们先定义以䋏数为模的

$$x^{2} \equiv a \quad(\bmod n)$$

1 ，如畢 $a$ 是二次余数模 $p$ ，
$-1$ ， 如果 $a$ 是二次非残差模 $p$ ，
0 ，如果 $p \mid a$.

$$\left(\frac{0}{7}\right)=0,\left(\frac{1}{7}\right)=1,\left(\frac{2}{7}\right)=1,\left(\frac{3}{7}\right)=-1,\left(\frac{4}{7}\right)=1,\left(\frac{5}{7}\right)=-1,\left(\frac{6}{7}\right)=$$ 如 [327, p. 279]) 。勒让德符号的值 $p=17$ 标记在图 $2.6$ 中。

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## 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。

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