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

First we recall a well-known theorem.
Theorem 2.4 Let $a, b, c$ be the lengths of sides of a triangle. Then this triangle is right with hypotenuse $c$ if and only if
$$a^{2}+b^{2}=c^{2}$$

The implication $\Rightarrow$ is the classical Pythagorean Theorem (see Fig. 2.1). The converse implication $\Leftarrow$, which is more important from a practical point of view, is unfortunately omitted in many textbooks.

Let $a^{2}+b^{2}=c^{2}$ for an ordered triple $\langle a, b, c\rangle$ of positive integers. Then this triple is called a Pythagorean and the corresponding triangle from Theorem $2.4$ is called a Pythagorean triangle. Moreover, if $a, b, c$ have no common divisor $d>1$, then $\langle a, b, c\rangle$ is called a primitive Pythagorean triple.

The following theorem is presented in Diophantus’s Arithmetic whose Latin version was published in 1621 by Claude-Gaspard Bachet de Méziriac (1581-1638). It shows Euclid’s fundamental formulae for generating primitive Pythagorean triples.

## 数学代写|数论作业代写number theory代考|Fermat’s Method of Infinite Descent

In this chapter we will introduce Fermat’s method of infinite descent, which is close to the principle of mathematical induction. It relies on the property that a set of positive integers $\mathbb{N}$ is well ordered, which means that any of its non-empty subsets has a smallest element.
Fermat’s method of infinite descent is based on the following theorem.
Theorem 2.7 Let $M$ be a subset of $\mathbb{N}$ and suppose that for an arbitrary $m \in M$ there exists $n \in M$ such that $n<m$. Then the set $M$ is empty.

Proof Assume to the contrary that $M$ is not empty. Since $\mathbb{N}$ is well ordered, there exists its smallest element $m$ of the set $M$. Then by the assumption of the theorem there exists an element $n \in M$, which is smaller than $m$. This is a contradiction of the minimality of $m$. Hence, $M=\emptyset$.

Theorem 2.7 is used mainly in proofs of the non-existence of positive integers with certain properties. We will show its nsefulness on two statements, which Pierre de Fermat himself dealt with around 1640 . The proofs of both statements will illustrate how Fermat’s infinite descent method is actually used. The first statement concerns Pythagorean triangles.
Since $a$ or $b$ in Theorem $2.5$ is even, the area
$$P=\frac{a b}{2}$$
of a Pythagorean triangle is always a positive integer. Now we will apply Theorems $2.5$ and $2.7$ to the proof of the following property.

# 数论作业代写

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

$$a^{2}+b^{2}=c^{2}$$

## 数学代写|数论作业代写number theory代考|Fermat’s Method of Infinite Descent

$$P=\frac{a b}{2}$$

## 有限元方法代写

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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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