# 数学代写|数论作业代写number theory代考|MATH2031

#### Doug I. Jones

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

Having just seen that the Diophantine equation $x^2+y^2=z^2$ has infinitely many primitive solutions (i.e., solutions in which $\operatorname{gcd}(x, y)=1$, it would be natural to expect that the equations $x^3+y^3=z^3, x^4+y^4=z^4$, and so on would also have infinitely many solutions. As we shall now see in this section and the next, this turns out to be not true, and in fact very strongly not true in the sense that we go from infinitely many solutions to no (nontrivial) solutions for each power $n>2$. This surprising fact was first conjectured in the 17-th century by Pierre de Fermat but was only proved in full generality about 350 years later in 1995 by Andrew Wiles. Wiles’ proof uses methods which are far beyond elementary number theory, but we shall see here that with some considerable effort we are able to use our elementary techniques to conclude that the Diophantine equation $x^4+y^4=z^4$ has no integer solutions beyond the “trivial” ones; i.e., the ones in which $x$ or $y$ is 0 . The exponent $n=4$ turns out to be easier to deal with than $n=3$ since with $n=4$, we are able, as we shall soon see, to make use of Theorem 9.2.

To get the desired result about $x^4+y^4=z^4$, we will first closely examine the related equation $x^4+y^4=z^2$, after which we can apply what we learned about it to achieve our main goal. So here is our central result:

Theorem 9.4. The only integral solutions of the Diophantine equation $x^4+y^4=z^2$ are the trivial solutions $\left{x=0, y, z=\pm y^2\right}$ and $\left{x, y=0, z=\pm x^2\right}$.

Proof. Our proof illustrates the method of “proof by descent” or “Fermat’s method of infinite descent,” as it is sometimes called. The idea is the following: Assume that the equation $x^4+y^4=$ $z^2$ has a non-trivial solution $\left{x_0, y_0, z_0\right}$ and then show that this assumption leads to another non-trivial solution $\left{x_1, y_1, z_1\right}$ with $z_1<z_0$. Repeating this process will then lead to a solution with a smaller $z_2$, and so on. Since starting with $z_0$ cannot produce an infinite strictly decreasing sequence of positive integers, our assumption that the original solution existed must have been false, and the theorem follows. So get ready for the tricky argument needed to produce the first new “smaller” solution $\left{x_1, y_1, z_1\right}$. In the course of this proof it will be necessary to introduce eight new positive integers: ${u, v}$ and later ${r, s}$ when applying Theorem $9.2 ;\left{z_1, t\right}$ and later $\left{x_1, y_1\right}$ when using the fact (already used once in the previous section) that if the product of two relatively prime integers is a perfect square, then each of the numbers must also be a perfect square.

## 数学代写|数论作业代写number theory代考|The Equation xn+yn=zn, n>2

Having now closely examined the specific Diophantine equations $x^2+y^2=z^2$ and $x^4+y^4=z^4$, and having reached totally different conclusions about the existence of integral solutions of them, we now turn to the general case: the equation $x^n+y^n=z^n$. Here is what Pierre de Fermat concluded about its solutions.

Conjecture 9.6. (Fermat’s Conjecture, 1637) If $n>2$, the Diophantine equation $x^n+y^n=z^n$ has no integral solutions except for the trivial solutions when one of $x$ or $y$ is 0 .

So Fermat stated that, even though $x^2+y^2=z^2$ has infinitely many integral solutions which we can explicitly describe (Theorem $9.2$ ), none of the corresponding equations with exponents greater than 2 have any non-trivial solutions. (We proved in the previous section, with considerable effort, that this is true of the equation $x^4+y^4=z^4$.) Fermat claimed to have a proof of his conjecture, but he never wrote his proof down, if indeed he really did have one. Though various special cases of the theorem were settled in the ensuing years (for example, in 1770 Leonhard Euler gave a proof for the case $n=3$ ), a general proof eluded mathematicians for centuries. Fermat’s Conjecture was finally resolved in 1995, when Andrew Wiles at Princeton University gave an extremely complicated proof requiring several hundred pages of very advanced mathematics. There remains hope that someone might discover an elementary proof of Fermat’s Conjecture (i.e., a proof using the kinds of arguments we used in the previous section to settle the matter for the case $x^4+y^4=z^4$ ), but this seems unlikely to occur because after more than 350 years no such elementary proof has been found.

We remark also that Fermat’s Conjecture is often referred to as “Fermat’s Last Theorem.” It has been given this name since Fermat had a habit of writing down conjectures but then not supplying the needed proofs. One by one these other conjectures were proved to be true by other mathematicians as the years passed, but no proof of Conjecture $9.6$ was found until 1995 . Hence it was, for many, many years, his “last theorem.”

# 数论作业代写

## 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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