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# 数学代写|数论代写Number Theory代考|MATH3170 Introduction

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## 数学代写|数论代写Number Theory代考|Introduction

In this final chapter, we hope to inspire students (and instructors) to continue and extend their interest in mathematics by considering and studying a number of open problems that are directly related to the material in this text. Often students of mathematics believe that “everything has been solved” and that they are simply learning about those solutions, but that is not the case. In particular, as we shall see, although there are various mysteries about prime numbers which mathematicians would love to unravel, thus far they have been unable to do so.

Before embarking on our brief exploration of open problems in number theory, we bring up two problems which are not open but which have very different histories. The first of these is, quite simply, Are there infinitely many primes? This one was answered in the affirmative long ago by Euclid (around $250 \mathrm{BC}$ ) when he gave the first known proof of this fact, as follows: Suppose the set of primes is finite, i.e., suppose $\left{2,3,5, \cdots, p_n\right}$ is the complete set of $n$ primes. Consider the number $N=\left(2 \cdot 3 \cdot 5 \cdots p_n\right)+1$. If $N$ were divisible by one of our primes $p_k$, then $1=N-\left(2 \cdot 3 \cdot 5 \cdots p_n\right)$ would also be divisible by $p_k$, which is impossible. Hence $N$ must be divisible some new prime, which contradicts the assertion that we had a list of all of the primes. We conclude then that the set of primes is infinite. There have been, in the ensuing centuries, many new proofs of this fact devised, but none as elegant as Euclid’s original proof.

On the other hand, an example of a problem which remained unsolved for a very long time before finally being resolved is Fermat’s Last Theorem, which we discussed in Chapter 9. The problem, first asked by Fermat in 1637, is, If $n$ is a positive integer greater than 2, are there are any solutions in positive integers of the Diophantine equation $x^n+y^n=z^n$ ? In the years since special cases of the problem were resolved (for example, in that chapter, and with significant effort, we showed that the case $x^4+y^4=z^4$ has no such solutions), but it was not until 1995, about 360 years after Fermat made his conjecture, that Andrew Wiles (1953 – ) was able to solve the general problem: For every $n>2, x^n+y^n=z^n$ has no non-trivial integer solutions. So mathematicians chipped away at that problem for a long time before there was finally full success. Will this be what happens with all the open problems we list below, or will some of them never be resolved? Time will tell.

## 数学代写|数论代写Number Theory代考|Open Problems

In 1644 Marin Mersenne (1588 – 1648) conjectured that there were infinitely many primes of the form $M_n=2^n-1$ where $n$ is a positive integer. Numbers of the form $2^n-1$ are called Mersenne numbers, and if for some $n$ this number $M_n$ is prime, then $M_n$ is called a Mersenne prime. Let’s look at the value of the first ten Mersenne numbers and see if any patterns emerge:

Since 511 is divisible by 7 and 1023 is divisible by 3 , we see that the numbers $M_1, M_4, M_6, M_8, M_9$, and $M_{10}$ are all composite (i.e., not prime). This could lead us to make the following conjecture: If $n$ is composite, then $M_n$ is also composite. This conjecture turns out to be true, and we ask you to prove this in Problem 11.1. On the other hand, once we check that 127 is prime, we see that $M_2, M_3$, $M_5$, and $M_7$ are prime, which tempts us to make this conjecture: If $n$ is prime, then $M_n$ is also prime. However, this conjecture is immediately proven to be false since $M_{11}=2047=23 \cdot 89$. It turns out, for example, that for the 15 primes $p$ up to 50 , eight of them satisfy that $M_p$ is a Mersenne prime (besides the four already listed, they are $M_{13}, M_{17}, M_{19}$, and $M_{31}$ ) and the other seven are not Mersenne primes.

Mersenne’s conjecture remains unsettled. As of the time of this writing there are only 51 Mersenne primes known, the largest being $2^{82,589,933}-1$, which has $24,862,048$ decimal digits. These very large Mersenne primes, quite obviously, have been discovered using computers. If you enjoy computer programming and might be interested in helping with the search for other Mersenne primes, you can do so by joining GIMPS (Great Internet Mersenne Prime Search) online. However, finding larger and larger Mersenne primes cannot settle our conjecture; that can be done only through a mathematical proof, and such a proof has not been discovered as of yet.

## 数学代写|数论代与Number Theory代考|Open Problems

1644 年，马林梅秫 (Marin Mersenne) (1588-1648) 猜想有无穷多个质数的形式 $M_n=2^n-1$ 在棴里 $n$ 是一个正整数。表格 否出现了任何模式:

## 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。