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# 数学代写|数论代写Number Theory代考|Christmas Day, 1640

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## 数学代写|数论代写Number Theory代考|Christmas Day, 1640

Bachet’s translation of Diophantus opened a new mathematical world for Fermat, and he spent the rest of his life exploring it. In this chapter we will look at several discoveries that Fermat made within the pages of the Arithmetica, but we begin with a topic that was central in the work of Diophantus: sums of squares.

Pierre de Fermat was one of the greatest mathematicians of all time, yet by today’s standards he was not a working mathematician; in his day such a concept did not even exist, though he might have considered himself a “geometer.” Fermat’s professional career was as a jurist at the High Court in Toulouse and at a nearby court in Castres.

In spite of this full-time job, Fermat did much of the groundbreaking work on tangents and on maxima and minima problems that would lead to the discovery of calculus by Newton and Leibniz later in the seventeenth century. He also made equally important contributions to physics, such as his discovery of the principle of least time, which says that light travels between two points by a path that minimizes the amount of time taken to travel between the two points. This principle, in turn, implies the familiar laws of reflection and refraction for light. Sadly, however, Fermat published almost nothing about his many results during his lifetime.

What we know of the work of Fermat we know only because of two people, his son Clémont-Samuel, and a French friar who lived in a monastery in Paris. After Fermat’s death in 1665 his son spent five years editing his father’s papers before publishing his work in two volumes in 1669 and 1670, the latter being an edition of the Arithmetica by Diophantus that included forty-eight of the marginal notes made by his father in his original copy of Bachet’s translation. This is why we know today of Fermat’s most famous marginal note: “I have a truly marvelous proof of this proposition which this margin is too narrow to contain.” His son included this observation in this 1670 edition.

Marin Mersenne was a French friar living in Paris during the first half of the seventeenth century. More important, he was also the center of a vast network of scientists and mathematicians spread throughout Europe. Much of the excitement and intellectual vigor of that period was due to the correspondence that took place among this circle of friends-many of whom never met; Fermat, for example, simply did not travel. Many of the letters that document this correspondence still exist today, letters written to one another by a truly remarkable group of men that included Mersenne, Descartes, Fermat, Desargues, Pascal, and Roberval. From these letters we have learned much about what Fermat knew, and when he knew it.

## 数学代写|数论代写Number Theory代考|Fermat’s Little Theorem

In this section, we will discuss the theorem that is known as Fermat’s little theorem in order to distinguish it from his “big” theorem-that is, Fermat’s last theorem. He discovered this theorem as a result of trying to factor numbers such as $2^{37}-1$. We will explain why he was interested in doing such a thing later, but for now just imagine how difficult this was. These days, we can factor $2^{37}-1=137438953471$ in the blink of an eye using a computer, and even a hand calculator can do it in a second or two, but in Fermat’s day this was a challenging problem to say the least. Perhaps 137438953471 is prime.

Here is how Fermat factored this large number once he discovered his “little” theorem. This theorem told him that if a prime $p$ divides $2^{37}-1$, then 37 must divide $p-1$. In other words, $p-1=37 n$, that is, $p=37 n+1$. But $p$ is an odd prime, which means that $p=$ $37(2 k)+1=74 k+1$. So now Fermat just tries all primes of the form $74 k+1$ to see whether they divide 137438953471 ; that is, he tries a list of possible prime divisors: $149,223,593, \ldots$, which is a far, far better strategy than simply trying all primes $2,3,5,7,11,13, \ldots$.

In this case, Fermat quickly discovered that $149 \times 137438953471$, but that 223|137438953471, since $137438953471=223$. 616318117 . Of course, Fermat was quite fortunate he found a divisor so early in this list (we will learn later that there could have been at most only about 780 primes on the list for him to check in any event). We will come back to why Fermat was interested in numbers such as $2^{37}-1$ in a moment, but let’s see how it led him to his theorem.

Fermat explained the idea in a letter to Bernard Frénicle de Bessy in 1640 using a geometric progression
$3, \quad 3^2=9, \quad 3^3=27, \quad 3^4=81, \quad 3^5=243, \quad 3^6=729, \ldots$,
where he takes as an example the prime number 13 , and says that 13 divides $27-1$, and then points out two things:
First, that the exponent for $27=3^3$ is 3 , and 3 divides 12 , which is $13-1$
Second, he points out that the next place in the progression where this happens is at 729 , because the exponent of $729=3^6$ is 6 , which is the next multiple of the exponent 3 .

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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。