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# 数学代写|数论代写Number Theory代考|FA20 Prime Numbers

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

A prime number is an integer or a whole number that has only two factors 1 and itself. In other words, a prime number can be divided only by 1 and itself. Also primes are greater than 1. For example, 3 is prime as it fails to be divided evenly by any number except for 1 and 3. However, 6 is not because it can be evenly divided by 2 and 3 .

The largest known prime number is $2^{82,589,933}-1$, a number which has $24,862,048$ digits when written in base 10. It was discovered by Patrick Laroche of the great internet Mersenne Prime search. Euclid recorded a proof that there does not exist any largest prime number and many mathematicians continue to search for large prime numbers.

In 1978, few researchers used prime numbers to scramble and unscramble coded messages. This early form of encryption smoothen the way for Internet security, putting prime numbers at the heart of electronic commerce. Publickey cryptography, or RSA encryption, has simplified secure transactions of all times. The security of this type of cryptography depends on the difficulty of factoring large composite numbers, which is the product of two large prime numbers. Also, in modern banking security systems depend on the fact that large composite numbers cannot be factored in a short amount of time. Two

primes are considered secure if they are 2,048 bits long, because the product of these two primes would be about 1, 234 decimal digits.

Prime numbers have shown its existence in nature. Cicadas insect spend most of their time hiding, only reappearing to mate every 13 or 17 years. Why this particular number? Scientists invented that cicadas reproduce in cycles that minimize possible interactions with predators. Any predator reproductive cycle that divides the cicada’s cycle evenly means that the predator will hatch out the same time as the cicada at some point. For instance, if the cicada evolved towards a 12-year reproductive cycle, predators who reproduce at the $2,3,4$ and 6 year intervals would find themselves with plenty of cicadas to eat. By using a reproductive cycle with a prime number of years, cicadas would be able to minimize contact with predators. Simulation models of 1,000 years of cicada evolution prove that there is a major advantage for reproductive cycle times based on primes.

## 数学代写|数论代写Number Theory代考|Primes \& Fundamental Theorem of Arithmetic

Positive divisors of an integer have a great importance in the study of number theory. The integer 1 has only one positive divisor which is 1 itself. Any other integers has more than one divisor. At Least two divisors of them are 1 and the integer itself. There are integers which have divisors other than 1 and itself. The numbers which have only two divisors 1 and itself are called prime numbers.
Definition 3.2.1. An integer $p>1$ is said to be a prime number or prime if its only divisors are 1 and $p$ itself.

An integer which is not prime is known to be a composite number, having more than two(what are those?) divisors.

Among the first ten positive integers 2,3,5,7 are prime numbers whereas $4,6,8,9,10$ are examples of composite numbers. Here 1 is a special type of integer which is neither prime nor composite. Here the study of prime numbers starts with the study of prime divisors. Here 5 is prime where $5 \nmid 3$ but $5 \mid 5$ itself together implies $5 \mid 15$, leads us to the following theorem:

Theorem 3.2.1. An integer $p>1$ is prime if and only if $p \mid a b$ implies $p \mid a$ or $p \mid b$

Proof. Let $p$ be a prime number such that for any two integers $a$ and $b, p \mid a b$ holds. If $p \mid a$, then we are done. Let $p \nmid a$ then the only divisors of $p$ are 1 and $p$ itself. As $p$ is prime we have $\operatorname{gcd}(p, a)=1$ implies there exists integers $r, t$ such that $1=r p+a t$. Then $b=b r p+t(a b)$. Now $p \mid a b$ and $p \mid p r b$ imply $p \mid b$.

Conversely, let $p$ satisfy the condition and $q, r$ be any integers such that $p=q r$ where $q<p$. Thus $p \mid q r$ and by the condition we can say either $p \mid q$ or $p \mid r$. But $q \mid p$ shows $p \mid r$ only. Therefore $r=p t$ for some integer $t$. Hence $p=q r=q p t$ implies $q t=1$ implies $q=1$. So 1 and $p$ are only divisors of $p$. This shows $p$ is prime.

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

1978 年，很少有研究人员使用素数来打乱和解读编码信息。这种早期的加密形式为 Internet 安全铺平了道路，将质数置于电子商务的核心。公钥加密或 RSA 加密一直以来都简化了安全交易。这种密码学的安全性取决于分解大复合数的难度，复合数是两个大素数的乘积。此外，现代银行安全系统依赖于这样一个事实，即不能在短时间内分解大的复合数。二

## 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。