# 数学代写|离散数学作业代写discrete mathematics代考|MATH-UA120

## 数学代写|离散数学作业代写discrete mathematics代考|Euclid’s Algorithm

Euclid’s ${ }^4$ algorithm is one of the oldest known algorithms, and it provides a procedure for finding the greatest common divisor of two numbers. It appears in Book VII of Euclid’s Elements, and the algorithm was known prior to Euclid (Fig. 3.8).
Lemma
Let $a, b, q$ and $r$ be integers with $b>0$ and $0 \leq r<b$ such that $a=b q+r$. Then $\operatorname{gcd}(a, b)=\operatorname{gcd}(b, r)$.

Proof Let $K=\operatorname{gcd}(a, b)$ and let $L=\operatorname{gcd}(b, r)$, and we therefore need to show that $K=L$. Suppose $m$ is a divisor of $a$ and $b$, then as $a=b q+r$ we have $m$ is a divisor of $r$ and so any common divisor of $a$ and $b$ is a divisor of $r$.

Similarly, any common divisor $n$ of $b$ and $r$ is a divisor of $a$. Therefore, the greatest common divisor of $a$ and $b$ is equal to the greatest common divisor of $b$ and $r$.

Theorem $3.4$ (Euclid’s Algorithm) Euclid’s algorithm for finding the greatest common divisor of two positive integers $a$ and $b$ involves applying the division algorithm repeatedly as follows:
$$\begin{array}{lc} a=b q_0+r_1 & 0<r_1<b \ b=r_1 q_1+r_2 & 0<r_2<r_1 \ r_1=r_2 q_2+r_3 & 0<r_3<r_2 \end{array}$$
\begin{aligned} &r_{n-2}=r_{n-1} q_{n-1}+r_n \quad 0<r_n<r_{n-1} \ &r_{n-1}=r_n q_n . \end{aligned}

## 数学代写|离散数学作业代写discrete mathematics代考|Distribution of Primes

We already have shown that there are an infinite number of primes. However, most integer numbers are composite and a reasonable question to ask is how many primes are there less than a certain number. The number of primes less than or equal to $x$ is known as the prime distribution function (denoted by $\pi(x)$ ) and it is defined by $\pi(x)=\sum_{p \leq x} 1$ (where $p$ is prime).
The prime distribution function satisfies the following properties:
(i) $\lim {x \rightarrow \infty} \frac{\pi(x)}{x}=0$ (ii) $\lim {x \rightarrow \infty} \pi(x)=\infty$.
The first property expresses the fact that most integer numbers are composite, and the second property expresses the fact that there are an infinite number of prime numbers.

There is an approximation of the prime distribution function in terms of the logarithmic function $\left({ }^x / \ln x\right)$ as follows:
$$\lim _{x \rightarrow \infty} \frac{\pi(x)}{x / \ln x}=1 \text { (Prime Number Theorem). }$$
The approximation $x / \ln x$ to $\pi(x)$ gives an easy way to determine the approximate value of $\pi(x)$ for a given value of $x$. This result is known as the Prime Number Theorem, and Gauss originally conjectured this theorem.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Euclid’s Algorithm

$$a=b q_0+r_1 \quad 0<r_1<b b=r_1 q_1+r_2 \quad 0<r_2<r_1 r_1=r_2 q_2+r_3 \quad 0<r_3<r_2$$
$$r_{n-2}=r_{n-1} q_{n-1}+r_n \quad 0<r_n<r_{n-1} \quad r_{n-1}=r_n q_n .$$

## 数学代写|离散数学作业代写discrete mathematics代考|Distribution of Primes

(i) $\lim x \rightarrow \infty \frac{\pi(x)}{x}=0$ (二) $\lim x \rightarrow \infty \pi(x)=\infty$.

$$\lim _{x \rightarrow \infty} \frac{\pi(x)}{x / \ln x}=1 \text { (Prime Number Theorem). }$$

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