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# 数学代写代考| Elementary Number Theory 离散数学

## 数学代写| Elementary Number Theory 代考

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## 离散数学代写

A square number (Fig. 3.3) is an integer that is the square of another integer. For example, the number 4 is a square number since $4=2^{2}$. Similarly, the number 9 and the number 16 are square numbers. A number $n$ is a square number if and only if one can arrange the $n$ points in a square. For example, the square numbers 4,9 and 16 are represented in squares as follows.

The square of an odd number is odd whereas the square of an even number is even. This is clear since an even number is of the form $n=2 k$ for some $k$, and so $n^{2}=4 k^{2}$ which is even. Similarly, an odd number is of the form $n=2 k+1$ and so $n^{2}=4 k^{2}+4 \mathrm{k}+1$ which is odd.

A rectangular number (Fig. 3.4) $n$ may be represented by a vertical and horizontal rectangle of $n$ points. For example, the number 6 may be represented by a rectangle with length 3 and breadth 2 , or a rectangle with length 2 and breadth 3 . Similarly, the number 12 can be represented by a $4 \times 3$ or a $3 \times 4$ rectangle.
A triangular number (Fig. 3.5) $n$ may be represented by an equilateral triangle of $n$ points. It is the sum of $k$ natural numbers from 1 to $k$. = That is,
$$n=1+2+\cdots+k$$
Parity of Integers
The parity of an integer refers to whether the integer is odd or even. An integer $n$ is odd if there is a remainder of one when it is divided by two, and it is of the form $n=2 k+1$. Otherwise, the number is even and of the form $n=2 k$.

The sum of two numbers is even if both are even or both are odd. The product of two numbers is even if at least one of the numbers is even. These properties are expressed as
\begin{aligned} &\text { even } \pm \text { even }=\text { even } \ &\text { even } \pm \text { odd }=\text { odd } \ &\text { odd } \pm \text { odd }=\text { even } \end{aligned}
$$\begin{gathered} \text { even } \times \text { even }=\text { even } \ \text { even } \times \text { odd }=\text { even } \ \text { odd } \times \text { odd }=\text { odd } \end{gathered}$$
Divisors
Let $a$ and $b$ be integers with $a \neq 0$ then $a$ is said to be a divisor of $b$ (denoted by $a$ । $b)$ if there exists an integer $k$ such that $b=k a$.

A divisor of $n$ is called a trivial divisor if it is either 1 or $n$ itself; otherwise, it is called a non-trivial divisor. A proper divisor of $n$ is a divisor of $n$ other than $n$ itself.
DEFINITION (PRIME NUMBER)
A prime number is a number whose only divisors are trivial. There are an infinite number of prime numbers.

The fundamental theorem of arithmetic states that every integer number can be factored as the product of prime numbers.
Mersenne Primes
Mersenne primes are prime numbers of the form $2^{p}-1$ where $p$ is a prime. They are named after Marin Mersenne (Fig. 3.6) who was the 17th French monk, philosopher and mathematician. Mersenne did some early work in identifying primes of this format, and there are 47 known Mersenne primes. It remains an open question as to whether there are an infinite number of Mersenne primes.

## 图论代考

$$n=1+2+\cdots+k$$

$$\开始{对齐} &\text { 偶数 } \pm \text { 偶数 }=\text { 偶数 } \ &\text { 偶数 } \pm \text { 奇数 }=\text { 奇数 } \ &\text { 奇数 } \pm \text { 奇数 }=\text { 偶数 } \end{对齐}$$
$$\开始{聚集} \text { 偶数 } \times \text { 偶数 }=\text { 偶数 } \ \text { 偶数 } \times \text { 奇数 }=\text { 偶数 } \ \text { 奇数 } \times \text { 奇数 }=\text { 奇数 } \结束{聚集}$$

$n$ 的除数如果是 1 或 $n$ 本身，则称为平凡除数；否则，它被称为非平凡除数。 $n$ 的真除数是 $n$ 的除数，而不是 $n$ 本身。

## 密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码