数学代写代考| Elementary Number Theory 离散数学代写 - 代考代写：100%准时可靠您的作业代写专家

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

平方数（图 3.3）是一个整数，它是另一个整数的平方。例如，数字 4 是一个平方数，因为 $4=2^{2}$。同样，数字 9 和数字 16 是平方数。一个数 $n$ 是一个平方数当且仅当人们可以将 $n$ 个点排列在一个正方形中。例如，正方形数字 4,9 和 16 用正方形表示如下。

奇数的平方是奇数，偶数的平方是偶数。这很清楚，因为对于某些 $k$，偶数的形式为 $n=2 k$，因此 $n^{2}=4 k^{2}$ 是偶数。类似地，奇数的形式为 $n=2 k+1$，因此 $n^{2}=4 k^{2}+4 \mathrm{k}+1$ 是奇数。

一个矩形数（图 3.4）$n$ 可以由一个垂直和水平的矩形 $n$ 个点来表示。例如，数字6可以用长3宽2的长方形来表示，也可以用长2宽3的长方形来表示。类似地，数字 12 可以用 $4\times 3$ 或 $3\times 4$ 矩形表示。
一个三角形数（图 3.5）$n$ 可以用一个由 $n$ 个点组成的等边三角形来表示。它是从 1 到 $k$ 的 $k$ 个自然数之和。 = 也就是说，
$$
n=1+2+\cdots+k
$$
整数奇偶校验
整数的奇偶性是指整数是奇数还是偶数。一个整数 $n$ 是奇数，如果它被 2 除后余数为 1，它的形式为 $n=2 k+1$。否则，该数为偶数，形式为 $n=2 k$。

两个数之和是偶数，即使两者都是偶数或都是奇数。两个数的乘积是偶数，即使其中至少一个数是偶数。这些属性表示为
$$
\开始{对齐}
&\text { 偶数 } \pm \text { 偶数 }=\text { 偶数 } \
&\text { 偶数 } \pm \text { 奇数 }=\text { 奇数 } \
&\text { 奇数 } \pm \text { 奇数 }=\text { 偶数 }
\end{对齐}
$$
$$
\开始{聚集}
\text { 偶数 } \times \text { 偶数 }=\text { 偶数 } \
\text { 偶数 } \times \text { 奇数 }=\text { 偶数 } \
\text { 奇数 } \times \text { 奇数 }=\text { 奇数 }
\结束{聚集}
$$
除数
令 $a$ 和 $b$ 为整数，$a \neq 0$ 则如果存在整数 $k$，则称 $a$ 是 $b$ 的除数（记为 $a$ । $b）$这样$b=ka$。

$n$ 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。
定义（质数）
素数是唯一的除数是微不足道的数。有无数个素数。

算术基本定理指出，每个整数都可以分解为素数的乘积。
梅森素数
梅森素数是 $2^{p}-1$ 形式的素数，其中 $p$ 是素数。它们以第 17 位法国僧侣、哲学家和数学家 Marin Mersenne（图 3.6）命名。梅森在识别这种格式的素数方面做了一些早期工作，已知有 47 个梅森素数。梅森素数是否有无限个仍然是一个悬而未决的问题。

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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

编码理论（英语：Coding theory）是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩、加密、纠错，最近也用于网络编码中。不同学科（如信息论、电机工程学、数学、语言学以及计算机科学）都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正（或检测）数据传输中的错误。

编码共分四类：^[1]