# 什么是数论？ EssayTA™为留学生提供论文和作业的最佳写作平台

## What Is Number Theory

Number theory is the study of the set of positive whole numbers
$$1,2,3,4,5,6,7, \ldots$$
which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here are some familiar and not-so-familiar examples:
$\begin{array}{ll}\text { odd } & 1,3,5,7,9,11, \ldots \ \text { even } & 2,4,6,8,10, \ldots \ \text { square } & 1,4,9,16,25,36, \ldots \ \text { cube } & 1,8,27,64,125, \ldots \ \text { prime } & 2,3,5,7,11,13,17,19,23,29,31, \ldots \ \text { composite } & 4,6,8,9,10,12,14,15,16, \ldots \ \text { 1 (modulo 4) } & 1,5,9,13,17,21,25, \ldots \ 3 \text { (modulo 4) } & 3,7,11,15,19,23,27, \ldots \ \text { triangular } & 1,3,6,10,15,21, \ldots \ \text { perfect } & 6,28,496, \ldots \ \text { Fibonacci } & 1,1,2,3,5,8,13,21, \ldots\end{array}$
Many of these types of numbers are undoubtedly already known to you. Others, such as the “modulo 4 ” numbers, may not be familiar. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4 , and similarly for the 3 (modulo 4) numbers. A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the

## Some Typical Number Theoretic Questions

The main goal of number theory is to discover interesting and unexpected relationships between different sorts of numbers and to prove that these relationships are true. In this section we will describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day.

Sums of Squares I. Can the sum of two squares be a square? The answer is clearly “YES”; for example $3^{2}+4^{2}=5^{2}$ and $5^{2}+12^{2}=13^{2}$. These are examples of Pythagorean triples. We will describe all Pythagorean triples in Chapter $2 .$

Sums of Higher Powers. Can the sum of two cubes be a cube? Can the sum of two fourth powers be a fourth power? In general, can the sum of two $n^{\text {th }}$ powers be an $n^{\text {th }}$ power? The answer is “NO.” This famous problem, called Fermat’s Last Theorem, was first posed by Pierre de Fermat in the seventeenth century, but was not completely solved until 1994 by Andrew Wiles. Wiles’s proof uses sophisticated mathematical techniques that we will not be able to describe in detail

Infinitude of Primes. A prime number is a number $p$ whose only factors are 1 and $p$.

• Are there infinitely many prime numbers?
• Are there infinitely many primes that are 1 modulo 4 numbers?
• Are there infinitely many primes that are 3 modulo 4 numbers?
The answer to all these questions is “YES.” We will prove these facts in Chapters 12 and 21 and also discuss a much more general result proved by Lejeune Dirichlet in $1837 .$

## 什么是数论

$$1,2,3,4,5,6,7, \ldots$$

odd $1,3,5,7,9,11, \ldots$ even $2,4,6,8,10, \ldots$ square $1,4,9,16,25,36, \ldots$ cube $1,8,27,64,125$ 许多这些类型的数字无疑已经为您所知。其他数字, 例如“模 4”数字, 可能并不孰悉。如果一个数除以 4 时余 数为 1, 则称该数与 1 (模 4) 全等, 对于 3 (模 4) 数也是如此。一个数字称为三角形, 如果该数囯的鹅卵石 可以排列成一个三角形, 顶部有一个鹅卵石, 下一行有两个鹅卵石, 依此类推。斐波那契数是从 1 和 1 开始创 建的。然后, 要获得列表中的下一个数字, 只需将前两个相加即可。最后, 一个数是完美的, 如果它的所有除 数之和, 除了它自己,

## 一些典型的数论问题

• 有无限多个素数吗?
• 是否有无限多个素数是 1 模 4 数?
• 有无限多个素数是 3 模 4 数吗?
所有这些问题的答亲都是“是”。我们将在第 12 章和第 21 章证明这些事实, 并讨论 Lejeune Dirichlet 在 1837 .