The crown of mathematics-number theory is a branch of mathematics, which studies the laws of numbers, especially the properties of integers. Like geometry, it is the oldest and most active field in mathematical research. The distribution of prime numbers is the earliest research topic in number theory. Euclid once proved that there are infinite prime numbers. Most mathematicians in history have studied number theory. For a long time, number theory has only basic properties in pure mathematics, but it is considered to have no direct application value. With the appearance and development of computers, great and profound changes have been brought to science and technology. This makes number theory widely used. No matter what problem must be discretized before numerical calculation can be carried out on the computer, so discrete mathematics becomes more and more important, and one of the foundations of discrete mathematics is number theory. Humans have been dealing with natural numbers since they learned to count. Later, due to the need of practice, the concept of number was further expanded. Natural numbers are called positive integers, while their opposites are called negative integers, and neutral numbers between positive and negative integers are called 0. They add up to an integer. For integers, four operations can be performed: addition, subtraction, multiplication and division, which are called four operations. Among them, addition, subtraction, multiplication and division can be carried out in an integer range without obstacles. That is to say, any two or more integers are added, subtracted and multiplied, and their sum, difference and product are still an integer. However, the division between integers may not be carried out smoothly within the integer range. In the application and research of integer operation, people are gradually familiar with the characteristics of integers. For example, integers can be divided into two categories-odd and even (usually called odd and even) and so on. Using some basic properties of integers, we can further explore many interesting and complex mathematical laws. It is the charm of these characteristics that has attracted many mathematicians to study and explore continuously throughout the ages. The subject of number theory begins with the study of integers, so it is called integer theory. Later, the theory of integers was further developed and called number theory. To be exact, number theory is a subject that studies the properties of integers. Introduction to the development of number theory Since ancient times, mathematicians have always attached great importance to the study of integer properties, but until the19th century, these research results were only recorded in arithmetic works of various periods in isolation, that is to say, a complete and unified discipline has not yet been formed. Since ancient China, many famous mathematical works have discussed the content of number theory, such as finding the greatest common divisor, pythagorean array, integer solutions of some indefinite equations and so on. Abroad, mathematicians in ancient Greece have systematically studied one of the most basic problems in number theory-divisibility, and a series of concepts such as prime number, sum number, divisor and multiple have also been put forward and applied. Mathematicians of past dynasties have also made great contributions to the study of integer properties, and gradually improved the basic theory of number theory. In the study of the properties of integers, the prince of mathematics-Gauss found that prime numbers are the basic "materials" that constitute positive integers. In order to study the properties of integers, it is necessary to study the properties of prime numbers. Therefore, some problems about the properties of prime numbers have always been concerned by mathematicians. By the end of18th century, the scattered knowledge about the properties of integers accumulated by mathematicians in past dynasties was very rich, and the conditions for sorting it out and processing it into a systematic discipline were completely mature. Gauss, a German mathematician, concentrated the achievements of his predecessors and wrote a book called Arithmetic Discussion, which was sent to the French Academy of Sciences in 1800, but the French Academy of Sciences rejected Gauss's masterpiece, so Gauss had to publish it himself in 180 1 year. This book initiated a new era of modern number theory. In On Arithmetic, Gauss standardized the symbols used to study the properties of integers in the past, systematized and summarized the existing theorems at that time, classified the problems to be studied and the methods of will, and introduced new methods. The basic content of number theory After number theory became an independent subject, with the development of other branches of mathematics, the methods of studying number theory are also developing. According to the research method, it can be divided into four parts: elementary number theory, analytic number theory, algebraic number theory and geometric number theory. Elementary number theory is a branch of number theory, which studies the properties of integers only by elementary methods without the help of other mathematical disciplines. For example, the famous "China's Remainder Theorem" in ancient China is a very important content in elementary number theory. Analytic number theory is a branch of solving number theory problems with mathematical analysis as a tool. Mathematical analysis is a mathematical discipline based on the concept of limit and taking function as the research object. Solving number theory problems by mathematical analysis was laid by Euler, and Russian mathematician Chebyshev also contributed to its development. Analytic number theory is a powerful tool to solve the problem of number theory. For example, for the proposition with infinite prime numbers, Euler gave the proof of analytical method, which used some knowledge about infinite series in mathematical analysis. In 1930s, vinogradov, a Soviet mathematician, creatively put forward the "triangle sum method", which played an important role in solving some difficult problems in number theory. Chen Jingrun, a mathematician in China, also solved Goldbach's conjecture with analytic number theory. Algebraic number theory is a branch that extends the concept of integer to algebraic integer. Mathematicians extended the concept of integer to the general algebraic number field, and accordingly established the concepts of prime number and divisibility. German mathematician and physicist Minkowski founded and laid the foundation of geometric number theory. The basic object of geometric number theory research is "spatial grid". What is a spatial grid? In a given rectangular coordinate system, the point whose coordinates are all integers is called the whole point; A set of all points is called a spatial grid. Spatial grid is of great significance to geometry and crystallography. Because of the complexity of the problems involved in geometric number theory, it needs a considerable mathematical foundation to study it in depth. Number theory is a highly abstract mathematical subject. For a long time, its development has been in the state of pure theoretical research, which has played a positive role in the development of mathematical theory. But for most people, its practical significance is not clear. With the development of modern computer science and applied mathematics, number theory has been widely used. For example, many research results within the scope of elementary number theory are widely used in calculation methods, algebraic coding, combinatorial theory, etc. It is also reported in the literature that some countries now use the "Sun Tzu Theorem" to measure the distance and calculate the discrete Fourier transform with the original root and the original exponent. In addition, many profound research results of number theory have also been applied in approximate analysis, difference set, rapid transformation and so on. Especially due to the development of computer, it is possible to approximate continuous quantity with the calculation of discrete quantity and achieve the required accuracy. The position of number theory in mathematics is unique. Gauss once said, "Mathematics is the queen of science, and number theory is the crown in mathematics". So mathematicians like to call some unsolved problems in number theory "crown jewels" to encourage people to "choose". Here are some "pearls": Fermat's last theorem, twin prime number problem, Goldbach conjecture, integer problem in a circle, perfect number problem ... In modern China, number theory is also one of the earliest branches of mathematics. Since the 1930s, he has made great contributions to analytic number theory, complexity equation and even distribution, and first-class number theory experts such as Hua, Min Sihe and Ke Zhao have emerged. Among them, Professor Hua is most famous for his research on trigonometric sum assignment and heap prime theory. After 1949, the study of number theory has made great progress. Especially in the research of "screening method" and "Goldbach conjecture", it has made outstanding achievements in the world. Especially after Chen Jingrun proved in 1966 that "a big even number can be expressed as the sum of the products of a prime number and no more than two prime numbers", it aroused strong repercussions in the international mathematics community, praising Chen Jingrun's paper as a masterpiece of analytical mathematics and the glorious culmination of screening method. So far, this is still the best result of Goldbach's conjecture. Other branches of mathematics include arithmetic, elementary algebra, advanced algebra, number theory, Euclidean geometry, non-Euclidean geometry, analytic geometry, differential geometry, algebraic geometry, projective geometry, topology, fractal geometry, calculus, real variable function theory, probability and mathematical statistics, complex variable function theory, functional analysis, partial differential equations, ordinary differential equations, mathematical logic, fuzzy mathematics and operational research.