First, the Millennium issue.
One of the Millennium Problems: P (polynomial algorithm) versus NP (non-polynomial algorithm)
On a Saturday night, you attended a grand party. It's embarrassing. You want to know if there is anyone you already know in this hall. Your host suggests that you must know Ms. Ross sitting in the corner near the dessert plate. You don't need a second to glance there and find that your master is right. However, if there is no such hint, you must look around the whole hall and look at everyone one by one to see if there is anyone you know. Generating a solution to a problem usually takes more time than verifying a given solution. This is an example of this common phenomenon. Similarly, if someone tells you that the numbers 13, 7 17, 42 1 can be written as the product of two smaller numbers, you may not know whether to believe him or not, but if he tells you that you can factorize it into 3607 times 3803, then you can easily verify this with a pocket calculator. Whether we write a program skillfully or not, it is regarded as one of the most prominent problems in logic and computer science to determine whether an answer can be quickly verified with internal knowledge, or it takes a lot of time to solve it without such hints. It was stated by StephenCook in 197 1.
The second part of "Thousands of Questions": Hodge conjecture
Mathematicians in the twentieth century found an effective method to study the shapes of complex objects. The basic idea is to ask to what extent we can shape a given object by bonding simple geometric building blocks with added dimensions. This technology has become so useful that it can be popularized in many different ways; Finally, it leads to some powerful tools, which make mathematicians make great progress in classifying various objects they encounter in their research. Unfortunately, in this generalization, the geometric starting point of the program becomes blurred. In a sense, some parts without any geometric explanation must be added. Hodge conjecture asserts that for the so-called projective algebraic family, a component called Hodge closed chain is actually a (rational linear) combination of geometric components called algebraic closed chain.
The third "Millennium mystery": Poincare conjecture
If we stretch the rubber band around the surface of the apple, then we can move it slowly and shrink it into a point without breaking it or letting it leave the surface. On the other hand, if we imagine that the same rubber belt is stretched in a proper direction on the tire tread, there is no way to shrink it to a point without destroying the rubber belt or tire tread. We say that the apple surface is "single connected", but the tire tread is not. About a hundred years ago, Poincare knew that the two-dimensional sphere could be characterized by simple connectivity in essence, and he put forward the corresponding problem of the three-dimensional sphere (all points in the four-dimensional space at a unit distance from the origin). This problem became extremely difficult at once, and mathematicians have been fighting for it ever since.
The fourth part of the Millennium puzzle: Riemann hypothesis
Some numbers have special properties and cannot be expressed by the product of two smaller numbers, such as 2, 3, 5, 7, etc. Such numbers are called prime numbers; They play an important role in pure mathematics and its application. In all natural numbers, the distribution of such prime numbers does not follow any laws; However, German mathematician Riemann (1826~ 1866) observed that the frequency of prime numbers is closely related to the behavior of a well-constructed so-called Riemann zeta function z(s$). The famous Riemann hypothesis asserts that all meaningful solutions of the equation z(s)=0 are on a straight line. This has been verified in the original 1, 500,000,000 solutions. Proving that it applies to every meaningful solution will uncover many mysteries surrounding the distribution of prime numbers.
The fifth of "hundreds of puzzles": the existence and quality gap of Yang Mill.
The laws of quantum physics are established for the elementary particle world, just as Newton's classical laws of mechanics are established for the macroscopic world. About half a century ago, Yang Zhenning and Mills discovered that quantum physics revealed the amazing relationship between elementary particle physics and geometric object mathematics. The prediction based on Young-Mills equation has been confirmed in the following high-energy experiments in laboratories all over the world: Brockhaven, Stanford, CERN and Tsukuba. However, they describe heavy particles and mathematically strict equations have no known solutions. Especially the "mass gap" hypothesis, which has been confirmed by most physicists and applied to explain the invisibility of quarks, has never been satisfactorily proved mathematically. The progress on this issue needs to introduce basic new concepts into physics and mathematics.
The Sixth Millennium Problem: Existence and Smoothness of Navier-Stokes Equation
The undulating waves follow our ship across the lake, and the turbulent airflow follows the flight of our modern jet plane. Mathematicians and physicists are convinced that both breeze and turbulence can be explained and predicted by understanding the solution of Naville-Stokes equation. Although these equations were written in19th century, we still know little about them. The challenge is to make substantial progress in mathematical theory, so that we can solve the mystery hidden in Naville-Stokes equation.
The seventh "Millennium Mystery": Burch and Swinerton Dale's conjecture.
Mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as x2+y2=z2. Euclid once gave a complete solution to this equation, but for more complex equations, it became extremely difficult. In fact, as a surplus. V.Matiyasevich pointed out that Hilbert's tenth problem is unsolvable, that is, there is no universal method to determine whether such a method has an integer solution. When the solution is a point of the Abelian cluster, Behe and Swenorton-Dale suspect that the size of the rational point group is related to the behavior of the related Zeta function z(s) near the point s= 1. In particular, this interesting conjecture holds that if z( 1) is equal to 0, there are infinite rational points (solutions); On the other hand, if z( 1) is not equal to 0, there are only a limited number of such points.
Second, the frontier problems in today's mathematics.
Brief introduction to frontier problems of mathematics
A Brief Comment on Mathematics Research in the 20th Century
Reporter: Hello, Mr. Lin. First of all, thank you very much for taking time out of your busy schedule to accept this interview and introduce some basic information about the frontier of mathematics to primary and secondary school teachers all over the country. Scientific research has entered the threshold of the new century. We can see that on the one hand, each discipline is reviewing its own development process, on the other hand, it is also looking forward to its own development prospects. You entered the Chinese Academy of Sciences from 1956 and formally engaged in mathematical research. It has been nearly half a century now. In this half century, you have been struggling at the forefront of mathematical research. Based on your research on mathematics for so many years, you can look back on the development of mathematics in the 20th century. What significant progress and achievements have been made in mathematical research during this course?
Lin Qun: According to what you said, from the mathematical point of view, the mathematics of the last century must be attributed to the famous book Mathematical Problems published by the 38-year-old German mathematician Hilbert (1862- 1943) at the second international congress of mathematicians held in Paris on August 6, 2000. According to the achievements and development trend of mathematical research in the past, especially in the 19th century, he put forward 23 most important mathematical problems. These 23 problems are collectively called Hilbert problems. This speech became a milestone in the development of world mathematics history and opened a glorious page for the development of mathematics in the 20th century. Among these 23 problems, the first six problems are related to the basis of mathematics, and the other 17 problems involve number theory, indefinite integral, quadratic theory, invariant theory, differential equation, variation and other fields.
1905, Einstein founded the special theory of relativity (in fact, two mathematicians, Poincare and Lorenz, also walked to the door of relativity), 1907, he found that the special theory of relativity was very successful in other fields of physics, except the problem of gravity. In order to solve this contradiction, Einstein turned to the study of general relativity, and soon established "general relativity" and "equivalence theory", but the difficulties encountered in mathematics made him make little progress for many years. About 19 1 1 years ago, Einstein finally found that the gravitational field is related to the geometric properties of space and is the result of space-time bending. So Einstein's mathematical tool is non-Euclidean geometry. 19 15 years, Einstein finally completed the general theory of relativity with the framework of Riemannian geometry and the language of tensor analysis.
More importantly, emmy noether 1882 ~ 1935, a German female mathematician, published the paper "Ideal in the Ring of Berich", which marked the beginning of the modernization of abstract algebra. She taught us to think with the simplest, most economical and most general concepts and terms: isomorphism, ideal, operator ring and so on.
There are many other great achievements in mathematics. To be lazy, the work of nearly 50 Fields Prize winners in mathematics in the 20th century is a great achievement within mathematics. However, from the perspective of promoting social development, computer algorithm research related mathematics may be more influential. This kind of research took place around the Second World War. There are three mathematicians (Turing, Godel and von Neumann), not engineers. Because they played a foundation and guiding role in the birth, design and development of computers, they were included in the list of "100 stars" in the 20th century. Two other pure mathematicians who won the Nobel Prize (Kantrovich and Nash) are also related to algorithm research (or military mathematics), and the latter has just won an Oscar. The work of Wu Wenjun, the first winner of the highest national science and technology prize (not the mathematics prize) in China, also includes the research of algorithms. Among the top ten scientific and technological advances in China, there was once a work by mathematician Dingzhu Du, which was also related to algorithms. It is worth noting that none of these people won the Fields Prize.
Related to algorithm research (or military mathematics) are learning, cryptography and large-scale scientific engineering calculation. How can I have a vague feeling (infected by Wu Wenjun? ), it seems that in the twentieth century, the mathematical research centered on algorithms had a quite direct impact on the outside world, science and technology and military affairs. Will this century (information, materials, biology) be like this again? Wait and see!
Second, the main problems in the field of mathematical research
Reporter: Just now, Academician Lin painted us a picture of mathematical research in the 20th century. It should be said that in the 20th century, the classical and emerging branches of mathematics have made great progress. However, we also see that in the process of mathematical research, there are many regrets and many problems have not been solved, or they have not been solved perfectly. Teacher Lin, what do you think are the main problems in the field of mathematics research?
Lin Qun: As for the difficult problem, it should be said that it requires great determination to solve it. I think we researchers can do our jobs well. Problems that were not solved in the last century may not be solved in this century. It should be said that the twentieth century is a century of great development of mathematics. According to the report, many important problems in mathematics have been solved, such as the proof of Fermat's last theorem and the completion of the classification of finite simple groups, thus making the basic theory of mathematics develop unprecedentedly. The appearance of computer is a great achievement in the development of mathematics in the 20th century, which greatly promoted the deepening of mathematical theory and the direct application of mathematics in the front line of society and productivity. Looking back on the development of mathematics in the 20th century, as you said, mathematicians are deeply grateful to David, the greatest master of mathematics in the 20th century. Hilbert. As we mentioned at the beginning, Hilbert put forward 23 mathematical problems in his famous speech at the Second World Congress of Mathematicians held in Paris on August 8, 1900. Hilbert problem has inspired the wisdom of mathematicians and guided the direction of mathematics in the past hundred years, and its influence and promotion on the development of mathematics is enormous and immeasurable.
Taking Hilbert as an example, many famous mathematicians in the contemporary world have sorted out and put forward new mathematical problems in the past few years, hoping to point out the direction for the development of mathematics in the new century.
Mathematics also likes to create some news effects. At the beginning of 2000, the Scientific Advisory Board of the Clay Institute of Mathematics in the United States selected seven "Millennium Prize Questions", and the board of directors of the Clay Institute of Mathematics decided to set up a grand prize fund of 7 million dollars, and each "Millennium Prize Question" could be awarded10 million dollars. The purpose of selecting the "Millennium Prize Problem" of Clay Institute of Mathematics is not necessarily to form a new direction of mathematics development in the new century, but to focus on the major problems that are of central significance to mathematics development and that mathematicians dream of and expect to solve.
On May 24th, 2000, the Millennium Mathematics Conference was held in the famous French Academy. At the meeting, 1998 Faldts Prize winner Gals gave a speech on the topic of "The Importance of Mathematics". Later, Tate and attiya announced and introduced these seven "Millennium Prize Issues". Clay Institute of Mathematics also invited experts in related research fields to elaborate on each issue. Clay Institute of Mathematics has made strict regulations on the answer and award of the "Millennium Prize Question". Every "Millennium prize problem" is not solved immediately. Any solution must be published in a world-renowned mathematical magazine for two years and recognized by the mathematical community before it can be examined and decided by the scientific advisory Committee of Clay Institute of Mathematics whether it is worth winning a million dollars.
The seven "Millennium Prize Problems" are NP Complete Problem, Hodge Conjecture, Poincare Conjecture, Riemann Hypothesis, Young-Mills Theory, Naville-Stokes Equation and BSD (Birch and Swinerton).
Since the publication of the "Millennium Prize", it has had a strong response in the field of mathematics. These problems are all about the basic theory of mathematics, but the solution of these problems will greatly promote the development and application of mathematical theory (the first problem is a basic theory of computer algorithm). Understanding and studying the "Millennium Prize" has become a hot spot in mathematics. Many countries, including mathematicians in China, are organizing joint research.
III. Major Problems in the Field of Mathematical Research (Continued)
Other problems in the field of mathematics can be said to be endless. According to the information you provided, there are at least the following simple ones:
The first is Goldbach's conjecture.
Goldbach is a German mathematician who was born in 1690. 1742, Goldbach found in teaching that every even number not less than 6 is the sum of two prime numbers (numbers that can only be divisible by themselves). For example, 6 = 3+3, 12 = 5+7 and so on.
1742 on June 7, Goldbach wrote to the great mathematician Euler at that time, and put forward the following conjecture:
(a) Any > even number =6 can be expressed as the sum of two odd prime numbers.
(& lt-emo & amp； b)-& gt； & lt-endemo->； Any odd number > 9 can be expressed as the sum of three odd prime numbers.
This is the famous Goldbach conjecture. In his reply to him on June 30th, Euler said that he thought this conjecture was correct, but he could not prove it. Describing such a simple problem, even a top mathematician like Euler can't prove it. This conjecture has attracted the attention of many mathematicians. Since Goldbach put forward this conjecture, many mathematicians have been trying to conquer it, but they have not succeeded. Of course, some people have done some specific verification work, such as: 6 = 3+3, 8 = 3+5, 10 = 5+5 = 3+7, 12 = 5+7,14 = 7+7 = 3+/kloc. Someone checked the even numbers within 33× 108 and above 6 one by one, and Goldbach conjecture (a) was established. But strict mathematical proof requires the efforts of mathematicians.
Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 200 years have passed and no one has proved it. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics. It was not until the 1920s that people began to approach it. 1920, the Norwegian mathematician Bujue proved and concluded with an ancient screening method that every even number greater than 36 can be expressed as (9+9). This method of narrowing the encirclement is very useful, so scientists gradually reduce the number of prime factors of each number from (9+9) until each number is a prime number, thus proving the Goldbach conjecture.
At present, the best result is proved by Chinese mathematician Chen Jingrun in 1966, which is called Chen Theorem. That is, "any large enough even number is the sum of a prime number and a natural number, and the latter is just the product of two prime numbers." This conclusion is usually called a big even number and can be expressed as "1+2".
Before Chen Jingrun, the progress of even numbers can be expressed as the sum of the products of S prime numbers and T prime numbers (referred to as the "s+t" problem) as follows:
1920, Bren of Norway proved "9+9".
1924, Rademacher proved "7+7".
1932, Esterman of England proved "6+6".
1937, Ricei of Italy proved "5+7", "4+9", "3+ 15" and "2+366" successively.
1938, Buhe of the Soviet Union? Byxwrao proved "5+5".
1940, Buhe of the Soviet Union? Byxwrao proved "4+4".
1948, Hungary's benevolence and righteousness proved "1+c", where c is the number of nature.
1956, Wang Yuan of China proved "3+4".
1957, China and Wang Yuan successively proved "3+3" and "2+3".
1962, Pan Chengdong of China and Barba of the Soviet Union proved "1+5". Soon, Pan Chengdong and Wang Yuan proved "1+4".
1965, Buhe of the Soviet Union? Xi Taibo (Byxwrao) and vinogradov Jr. and Italian Bambier proved "1+3".
1966, China Chen Jingrun proved "1+2".
Who will finally overcome the problem of "1+ 1"? It is still unpredictable. However, Wang Yuan recently delivered a speech saying that British mathematicians were discussing it in a detour. I hope there is hope.
Figure 1 the great mathematician Euler
Figure 2 Young people's role models,
Chen Jingrun, a famous mathematician in China
Figure 3 Famous mathematician Wang Yuan
Fig. 4 Veda, a French mathematician
Figure 6 French mathematician D'Alembert
The second is the mystery of continuum.
(Note: In this article, Allaf is marked as alf(0), Allaf is marked as alf( 1), and so on ...)
Since alf(0) is an infinite radix and Allaf is a magical operation different from finite operation, it is not surprising that the following results:
alf(0)+ 1 = alf(0)
alf(0) + n = alf(0)
alf(0) + alf(0) = alf(0)
alf(0) n = alf(0)
Alf (0) Alf (0) = Alf (0)
Alf(0) is the cardinal number of the set of natural numbers. An infinite radix, as long as it is a countable set, its radix must be alf(0). From the orderability, we can know that the cardinality of integer set and rational number set is alf (0); Or if their cardinality is alf(0), they are countable sets. But the uncountable real number set (which can be disproved by Cantor dust line) deduces that it has a larger cardinality than alf(0). Multiplication cannot break through alf(0), but power set can break through: = alf( 1). It can be proved that cardinal number card (r) of real number set = alf (1). In addition, Allaf's "family" broke out:
= alf(2)； = alf(3)； ……
What is the meaning of alf(2)? People think hard and get the number of all curves in space. But the next alf(3), human beings have racked their brains and have not been able to figure it out so far. In addition, there is a puzzling continuum mystery: "Is there another cardinal number between alf(0) and alf( 1)?"
In 1878, Cantor put forward the conjecture that there is no other cardinality between alf(0) and alf( 1). But at that time, Cantor himself could not confirm this.
In 1900, at the Second International Congress of Mathematicians held in Paris, Hilbert, a professor at the University of G? ttingen in Germany, put forward 23 world-famous mathematical problems to be solved in the 20th century, and the continuum hypothesis ranked first. However, the final result of this problem is completely unexpected.
In AD 1938, Austrian mathematician G? del proved that "the continuum hypothesis will never lead to contradictions", which means that it is impossible for human beings to find out what is wrong with the continuum hypothesis. 1963, American mathematician Cohen proved that "the continuum hypothesis is independent", which means it is impossible to prove the continuum hypothesis.
Godel's work is so important that von Neumann was influenced by him to design the computer.
Coloring with four colors; And then push it to 50 countries. It seems that this progress is still very slow. After the emergence of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid improvement of calculation speed and the emergence of man-machine dialogue. 1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. The computer proof of the four-color conjecture has caused a sensation in the world. It not only solved a problem that lasted for more than 100 years, but also may become the starting point of a series of new ideas in the history of mathematics. However, many mathematicians are not satisfied with the achievements made by computers, and they are still looking for a simple and clear written proof method.
The fourth is the three major problems of geometry.
Plane geometric drawing is limited to rulers and compasses. The so-called ruler here refers to a ruler that can only draw straight lines without scale. Of course, many kinds of figures can be made with rulers and compasses, but some figures, such as regular heptagon and regular nonagon, can't be made. Some problems seem simple, but they are really difficult to solve. The most famous of these problems are the so-called three major problems.
The three main problems in geometry are:
1. Turn a circle into a square: find a square and make its area equal to the known circle;
2. Divide any corner into three equal parts;
3. Double cube: Find a cube and make it twice the volume of the known cube.
Circle and square are common geometric figures, but how to make a square with the same area as the known circle? If the radius of a circle is known as 1 and its area is π, then the problem of turning a circle into a square is equivalent to finding a square with an area of π, that is, making a line segment (or a line segment of π) with a ruler.
The second of the three major problems is the problem of bisecting an angle. For some angles, it is not difficult to divide into three parts, but can all angles be divided into three parts? For example, if the angle that can be made can be divided into three equal parts, then a regular 18 polygon and a regular nonagon can also be made (note: the circumferential angle of each side of a regular octagon in a circle is). In fact, the problem of angle trisection is caused by the problem of finding regular polygons.
The third problem is cubes. Eratoseni (276 BC ~ 65438 BC+095 BC) once described a myth that a prophet had to double the size of the cube altar when he got the Oracle. Some people advocate doubling the length of each side, but we all know that this is wrong, because the size has been eight times the original.
These problems have puzzled the mathematician 1000 years, but in fact, none of these three problems can be solved by a ruler and compass through limited steps.
After Descartes founded analytic geometry in 1637, many geometric problems can be transformed into algebraic problems to study. In 1837, Wantzel gave a proof that it is impossible to draw any angle and cube with a ruler. In 1882, Lin Deman also proved the transcendence of π (that is, π is not the root of any integer coefficient multiple), and the impossibility of changing the square of a circle is established.
Main problems in mathematical research of verb (verb's abbreviation) (continued)
The fifth is Fermat's last theorem.
On June 24th, 1993, The New York Times, an authoritative newspaper recognized by the world, published a news about solving mathematical problems. The headline of the news is "In the ancient mathematical dilemma, someone finally said" I found it ". The opening article of the first edition of The Times is accompanied by a photo of a man with long hair and wearing a medieval European robe. This ancient man was the French mathematician Pierre de Fermat (please refer to the appendix of Fermat's biography). Fermat is one of the most outstanding mathematicians in17th century. He has made great contributions in many fields of mathematics because he is a professional lawyer. In recognition of his mathematical attainments, the world called him "amateur prince". One day more than 360 years ago, Fermat was reading a math book by Diofendos, an ancient Greek mathematician. Suddenly, on a whim, he wrote a seemingly simple theorem in the margin of the page. The content of this theorem is about the positive integer solution of an equation. When n=2, it is the well-known Pythagorean Theorem (also called Pythagorean Theorem in ancient China):, where z represents the hypotenuse of a right triangle, and X and Y are its two branches, that is, the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two branches. Of course, this equation has integer solutions (in fact, there are many), such as x=3, Y = x=6, y=8, z =10; X=5, y= 12, z = 13 and so on. Fermat claimed that when n>2, it could not find a satisfactory integer solution, for example, an equation could not find an integer solution.
At that time, Fermat did not explain why. He just left this narrative, saying that he found a wonderful way to prove this theorem, but there was not enough space on the page to write it down. Fermat, the initiator, thus left an eternal problem. For more than 300 years, countless mathematicians have tried in vain to solve this problem. This Fermat's last theorem, known as the century's difficult problem, has become a big worry in mathematics and is extremely eager to solve it.
19th century, Francis Institute of Mathematics in France provided a gold medal and 300 francs to anyone who solved this problem twice 18 15 and 1860. Unfortunately, no one was rewarded. German mathematician Wolfskeil (p? Wolfskehl) provides 100000 marks in 1908 to those who can prove the correctness of Fermat's last theorem, and the validity period is100 years. In the meantime, due to the Great Depression, the bonus amount has been devalued to 7500 marks, but it still attracts many "math idiots".
After the development of computers in the 20th century, many mathematicians can prove that this theorem holds when n is large. 1983, the computer expert Sloansky ran the computer for 5782 seconds, which proved that Fermat's last theorem was correct when n was 286243- 1 (Note 286243- 1 is an astronomical figure with about 25960 digits).
Nevertheless, mathematicians have not found a universal proof. However, this 300-year-old math unsolved case has finally been solved. Andrew wiles, an English mathematician, solved this mathematical problem. In fact, Willis proved this point with the achievements of abstract mathematics development in the past 30 years of the twentieth century.
In 1950s, Yutaka Taniyama, a Japanese mathematician, first put forward a conjecture about elliptic curvature, which was later developed by Goro Shimamura, another mathematician. At that time, no one thought that this conjecture had anything to do with Fermat's last theorem. In 1980s, German mathematician Frey linked Yutai Taniyama's conjecture with Fermat's last theorem. What Willis did was to prove that one form of Yutai Taniyama's conjecture was correct according to this connection, and then deduced Fermat's last theorem. This conclusion was officially published by Willis at the seminar of Newton Institute of Mathematics, Cambridge University, UK on June 2 1, 1993. This report immediately shocked the whole mathematics field, and even the public outside the mathematics door paid infinite attention. However, Willis' proof was immediately found to have some defects, so it took Willis and his students 14 months to correct it. 1September 1994 19 They finally handed over a complete and flawless scheme, and the nightmare of mathematics finally ended. 1In June, 1997, Willis won the Wolfskeil Prize at the University of G? ttingen. At that time,1100,000 grams was about $2 million, and when Willis received it, it was only worth about $50,000, but Willis has been recorded in the history books and will be immortal.
In order to prove that Fermat's last theorem is correct (that is, there is no positive integer solution for n>3), it is only necessary to prove that sum (p is an odd prime number) has no integer solution.
Six, the major problems in the field of mathematical research (continued)
Sixth, the problem of seven bridges (a problem)
When Euler visited Konigsberg and Prussia (now Kaliningrad, Russia) on 1736, he found that local citizens were engaged in a very interesting pastime. In konigsberg, there is a river named Pregel running through it, and there are seven bridges on the river, as shown in the figure:
This interesting pastime is to walk across all seven bridges on Saturday. Each bridge can only cross once, and the starting point and the ending point must be in the same place. Euler regards each piece of land as a point, and the bridge connecting the two pieces of land is represented by a line, and the following figure is obtained:
It was later inferred that such a move was impossible. His argument is this: In addition to the starting point, every time a person enters a piece of land (or point) from one bridge, he or she also leaves the point from another bridge. So every time you pass a point, two bridges (or lines) are counted, and the line leaving from the starting point and the line finally returning to the starting point are also counted, so the number of bridges connecting each piece of land and other places must be even. The graph formed by the seven bridges does not contain even numbers, so the above task is impossible. That's about all I have to say about math problems.