Four-color conjecture

Goldbach's Conjecture

1. Originated from Fermat's Last Theorem more than 300 years ago, it has challenged human beings for three centuries, shocked the world many times, exhausted the energy of many of the most outstanding human brains, and fascinated thousands of amateurs in Qian Qian. It was finally conquered by andrew wiles in 1994. Diophantine in ancient Greece wrote a famous work Arithmetic. After the ignorance and darkness of the Middle Ages and Renaissance, the remnants of "arithmetic" were rediscovered and studied.

1637, Pierre de Fremat, a great amateur mathematician in France, wrote a conjecture on the edge of the Pythagorean number in Arithmetic: a+b=c is impossible (where n is greater than 2; A, b, c and n are all non-zero integers). This conjecture was later called Fermat's last theorem. Fermat also wrote, "I have a wonderful proof of this, but the margin of this page is too narrow to write." It is generally believed that he could not have the correct proof at that time. After the conjecture was put forward, through the efforts of several generations of genius such as Euler, only four cases of n = 3, 4, 5 and 7 were solved in 200 years. In 1847, Kumar founded the modern important discipline "Algebraic Number Theory", which proved that Fermat's Last Theorem was valid for many n's (such as 100), which was a great leap.

In history, Fermat's Last Theorem had its climax and legend. Its amazing charm saved the lives of suicide youths at the last moment. He is Wolff Skerle, a German, who later offered a reward of 65,438+million marks (equivalent to more than 65,438+600,000 dollars now) for Fermat's Last Theorem, with a term of 1, 908-2007. Countless people have exhausted their efforts and left empty sighs. The most modern computer and mathematical skills have verified N within 4 million, but this is of no help to the final proof. Faltings proved in 1983 that there are only a limited number of A, B and C vibrating worlds for any fixed N, and won the Fields Prize (the highest prize in mathematics).

In the summer of 1986, a new turning point took place in history. Becker Ripper proved that Fermat's Last Theorem was included in the "Taniyama Yutai-Zhicun Goro conjecture". Wiles, who was obsessed with this in his childhood, immediately devoted himself to the research on the top floor, which lasted for seven years and brought together all the breakthrough achievements of number theory in the 20th century. Finally, Fermat's Last Theorem was proved at the end of "Words of the Century" written by Newton Institute of Cambridge University on June 23rd, 1993. Shake the world immediately and celebrate with the whole world. Unfortunately, after a few months, it was gradually discovered that there were loopholes in this certificate, which became the focus of attention of the world for a time. This proof system is a logical network, which consists of thousands of profound mathematical inferences connected with thousands of the most modern theorems, facts and calculations. Problems in any link will lead to all previous efforts being wasted. Tricks struggle to the death, there is no way out. 1September, 994 19, Monday morning, wiles suddenly found the lost key in the lightning of thinking: the solution is in the ruins! Tears filled his eyes. Wiles's historic long article "Modular Elliptic Curve and Fermat's Last Theorem" was published in1American Mathematical Yearbook (Volume 142) in May, 1995, which actually occupied the whole volume, including five chapters and 130 pages. 1On June 27th, 997, wiles was awarded a prize of 65,438+million marks by Wolff Kirli. From the deadline 10 years, the historical dream has come true. He also won the Wolf Award (1996.3), the National Academy Award (1996.6) and the Fields Special Award (1998.8).

2. The content of the four-color problem is: "Any map with only four colors can make countries with the same border have different colors." Expressed in mathematical language, it means "divide the plane into non-overlapping areas at will, and each area can always be marked with one of the four numbers 1, 2, 3 and 4, without making two adjacent areas get the same number." (right)

The adjacent area mentioned here means that there is a whole section of boundary that is common. Two regions are not adjacent if they intersect at only one point or a limited number of points. Because painting them the same color won't cause confusion.

The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so countries with the same border will be colored with different colors." Can this phenomenon be strictly proved by mathematical methods? He and his younger brother, Grace, who is in college, are determined to give it a try. The manuscript papers used by the two brothers to prove this problem have been piled up, but the research work has not progressed.

1852, 10 year123 October, his younger brother asked his teacher, the famous mathematician de Morgan, for proof of this problem. Morgan couldn't find a solution to this problem either, so he wrote to his good friend, Sir Hamilton, a famous mathematician, for advice. Hamilton demonstrated the four-color problem after receiving Morgan's letter. But until the death of 1865 Hamilton, this problem was not solved.

1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. During the two years from 1878 to 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. Everyone thought that the four-color conjecture was solved from now on.

Kemp's proof is as follows: First of all, it is pointed out that if no country surrounds other countries, or no more than three countries intersect at one point, this map is called "regular" (left picture). If it is a regular diagram, otherwise it is an irregular diagram (right). A map is often linked by a regular map and an informal map, but the number of colors required by an informal map generally does not exceed the number required by a regular map. If a map needs five colors, it means that its regular map is five colors. To prove the four-color conjecture, it is enough to prove that there is no regular five-color map.

Kemp proved this point by reducing to absurdity, to the effect that if there is a regular five-color map, there will be a "minimal regular five-color map" with the least number of countries. If there is a country with less than six neighboring countries in the minimal regular five-color graph, then there will be a regular graph with fewer countries that is still five-colored, so there will be no country with minimal five-color graph, and there will be no regular five-color graph. So Kemp thought he had proved the "four-color problem", but later people found him wrong.

However, Kemp's proof clarifies two important concepts and provides a train of thought for solving problems in the future. The first concept is "configuration". He proved that in every regular graph, at least one country has two, three, four or five neighbors, and there is no regular graph in which every country has six or more neighbors. In other words, a set of "configurations" consisting of two neighboring countries, three neighboring countries, four or five neighboring countries is inevitable, and each map contains at least one of these four configurations.

Another concept put forward by Kemp is reducibility. The use of the word "negotiable" comes from Kemp's argument. He proved that as long as there is a country with four neighboring countries in the five-color map, there will be a country with fewer five-color maps. Since the concepts of "configuration" and "reducibility" were put forward, some standard methods for testing configurations to determine whether they are reducible have been gradually developed, and the inevitable groups of reducible configurations can be found, which is an important basis for proving the "four-color problem". However, to prove that a large configuration is negotiable, many details need to be checked, which is quite complicated.

1 1 years later, that is, 1890, only 29-year-old, studying at Oxford University, pointed out the loopholes in Kemp's certificate with his own accurate calculation. He pointed out that Kemp's reason that a country without a minimum five-color map cannot have five neighboring countries is flawed. Soon, Taylor's proof was also denied. People found that they actually proved a weak proposition-the five-color theorem. In other words, it is enough to paint the map with five colors. Later, more and more mathematicians racked their brains for this, but found nothing. As a result, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture.

Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. 19 13. boekhoff, a famous American mathematician and Harvard University, used Kemp's ideas and combined his new ideas. It is proved that some large configurations are reducible. Later, American mathematician Franklin proved in 1939 that maps below 22 countries can be colored in four colors. 1950 someone has been promoted from 22 countries to 35 countries. 1960 proves that maps below 39 countries can be colored with only four colors; And then push it to 50 countries. It seems that this progress is still very slow.

The invention of high-speed digital computer urges more mathematicians to study the "four-color problem". Heck, who began to study the four-color conjecture from 1936, publicly declared that the four-color conjecture can be proved by finding the necessary group of reducible graphs. His student Toure wrote a calculation program. Heck can not only prove the reducibility of the configuration with the data generated by this program, but also describe the reducible configuration by transforming the mapping into a shape called "duality" in mathematics.

He marked the capital of each country, and then connected the capitals of neighboring countries with a railway crossing the border. Except for the capital (called vertex) and the railway (called arc or edge), all other lines have been erased, and the rest are called dual graphs of the original graph. In the late 1960s, Heck introduced a method similar to moving charges in an electrical network to find an inevitable set of configurations. The "discharge method", which first appeared in a rather immature form in Heck's research, is a key to the future study of inevitable groups and a central element to prove the four-color theorem.

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. Harken of the University of Illinois began to improve the "discharge process" in 1970, and then compiled a good program with Appel. 1June, 976, they spent 1200 hours on two different electronic computers of the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem, which caused a sensation in the world.

This is a great event that 100 has attracted many mathematicians and math lovers for more than a year. When two mathematicians published their research results, the local post office stamped all the mails sent that day with a special postmark of "four colors are enough" to celebrate the solution of this problem.

The "four-color problem" proved to be only a solution to a difficult problem that lasted for more than 100 years, and it became the starting point of a series of new ideas in the history of mathematics. In the research process of "four-color problem", many new mathematical theories have emerged and many mathematical calculation skills have been developed. For example, turning the coloring problem of maps into a graph theory problem enriches the content of graph theory. Moreover, the "four-color problem" has also played a role in effectively designing airline flight schedules and designing computer coding programs.

However, many mathematicians are not satisfied with the achievements made by computers. They think there should be a simple and clear written proof method. Today, many mathematicians and math lovers are still looking for a more concise proof method.

3. Among the mathematical conjectures related to prime numbers in history, the most famous one is Goldbach's conjecture.

1742 On June 7, the German mathematician Goldbach put forward two bold conjectures in a letter to the famous mathematician Euler:

1. Any even number not less than 6 is the sum of two odd prime numbers;

2. Any odd number not less than 9 is the sum of three odd prime numbers.

This is the famous Goldbach conjecture in the history of mathematics. Obviously, the second guess is the inference of the first guess. So it is enough to prove one of the two conjectures.

On June 30th of the same year, Euler made it clear in his reply to Goldbach that he was convinced that both Goldbach's conjectures were correct theorems, but Euler could not prove them at that time. Because Euler was the greatest mathematician in Europe at that time, his confidence in Goldbach's conjecture influenced the whole mathematics field in Europe and even the world. Since then, many mathematicians are eager to try and even devote their lives to proving Goldbach's conjecture. However, until the end of 19, there was still no progress in proving Goldbach's conjecture. The proof of Goldbach's conjecture is far more difficult than people think. Some mathematicians compare Goldbach's conjecture to "the jewel in the crown of mathematics".

Let's start with 6 = 3+3, 8 = 3+5, 10 = 5+5, ...,100 = 3+97 =1+89 =17+89. In the 20th century, with the development of computer technology, mathematicians found that Goldbach conjecture still holds true for larger numbers. However, natural numbers are infinite. Who knows if a counterexample of Goldbach's conjecture will suddenly appear on a sufficiently large even number? So people gradually changed the way of exploring problems.

1900, Hilbert, the greatest mathematician in the 20th century, listed Goldbach conjecture as one of the 23 mathematical problems at the International Mathematical Congress. Since then, mathematicians in the 20th century have "joined hands" to attack the world's "Goldbach conjecture" fortress, and finally achieved brilliant results.

The main methods used by mathematicians in the 20th century to study Goldbach's conjecture are screening method, circle method, density method, triangle method and so on. The way to solve this conjecture, like "narrowing the encirclement", is gradually approaching the final result.

1920, the Norwegian mathematician Brown proved the theorem "9+9", thus delineating the "great encirclement" that attacked "Goldbach conjecture". What is this "9+9"? The so-called "9+9", translated into mathematical language, means: "Any large enough even number can be expressed as the sum of two other numbers, and each of these two numbers is the sum of nine odd prime numbers." Starting from this "9+9", mathematicians all over the world concentrated on "narrowing the encirclement", and of course the final goal was "1+ 1".

1924, the German mathematician Redmark proved the theorem "7+7". Soon, "6+6", "5+5", "4+4" and "3+3" were captured. 1957, China mathematician Wang Yuan proved "2+3". 1962, China mathematician Pan Chengdong proved "1+5", and cooperated with Wang Yuan to prove "1+4" in the same year. 1965, Soviet mathematicians proved "1+3".

1966, Chen Jingrun, a famous mathematician in China, conquered "1+2", that is, "any even number large enough can be expressed as the sum of two numbers, one of which is an odd prime number and the other is the sum of two odd prime numbers." This theorem is called "Chen Theorem" by the world mathematics circle.

Thanks to Chen Jingrun's contribution, mankind is only one step away from the final result of Goldbach's conjecture "1+ 1". But in order to achieve this last step, it may take a long exploration process. Many mathematicians believe that to prove "1+ 1", new mathematical methods must be created, and the previous methods are probably impossible.