One of the three major mathematical problems in the modern world. The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research institute to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, which makes countries with the same border painted with different colors." Can this conclusion 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, 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.
1 1 years later, that is, 1890, the mathematician Hurwood pointed out that Kemp's proof and his accurate calculation were wrong. Soon, Taylor's proof was also denied. Later, more and more mathematicians racked their brains for this, but found nothing. Therefore, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture: the efforts of previous mathematicians paved the way for later mathematicians to uncover the mystery of the four-color conjecture.
Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. 19 13 years, boekhoff introduced some new skills on the basis of Kemp, and American mathematician Franklin proved in 1939 that maps in 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. 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.
Goldbach conjecture (the second of the three major mathematical problems)
One of the three major mathematical problems in the modern world. Goldbach is a German middle school teacher and a famous mathematician. He was born in 1690, and was elected as an academician of Russian Academy of Sciences in 1725. 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.
(b) 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 Fermat 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. However, the mathematical proof of lattice test needs 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 by an ancient screening method, and reached the conclusion that every even number with larger ratio can be expressed as (99). This method of narrowing the encirclement is very effective, so scientists gradually reduced the number of prime factors in each number from (99) until each number is a prime number, thus proving "Goldbach".
At present, the best result is proved by China mathematician Chen Jingrun in 1966, which is called Chen's theorem? "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 result is often 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, Byxwrao of the Soviet Union proved "5+5".
1940, Byxwrao of the Soviet Union 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", and Wang Yuan of China proved "1+4".
1965, Byxwrao and vinogradov Jr of the Soviet Union and Bombieri of Italy proved "1+3".
1966, China Chen Jingrun proved "1+2".
Who will finally overcome the problem of "1+ 1"? It is still unpredictable.
Fermat's last theorem and its proof (one of the three major mathematical problems)
Modern mathematics is like a towering tree, flourishing. There are countless math problems hanging on this vigorous tree. Among them, the four-color map problem, Fermat's last theorem and Goldbach's conjecture are the most dazzling. They are called the three major mathematical problems in modern times.
For more than 300 years, Fermat's Last Theorem has made many famous mathematicians in the world exhausted their efforts, and some even spent their whole lives. The mysterious veil of Fermat's Last Theorem was finally unveiled at 1995, which was proved by 43-year-old British mathematician Wells. This is considered as "the greatest mathematical achievement of the 20th century".
The origin of Fermat's last theorem
The story involves two mathematicians separated by 1400 years, one is Diophantine in ancient Greece and the other is Fermat in France. Diophantine activities took place around 250 AD.
1637, Fermat in his thirties was reading the French translation of Diophantine's masterpiece Arithmetic. In his book, he wrote all positive integer solutions of the indefinite equation x2+ y2 =z2 in Latin: "No cube of a number can be divided by the sum of two cubes; The fourth power of any number cannot be divided by the sum of the fourth powers of two numbers. Generally speaking, it is impossible to divide a power higher than a quadratic power by the sum of two powers of the same power. I found a wonderful proof of this judgment, but unfortunately the space here is too small to write. "
After Fermat's death, people found this passage written on their eyebrows when sorting out his relics. 1670 His son published Fermat's pagination note, and everyone knows this problem. Later, people called this assertion Fermat's Last Theorem. In mathematical language, it is an equation in the form of xn +yn =zn. When n is greater than 2, there is no positive integer solution.
Fermat is an amateur mathematician and is known as the "king of amateur mathematicians". 160 1 year, he was born in a leather merchant's family near Toulouse in southern France. I was educated at home when I was a child. When he grew up, his father sent him to university to study law and became a lawyer after graduation. Member of Toulouse City Council, tel. 1648.
He loves mathematics very much and spends all his spare time studying mathematics and physics. Because of his quick thinking, strong memory and indomitable spirit necessary for studying mathematics, he has achieved fruitful results, making him one of the great mathematicians in the17th century.
Difficult exploration
At first, mathematicians tried to rediscover Fermat's "wonderful proof", but no one succeeded. Euler, a famous mathematician, proved that equations x3+ y3 =z3 and x4+y4 = z4 cannot have positive integer solutions.
Because any integer greater than 2, if not a multiple of 4, must be an odd prime number or a multiple of it. So as long as it can be proved that n = 4 and n is an arbitrary odd prime number, the equation has no positive integer solution, and Fermat's last theorem is completely proved. The case of n = 4 has been proved, so the problem focuses on proving that n is equal to an odd prime number.
After Euler proved that n = 3 and n = 4, Legendre and Dirichlet independently proved the case of n = 5 in 1823 and 1826, and Lame proved the case of n = 7 in 1839. In this way, the long March to prove odd prime numbers began.
Among them, the German mathematician Cuomo made an important contribution. He introduced the concepts of "ideal number" and "cyclotomic number" invented by himself by means of modern algebra, and pointed out that Fermat's Last Theorem may be incorrect only when n equals some values called irregular prime numbers, so it is only necessary to study these numbers. Such figures, within 100, are only 37,59,67. He also proved that when n = 37, 59, 67, the equation Xn+Yn = Zn cannot have a positive integer solution. This pushes Fermat's last theorem to the point where n is within 100. Cuomo proved the theorem in batches, which was regarded as a major breakthrough. 1857 won the gold medal of the Paris Academy of Sciences.
Although this "Long March" proof method keeps refreshing records, such as 1992, or even advanced to n = 1000000, it does not mean that the theorem has been proved. It seems that we need to find another way.
Who will win the 65438+ million mark prize?
Since Fermat's time, the Paris Academy of Sciences has twice offered medals and prizes to those who proved Fermat's Last Theorem, and the Brussels Academy of Sciences has also offered a large prize, but all of them have no results. 1908, when Wolfskeil, a German mathematician, died, he presented his 65,438+ten thousand marks as a prize for solving Fermat's last theorem to the Gottingen Scientific Society in Germany.
The Gottingen Science Society announced that the prize is valid for 100 years. The G? ttingen Scientific Association is not responsible for reviewing manuscripts.
65438+ million marks was a lot of wealth at that time, and Fermat's last theorem was a problem that all primary school students could understand. Therefore, not only people who specialize in mathematics, but also many engineers, priests, teachers, students, bank employees, government officials and ordinary citizens are studying this problem. In a short time, there are thousands of certificates issued by various publications.
At that time, a German magazine named Records of Mathematics and Physics volunteered to review papers in this field. From 19 1 1, * * has approved1"proofs", all of which are wrong. Later, I really couldn't stand the heavy burden of peer review and announced that I would stop this review and appraisal work. However, the wave of proof is still surging. Although the German currency depreciated many times after the two world wars, it is of little value to exchange the original 654.38+ million marks for later marks. However, the precious spirit of loving science still encourages many people to continue this work.
Belated evidence
Through the efforts of predecessors, the proof of Fermat's Last Theorem has achieved a lot, but there is no doubt that there is still a long way to go to prove the distance theorem. What shall we do? In the future, we must use a new method. Some mathematicians have already used the traditional method-transformation problem.
People associate the solution of Diophantine equation with a point on algebraic curve, which becomes a transformation of algebraic geometry, and Fermat problem is only a special case of Diophantine equation. On the basis of Riemann's work, 1922, the British mathematician Mo Deer put forward an important conjecture. Let F(x, y) be a rational coefficient polynomial of two variables x, y, then when the genus of the curve F(x, y) = 0 (a quantity related to the curve) is greater than 1, the equation F(x, y)= 0 has at most a finite set of rational numbers. 1983, the 29-year-old German mathematician Fortins proved Mo Deer's conjecture in the Soviet Union with a series of results of Shafara's algebra and geometry. This is another major breakthrough in the proof of Fermat's last theorem. Fielding won the Fields Prize with 1986.
Wells still uses algebraic geometry to climb. He connected other people's achievements wonderfully, drew lessons from the conquerors who had gone through this road, and noticed a brand-new circuitous path: if the Taniyama-Zhicun conjecture holds, then Fermat's Last Theorem must hold. This was discovered in 1988 by German mathematician Ferrer when he studied a conjecture of Japanese mathematician intellectual village about 1955 elliptic function.
Wells was born into a theological family in Oxford, England. He was very curious and interested in Fermat's last theorem since he was a child. This wonderful theorem led him into the palace of mathematics. After graduating from college, he began his childhood fantasy and decided to fulfill his childhood dream. He studied Fermat's last theorem in great secrecy and kept his mouth shut.
After seven years of poverty, I persisted until June 23 1993. On this day, a regular academic report meeting is being held in the hall of Newton Institute of Mathematics at Cambridge University. The speaker Wells gave a two-and-a-half-hour speech on his research results. 10: 30, at the end of the report, he calmly announced: "Therefore, I proved Fermat's last theorem". This sentence, like a thunder, set many hands that only need to do routine work in the air, and the hall was silent. After half a minute, thunderous applause seemed to overturn the roof of the hall. British scholars are ecstatic regardless of their elegant gentlemanly manners.
The news quickly caused a sensation all over the world. Various mass media have reported it and called it "the achievement of the century". It is believed that Wells finally proved Fermat's Last Theorem and was listed as one of the top ten scientific and technological achievements in the world in 1993.
But soon, the media quickly reported an "explosive" news: Wells' 200-page paper was found to be flawed when it was submitted for trial.
In the face of setbacks, Wells did not stop. It took him more than a year to revise the paper and correct the loopholes. At this time, he was "haggard for Iraq", but "his clothes became wider and he had no regrets." 1September 1994, he wrote a new paper with a page of 108 and sent it to the United States. The paper passed the examination successfully, and the Yearbook of Mathematical Journal published his paper in May 1995. Wells obtained the wolf prize in mathematics of 1995 ~ 1996.
After more than 300 years of continuous struggle, mathematicians have made many important discoveries around Fermat's last theorem, which has promoted the development of some branches of mathematics, especially the progress of algebraic number theory. The core concept "ideal number" in modern algebraic number theory is put forward to solve Fermat's last theorem. No wonder Hilbert, a great mathematician, praised Fermat's Last Theorem as "a hen that lays golden eggs".