One, four-color conjecture

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.

Second, Goldbach conjecture.

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.

Third, Fermat conjecture.

Also known as Fermat's last theorem, it was put forward by the French mathematician Fermat.

It asserts that when the integer n > 2, the equation x n+y n = z n about x, y and z has no positive integer solution.

After it was put forward, it was proved by British mathematician andrew wiles in 1995 after more than 300 years of history.

Wolfsk, Germany, announced that it would award 65,438+ten thousand marks to the first person who proved the theorem within 100 years after his death, which attracted many people to try and submit their "proofs". After World War I, the mark depreciated sharply, and the charm of this theorem also declined greatly.

Fourth, Qiu Chengtong conjecture.

According to the "string" theory, the universe is ten-dimensional space-time, that is, the usual four-dimensional space-time and a very small six-dimensional space.

Calabi, a famous Italian geometer, pointed out that a complex high-dimensional space is "glued" together by several simple multi-dimensional spaces, which means that the high-dimensional space can be assembled by some simple geometric models.

1975, mathematicians Qiu Chengtong and others conquered the "Calabi conjecture" with negative Chen category and zero Chen category, but failed to solve the problem that the first Chen category was positive. Qiu Chengtong put forward the stability problem that can be transformed into algebraic geometry, which is the "Qiu Chengtong conjecture" that has puzzled international academic circles for decades.

From 2065438 to May 2004, Chen Xiuxiong, Donaldson and Sun Song gave a complete proof of "Qiu Chengtong conjecture".

5. Riemann conjecture

Riemann conjecture is a conjecture about the zero distribution of Riemann zeta function zeta (s), which was put forward by the mathematician Riemann in 1859. Hilbert put forward 23 mathematical problems that mathematicians should try to solve in the 20th century at the 2nd International Congress of Mathematicians, which are considered as the commanding heights of mathematics in the 20th century, including Riemann hypothesis. Riemann conjecture has also been included in the seven major mathematical problems in the world today.

Compared with Fermat's conjecture, which took more than three and a half centuries to solve, and Goldbach's conjecture, which took more than two and a half centuries to survive, Riemann's conjecture is far from being recorded for only a century and a half, but its importance in mathematics is far greater than these two conjectures with higher public awareness. Riemann conjecture is the most important and anticipated mathematical problem in the field of mathematics today.

So far, no one has given a convincing and reasonable proof of Riemann conjecture.